Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations

1.   Introduction

The theory of symmetry – i.e., what is now known as Lie theory – was developed by Sophus Lie in his attempt to solve (nonlinear) differential equations, generalizing the approach by Evariste Galois for algebraic equations.

The Lie approach, and more generally symmetry methods for differential equations, produced a number of results by Lie and his pupils, but afterwards the theory remained at the same level for quite a long time, until it was taken up by Birkhoff in the USA and by Ovsjannikov in USSR in the sixties ^[1,2]. One of the reason for this "long sleep" lies in that the applications of the Lie theory do in general require quite extensive computations; these can nowadays be routinely performed on a computer by algebraic manipulation programs, but were quite prohibitive in earlier times.

After the reviving by Birkhoff and Ovsjannikov, the theory developed quickly, and underwent various generalizations; the reader can consult some of the by now standard texts ^{[3,4,5,6,7,8,9]} for the basics as well as for the most common extensions and generalizations.

Here we are concerned with two of these extensions and generalizations. That is, on the one side with the generalization to conditional ^[10], partial ^[11], and asymptotic ^[12] symmetries; and on the other side with the extension of the theory to the framework of stochastic differential equations ^{[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]} (by this we always mean equations of Ito or Stratonovich type ^{[31,32,33,34,35,36,37]}, see below).

Our goal is to put the two together, i.e., to consider conditional and asymptotic symmetries for stochastic differential equations. Needless to say, our main interest is not in the abstract definition of these, but in showing that they can be of concrete use in determining (conditional or asymptotic) solutions to the concerned equations.

In the first part of the paper we establish some background. In particular, we quickly recall some of the main points of the theory of symmetry for (deterministic) differential equations, mainly to fix notation (Section 2), then recall the (well-known) notion of conditional symmetry (Section 3) and of (less well-known) asymptotic symmetry (Section 4) together with their use. We will also recall the basics of the theory of symmetry for stochastic differential equations (Section 5), and the method of invariants for these (Section 6).

We then proceed to merge the two, i.e., we develop conditional and asymptotic symmetries and invariants for stochastic differential equations; this will be the subject of Sections 7 and 8. We also provide, in Section 9, a number of concrete examples, chosen to be computationally simple in order to keep the conceptual issues in focus.

Our main result can be shortly stated saying that the theory of conditional, partial and asymptotic symmetries can be extended to the framework of stochastic differential equations. The reader will not find "big theorems" in our discussion, rather a number of lemmas; this corresponds to the fact we are not overcoming any special technical difficulty here. We are instead proposing a point of view that has not been considered previously in the literature, i.e., building a method which – we believe – is useful in characterizing solutions to stochastic equations via their asymptotic properties, in the same way as the corresponding deterministic approach proved to be precious in analyzing solutions to deterministic equations having certain asymptotic properties.

As we will discuss in more detail in the following (see Sections 3 and 4), the key observation triggering the present approach is that while in several physical cases one has equations which are invariant – say, for the sake of concreteness, under a rotation or scaling transformation – and correspondingly for these there is a special set of solutions displaying such symmetry, it is of more general interest the case where either $(i)$ the solutions do have such symmetry only asymptotically in time and/or space (think e.g., to the case where the rotationally invariant solutions are stable, so that solutions which are not rotationally invariant at $t = 0$ but which are sufficiently near to those will, in the long run, acquire the rotational symmetry); or $(ii)$ the equations do not have such a symmetry, but there are solutions which, at least asymptotically in time and/or space, display such a symmetry.

The trivial, but not meaningless example, is that of the zero solution, which surely has a very high degree of symmetry, and which can be a valid – and maybe also stable or even attractive – solution for equations with no symmetry.

A less extreme, more concrete and also physically relevant, example is provided, in the deterministic framework, by the Boussinesq equation of Fluid Dynamics: in this case a number of physically relevant – and routinely observed in experiments – solutions do have symmetry properties which are not shared by the equation; and they can be analytically determined though the conditional symmetries approach. Similarly, stable solutions to several types of reaction-diffusion equations (including the prototypical Fisher-Kolmogorov equation) can be analytically determined through the method of asymptotic symmetries.

It must be stressed that in general, in this way one is able to obtain exact analytical expression for these symmetric class of solutions albeit there is no way to obtain a general solution to the equation under study in analytical terms. We will thus extend these approaches, developed in the framework of deterministic equations; and propose a general method to study special classes (characterized indeed by symmetry-like properties) of solutions to stochastic differential equations.

As already mentioned, we will illustrate the method by a number of concrete examples; these will however be chosen to be computationally simple in order to keep the conceptual issues in focus.

Throughout the paper, we will use the Einstein summation convention. We will denote by $t$ the time variable and by $x^i$ the spatial ones, while the $w^k$ will be Wiener processes; we moreover use the shorthand notation

and denote the time derivative of a function $x (t)$ as either $x_t$ or ${{\dot x}}$, depending on typographical convenience.

2.   Symmetry of differential equations

In this section we recall some basic notions concerning symmetries of deterministic differential equations. In particular we consider first order ODEs, i.e., dynamical systems (DS), as this is the deterministic analogue of Ito or Stratonovich equations to be considered in the following.

For a dynamical system

with $t \in R$, $x \in M_0 {\subseteq} R^n$, we define $M = M_0 \times R$ and consider maps $\Phi : M \to M$. In particular, we consider one-parameter groups of such maps, with generator

Note that with $\pi_0 : M \to R$ the projection $\pi_0 (m_0, r) = r$, the full phase bundle $M = M_0 \times R$ has a natural structure of fiber bundle over the time real axis, $(M, \pi_0, R)$.

Under the action of $X$ – or more precisely of the maps $\Phi = \exp[ {\varepsilon} X]$ in the one-parameter group generated by $X$ – we have, at first order in ${\varepsilon}$,

Once we define the transformation of the independent ($t$) and the dependent ($x^i$) variables, we have also implicitly defined the transformation of the derivatives of the latter with respect to the former. We have indeed (always working at first order in ${\varepsilon}$)

where $\psi^i$ is given by the prolongation formula ^{[3,4,5,6,7,8,9]}

and $D_t$ is the total time derivative (in this case, this is just $D_t = {\partial}_t + {{\dot x}}^j {\partial}_j$).

With this, the vector field $X$ on $M$ is extended (prolonged) to a vector field $X^{(1)}$ in $J^1 M$ (in which we consider local coordinates $\{t; x^1, ..., x^n; {{\dot x}}^1, ..., {{\dot x}}^n\}$),

Here and below $J^1 M$ is the (first) jet bundle associated to $M$ ^{[3,4,5,6,7,8,9]}. Note this can be seen both as a bundle over $M$ (the fiber corresponding to first derivatives ${{\dot x}}$) with projection $\widetilde {\pi}_1$ and as a bundle over $R$ (the fiber corresponding to variables $x$ and first derivatives ${{\dot x}}$) with projection $\pi_1$,

The projections satisfy $\pi_1 = \pi_0 \circ \widetilde {\pi}_1$, i.e. the diagram above is a commutative one.

The DS (1) defines a submanifold in $J^1 M$, also called the solution manifold $\Sigma_f$ for the Eq (1).

A function $x = \xi (t)$ is identified with its graph

i.e., with a section ${\gamma}_\xi$ of the bundle $(M, \pi_0, R)$; this is naturally extended to a section ${\gamma}_\xi^{(1)}$ in $J^1 M$. In local coordinates, we have

The function $x = \xi (t)$ is a solution to (1) if and only if

The vector field (2) is a Lie-point symmetry (generator) for the Eq (1) if and only if

in this case it also maps (sections of $(M, \pi_0, R)$ representing) solutions to (1) into – generally, but not necessarily, different – (sections of $(M, \pi_0, R)$ representing) solutions to (1) ^{[3,4,5,6,7,8,9]}.

Note that we will adhere to the usual abuse of notation consisting in calling "symmetries" the vector fields which are actually generators for the symmetry group.

It may be worth noting that the action of the vector field (2) on sections is given by

To avoid any misunderstanding, we specify that this notation means that the section ${\Gamma}_f$ given by ${\Gamma}_f = \{ (t, x = f(t))\}$ is mapped into the section ${\Gamma}_{\widetilde {f}}$ given by ${\Gamma}_{\widetilde {f}} = \{ (t, x = \widetilde {f} (t))\}$.

Actually this is also an equivalent characterization of symmetries: that is, symmetries can be defined as maps $M \to M$ which transform solutions into (generally, but not necessarily, different) solutions ^{[3,4,5,6,7,8,9]}.^*

^*For a very readable discussion of different – but equivalent – definitions of symmetries, see e.g., Chapter 3 in ref.^[8] or Section 2.2. in ref.^[6].

If $x = \xi (t)$ is a solution to (1) and if $X$, satisfying (8), leaves ${\gamma}_\xi$ invariant, we say that $x = \xi (t)$ is an $X$-invariant solution to (1).

Vector fields that are symmetries for a given equations, such as the evolution Eq (1), or of more general form, – or for equations, be these ODEs or PDEs, of more general form – are characterized as solutions to a set of linear PDEs for the coefficients of the vector field, known as determining equations ^{[3,4,5,6,7,8,9]}. For equations of the form (1) and vector fields expressed as in (2) with $\theta = 0$ (these are the simple vector fields to be considered below), these are written as

Remark 2.1. It should be mentioned that this sketchy discussion refers to proper Lie-point symmetries. In the case of deterministic Dynamical Systems, it turns out that so-called orbital symmetries are also quite useful ^[38,39,40]; as far as we know, these have not been investigated in the context of stochastic dynamical systems. $\odot$

Remark 2.2. The prolongation formula for ODEs, Eq (4), is a special occurrence of the general prolongation formula, which – in the multi-index notation used in most of the literature, and in particular denoting by $x^i$ the independent variables, and by $u^a$ the dependent ones – reads

We refer to ^{[3,4,5,6,7,8,9]} for discussion and details. $\odot$

2.1.   Dynamically invariant submanifolds

The evolution Eq (1) defines a dynamics, and a dynamical vector field $Z$, in all of $M$; when $f$ is time-autonomous, we actually also have a vector field $Y$ in $M_0$ and in this case $Z = {\partial}_t + Y$; symmetries $X$ are then characterized by $[X, Z] = 0$ (so called orbital symmetries are characterized by $[X, Z] = {\lambda} Z$, for their properties and applications see e.g., ^[38,39,40]). We are specially interested in this latter case of $f$ time-autonomous, both because the analysis is slightly simpler and due to its relevance in applications.

It may happen that there exist submanifolds of $M$ being invariant under $Z$. In particular, for $f$ time autonomous, there may be proper submanifolds $U {\subset} M_0$ which are invariant under $Y$. The simplest example is that of fixed points for $f$, i.e., points $x_0$ such that $f(x_0, t) = 0$. Other simple but significant examples are provided by periodic or multi-periodic solutions, by heteroclinic or homoclinic orbits, or by the stable and unstable manifolds of hyperbolic fixed points ^{[41,42,43,44]}.

One would expect that symmetries are constrained by the need to somehow respect the structure of such invariant sets; this is indeed the case, provided one considers time-independent symmetries ^[4,45]. In particular, it can be shown that Lie-point time independent (LPTI) symmetries maps solutions into solutions whose trajectories in $M_0$ have the same topology.

More specifically, we have the following results (these embody several Lemmas given in ^[4,45]; see there for proofs; the Corollary makes use of standard results in Topology, see e.g., ^[46]). We will denote by $K(X) {\subseteq} M$ the subset of $M$ which is point-wise invariant under a vector field $X$ (or $K(\mathcal{A}) {\subseteq} M$ for the set invariant under an algebra $\mathcal{A}$ of vector fields), by $Z$ the vector field defining the dynamics of the (autonomous) dynamical system under study, and by $\mathcal{G}$ the Lie algebra of symmetries for $Z$.

Lemma 2.1. The LPTI symmetry algebra of an autonomous ODE transforms stationary solutions into stationary solutions, periodic solutions into periodic solutions of the same period.

Lemma 2.2. Assume there is a compact set $K (Z) {\subset} M$ invariant under the flow of $Z$. Then for any subalgebra ${{\mathcal G}}_0 {\subseteq} {{\mathcal G}}$, the submanifold $K^0 ({{\mathcal G}}_0) : = (K({{\mathcal G}}_0) \cap K (Z)) {\subseteq} M$ is either empty or compact and invariant under the flow of $Z$.

Corollary. If $K^0 ({{\mathcal G}}_0)$ has components isomorphic to an even-dimensional sphere $S^{2n}$, then on each of these lie stationary solutions to the dynamical system ${{\dot x}} = Z (x)$. If $K^0 ({{\mathcal G}}_0)$ has components isomorphic to a disk $D^{2n+1}$, then on each of these lie stationary solutions to ${{\dot x}} = Z (x)$.

2.2.   Asymptotic solutions

In several cases one meets dynamical systems or PDEs which cannot be solved exactly, but whose asymptotic behavior (e.g., for $t \to \infty$ and/or for $|x| \to \infty$ or for $|x|\to 0$) is amenable to exact treatment.

We just mention here two very simple examples of use in the following (these can be readily generalized).

Example 2.1. For $x \in R$, consider

with $\Phi (x) = \Phi (-x)$ a convex even function, non-degenerate in $x = 0$. Then for any initial datum $x(0) = x_0$ the dynamics is attracted to $x_* = 0$, and in the region near $x_*$ (and thus asymptotically in time) the dynamics is well described by the linear equation

the solutions to this are of course

Example 2.2. In $R^2$ and with polar coordinates $(\rho, \vartheta)$, consider the system

with $h (\rho, \vartheta)$ an arbitrary smooth function. The system can not be solved in general, but we know that the dynamics is attracted to the circle $\rho = 1$, and on this attracting manifold we have a trivial dynamics $\dot{\vartheta} = 1$. Thus any solution is asymptotically described by $\rho = 1$, $\vartheta (t) = k_0 + t$ for some phase $k_0$. $\diamondsuit$

3.   Conditional and partial symmetries

As mentioned above, symmetries map (any) solution into a (generally different) solution. It may happen, however, that there are maps such that this property (i.e., mapping solutions into solutions) holds only for a subset of solutions.

3.1.   Conditional symmetries

The most common case is that of equations admitting solutions which are invariant under maps that are not a symmetry of the equation. For instance, any equation of the form (1) with $f$ linear in $x$, $f(x, t) = A(t) x$, will admit the origin as a fixed point, and this in turn will be invariant under any linear (possibly time-dependent) map $x \to B(t) x$. But this will be a symmetry of the equation if and only if

which is definitely a non-generic property.

In this case, and in similar but less trivial ones, one speaks also of conditional symmetries ^[10]; see also ^{[47,48,49,50,51,52,53,54]}. Let us discuss these in some more detail (the reader can see the references given above for a more complete discussion), referring to the case of PDEs for greater generality; we denote by $x^1, .., x^q$ the independent variables, and by $u^1, ..., u^p$ the dependent ones. Our goal is to determine $X$-invariant solutions for a differential equation (note that by this we will always mean possibly a vector one, i.e., a system of scalar differential equations) ${\Delta} = 0$, where

is not necessarily a symmetry of ${\Delta}$; at the same time, we want to determine the $X$ which admit such solutions.

In order to do this, we complement ${\Delta}$ with an equation ${\Delta}_X = 0$ expressing the requirement of $X$-invariance of the function $u = u(x)$. This is just, see (9) above,

We are thus led to consider the system

It is clear that the solutions of this, if they exist, are all and only the solutions to ${\Delta} = 0$ which are also $X$-invariant (i.e., which also solve ${\Delta}_X = 0$). Thus symmetries for the system $\mathcal{E}_X$, which transform solutions into solutions, will transform $X$-invariant solutions to ${\Delta}$ into $X$-invariant solutions to ${\Delta}$. In particular this apply for the vector field $X$ itself, which by construction will leave each of such solutions invariant.

We are thus at first sight reduced to a standard problem in symmetry analysis; however we should look at this with some more care. One possibility is that we fix in advance $X$ and look for solutions to ${\Delta}$ which are $X$-invariant. In this case ${\Delta}_X$ reduces to an ansatz on the functional form of $u(x)$ which can then be inserted into ${\Delta}$, and indeed nothing new is here.

On the other hand, the interesting case is the one where we do not know in advance what $X$ should or could be, and we study the system (12) in order to determine $X$ and the associated invariant solutions, if any.

In this case we get something new: in fact, the usual determining Eq (10) – or more generally their version for PDEs – are linear in the $X$ coefficients $\{ \xi^i (x, u), {\varphi}^a (x, u) \}$, while now we will have determining equations which are nonlinear (in particular, quadratic) in these.

We will not enter into the details of how such a system of nonlinear (determining) equations can be solved thanks to the peculiar structure of the problem, referring the reader to the original paper by Levi and Winternitz for this matter ^[10].

We instead stress a most relevant point: symmetries have to map solutions into solutions, but when we apply symmetry analysis to the system (12), this applies to the common solutions to ${\Delta} = 0$ and to ${\Delta}_X = 0$, i.e., only to invariant solutions. Thus these are symmetries of ${\Delta}$ conditional upon the additional invariance condition ${\Delta}_X$.

Remark 3.1. We should mention that one could also investigate the inverse problem for conditional symmetries: that is, for a given number of dependent and independent variables and a given order of the equation, and given a certain vector field $X$, determine all the differential equations ${\Delta}$ which admit $X$ as a conditional symmetry. This was considered by Levi, Rodriguez and Thomova in ^[55], and more recently by Pucci and Saccomandi ^[56,57]. We are not able to provide stochastic extensions of this analysis at the moment. $\odot$

3.2.   Partial symmetries

A slightly more general case is also possible, i.e., that we have a subset of solutions which are not individually invariant under the map, but which are indeed transformed one into another by the map – which again is not a symmetry of the original equation ${\Delta} = 0$. In this case one speaks of partial symmetries ^[11] (see also ^[58]).

A specially relevant case in our dynamical systems context is where the solutions in this set are characterized by living on a manifold which is dynamically invariant (i.e., invariant under the dynamics). The situation is characterized in a way which looks quite similar to the characterization provided above for conditional symmetries. That is, we consider a differential equation (again, in general this can be a system of PDEs) ${\Delta} = {\Delta}_0$, and a vector field $X$ which is not a symmetry for ${\Delta}_0$, thus such that – writing $S_0 : = S_{{\Delta}_0}$ for ease of writing,

We consider then an auxiliary equation

If we look at solutions to the system

which are globally invariant under $X$ (that is, the set of solutions is mapped into itself, albeit the single solutions are possibly not invariant), this set is characterized precisely by the property

here $S^{(1)} = S_0 \cap S_1$, with of course $S_1 = S_{{\Delta}_1}$, is the set of solutions to the system $\mathcal{E}$; and we wrote $X^{(*)}$ to mean the prolongation of $X$ of appropriate order, i.e., the same order as the equation ${\Delta}_1$. In this case, i.e., if (13) is satisfied, we say that $X^{(*)}$ is a partial symmetry of order one for ${\Delta}$, as it maps a subset of solutions into solutions (belonging to the same subset).

We stress that (13) may or may not be satisfied, depending on ${\Delta}$ and on our choice of the vector field $X$. If it is not satisfied, we can iterate our procedure and consider

and look for $S^{(2)} = S_0 \cap S_1 \cap S_2$, enquiring if – with an obvious notation – the equation

is satisfied. If this is the case, we say that $X$ is a partial symmetry of order two for ${\Delta}$; otherwise we can still iterate our procedure and so on.

We set this in the form of a Proposition, which is quoted from ^[11].

Proposition 3.1. Given the general differential problem ${\Delta} = {\Delta}_0 = 0$ and a vector field $X$ on $M$, define ${\Delta}_{r+1} : = X^{(*)} ({\Delta}_r)$. Denote by $S^{(r)}$ the set of simultaneous solutions of the system

and assume that this is not empty for $r \le s$. Assume moreover that

Then the set $S^{(n)}$ provides a family of solutions to the initial problem ${\Delta}$ which is mapped into itself by the transformations generated by $X$.

Remark 3.2. When the situation depicted in Proposition 3.1 is met, we shall say that $X$ is a partial symmetry, or P-symmetry for short, for the problem ${\Delta}$, and that the globally invariant subset of solutions $S^{(n)}$ obtained in this way is a $X$-symmetric set. We also refer to the number $n$ appearing in the statement as the order of the P-symmetry. $\odot$

The discussion can be specialized to the case of dynamical system; here we just recall a result for this case, quoting again from ^[11], to which the reader is referred for further detail. Here ${\mathcal{L}}_{({\lambda})} (f)$ denotes the linearization of $f$ at $x_{({\lambda})}$ (see below), i.e.,

Proposition 3.2. Assume that the dynamical system ${{\dot x}} = f(x)$ admits a LPTI partial symmetry $X = {\varphi}^i(x) {\partial}_i$, and let $x_{({\lambda})} = x_{({\lambda})} (t)$ be an orbit of solutions obtained under the action of the group generated by $X$, with ${\lambda}$ the group parameter. Then $\Phi : = {\varphi} (x_{({\lambda})})$ satisfies the equations

3.3.   Examples

We give here two Examples (examples 3.1 and 3.2) of conditional symmetries, and three Examples (examples 3.3 through 3.5) dealing with partial symmetries. These are taken respectively from ^[10] and from ^[11]. Some Examples deal with PDEs, as the essential of conditional and partial symmetries is better grasped in this context.

Example 3.1. The Boussinesq equation ^[59] in one spatial dimension, for a function $u = u(x, t)$, reads

If we consider the vector field

this is not a symmetry for the equation, but it turns out to be a conditional symmetry for it.

Under the action of $X_1$ we have a one-parameter flow (group parameter ${\lambda}$) which acts as

Assuming that $u = u(x, t)$ is a solution to (15), we have that $\widetilde {u} = \widetilde {u} (\widetilde {x}, \widetilde {t})$ is a solution if it satisfies the auxiliary condition (we omit the tildes from now on)

Now we note that the invariance condition ${\Delta}_X = 0$, see (11), reads in this case (i.e., for the $X$ we are now considering)

Thus (16) is just a differential consequence of this, $({\Delta}_1)_x = 0$. $\diamondsuit$

Example 3.2. Consider again the Boussinesq Eq (15), and now

again this is not a symmetry for Eq (15). Now we get

On the other hand, the invariance condition (11) is now

If $u(x, t)$ is a solution to (15), then the condition for $\widetilde {u} (\widetilde {x}, \widetilde {t})$ to be also a solution reads

where $A_i$ are certain smooth polynomial functions, and

Thus if $u(x, t)$ is a solution to the Boussinesq equation and $F = 0$, the transformed function $\widetilde {u} (\widetilde {x}, \widetilde {t})$ is also a solution. But looking at (17) we see that $F = 0$ is just the invariance condition under $X_2$. $\diamondsuit$

Example 3.3. Consider the modified Laplace equation

with $g(u)$ an arbitrary smooth function. We also consider the rotation vector field

this is a symmetry for ${\Delta}$ in the case $g(u) = 0$, but not in general. We will assume $g(u) \not = 0$.

Implementing our procedure, we get

The last equation is identically satisfied on common solutions to the previous ones, thus we have a partial symmetry of order five.

The set $S^{(5)}$ has the following form, with $k_{1}, \dots, k_{6}$ arbitrary constants:

that is, functions which are at most quadratic in the $(x, y)$ variables. Note that for these functions the Eq (18) reduces to the standard Laplace equation, and for this it is obvious that (19) is a symmetry; thus this seemingly complicated computation yields a rather obvious result. $\diamondsuit$

Example 3.4. We look again at the Boussinesq Eq (15), which will be our ${\Delta}_0$, and consider the vector field

Now we have (disregarding some inessential nonzero multiplicative constants)

We have thus to look for simultaneous solutions to ${\Delta}_0 = 0$ and to ${\Delta}_1 = 0$. Solving for ${\Delta}_1 = 0$ yields

with $v$ and $w$ arbitrary functions. Note that the functions (21) are $X$-invariant if and only if $w'(t) = 0$ (this is why the term $- t^2$ has not been absorbed into the arbitrary function $w(t)$ in writing (21)); thus we are considering a setting which is really a generalization of the conditional symmetries approach, even in this case.

Inserting now (21) into ${\Delta}_0$, i.e. into the Boussinesq equation (15), and writing $z = x -t^2/2$, we get that $v$ and $w$ must satisfy

We are specially interested, as mentioned above, in the case $w (t)\not = 0$ (otherwise we recover the Levi-Winternitz family of $X$-invariant solutions). E.g., for $w(t) = K$ a nonzero constant, we get solutions $u(x, t) = K+v(z)-t^2$ with $v$ a solution to

Example 3.5. We now consider an example involving a Dynamical System in $R^3$ with Cartesian coordinates $(x, y, z)$; we will also write $\rho^2 = x^2 + y^2$. Consider

with $g_i$ smooth functions. If we also consider the rotation vector field

then the partial symmetry condition reads

with $G_{\alpha}$ certain functions which are non-zero for generic $g_i$: e.g.,

Thus the system exhibits – as rather obvious from its expression – rotational $(x, y)$ symmetry only if restricted to the plane $z = 0$. Moreover, in this plane one can find three different families of solutions (beside the null one, which is rotationally invariant) which are mapped into themselves by the rotations, without being rotationally invariant themselves: the trajectories lying in $\rho < 1$, and respectively in $\rho > 1$, spiralling towards the limit cycle $\rho = 1$; and the solutions running on the single trajectory $\rho = 1$ which is left fixed by the partial symmetry (i.e., the limit cycle). $\diamondsuit$

4.   Asymptotic symmetries

In many cases, one has equations whose solutions have a complicated behavior, which becomes simple in some asymptotic regime. This may be asymptotic in time (thus for $t \to \pm \infty$, possibly with different behavior in the two limits or with a simple behavior only in one of the limits) or asymptotically in space (i.e., for $|x| \to \infty$, or in the one dimensional case for $x \to \pm \infty$, with the same notes as for $t$); or in a suitable combination of the two.

It is remarkable that in such cases one can adopt a renormalization-group (RG) like approach ^[60,61]; this in turn can be combined with symmetry considerations ^[12,62,63]. In particular, we can have asymptotic symmetries, i.e., maps which are symmetries (hence in particular map solutions into solutions) only in this asymptotic regime, being otherwise (that is, in the non-asymptotic regime) only approximate symmetries, at most.

4.1.   Asymptotic symmetries in the PDE setting

In order to discuss asymptotic symmetry properties (at least in the deterministic case) we consider the general case, i.e., possibly PDEs (of any order $n$); the independent variables live on a manifold $B$, the dependent ones take values in a manifold $U$, and we consider $M = U \times B$. We will follow the discussion given in ^[4], referring to ^[12,62,63] for further detail.

It is then convenient to use the geometrical representation of functions as cross sections of the bundle $(M, \pi_0, B)$, and of equations as submanifolds in $J^n M$; in the case of evolution equations, these can also be seen as sections of a suitable fiber bundle (with time derivatives in the fiber, i.e., in the vertical space), and on the other hand some care is needed in order to take into account the special role of the time variable in this case.

Here and in the following we write ${\partial}_a^J = ({\partial}/ {\partial} u^a_J)$; moreover $\psi^a_0 = {\varphi}^a$, and $u^{[n]}$ will denote $(u, u^{(1)}, ..., u^{(n)})$. Recall also that $|J|$ is the order of the multi-index $J$.

A vector field

in $M$, and its prolongation

to $J^n M$, will then induce an action on the space of sections in $(M, \pi_0, B)$, hence on functions; and on submanifolds in $J^n M$, hence on equations.

In particular, a (vector) function $u^a = f^a (x)$ is then mapped by the infinitesimal action of the vector field (with parameter ${\varepsilon}$) into

this infinitesimal action defines a flow (with parameter ${\lambda}$, not necessarily small) ${\gamma} ({\lambda}) = \Phi({\lambda}, {\gamma}_0)$ in the set ${\Gamma} (M)$ of sections in $(M, \pi_0, B)$, where ${\lambda}$ is the flow parameter and ${\gamma}_0$ is the initial point for the flow. From this one also computes the transformation rules for derivatives $u^a_J$, i.e., on sections of $(J^n M, \pi_n, B)$, and hence for equations; this also defines a flow ${\gamma}^{(n)} ({\lambda}) = \Phi^{(n)} ({\lambda}, {\gamma}^{(n)}_0)$ in the set ${\Gamma}^{(n)} (M)$ of sections in $(J^n M, \pi_n, B)$, where ${\lambda}$ is the flow parameter and ${\gamma}^{(n)}_0$ is the initial point for the flow.

We will also write these flow as

for ease of writing.

We say that a function $u = f(x)$ is symmetric under $Y$ if and only if the corresponding section ${\gamma}_f \in {\Gamma} (M)$ is invariant under (the flow $\Phi$ in ${\Gamma}$ induced by) $Y$; and similarly that the equation ${\Delta}$ in $J^n M$ is symmetric under $Y$ if and only if the corresponding submanifold $S_{\Delta}$ is invariant under (the flow $\Phi^{(n)}$ in $J^n M$ induced by) the prolongation $Y^{(n)}$.

We are specially interested in the case where the flow drives functions (or equations) which are not invariant to a fixed point, i.e., to a limit function (or an equation) which is invariant.

In particular, consider the case where the limit

exists. Then we say that $Y$ is an ${\omega}$-asymptotic symmetry for (the section ${\gamma}_f$ and hence) the function $f:B \to U$. (In the case where the limit $\lim_{{\lambda} \to - \infty} \Phi ({\lambda}, {\gamma}_f) \ = \ {\gamma}_f^{-\infty}$ exists, we say that $Y$ is an ${\alpha}$-asymptotic symmetry for $f$. This notation recalls the notions of ${\omega}$-limit and of ${\alpha}$-limit in use in Dynamical Systems theory ^{[41,42,43,44]}.)

Similarly, in the case where the limit

exists, we say that $Y$ is an ${\omega}$-asymptotic symmetry for (the submanifold $S_{\Delta}$ and hence) the equation ${\Delta} = 0$; and similarly considering the limit for ${\lambda} \to -\infty$.

We have the following results, which we quote from ^[4] (Lemma IX.1 therein):

Lemma 4.1. Let $u = f_0 (x)$ be a solution to the differential equation ${\Delta}_0$; and let the limits $f_{\pm \infty}$, ${\Delta}_{\pm \infty}$ for the flows

exist. Then $f_{\pm \infty}$ is a solution to ${\Delta}_{\pm \infty}$.

Corollary. If ${\Delta}_0 \to {\Delta}_*$ for ${\lambda} \to \pm \infty$, then under the flow $\Phi({\lambda}, f)$ all solutions $f_0$ to ${\Delta}_0$ go into solutions (not necessarily invariant) to ${\Delta}_*$ for ${\lambda} \to \pm \infty$.

Remark 4.2. We stress that here "asymptotic" refers to the flow parameter ${\lambda}$, i.e., to the flow $\Phi$ and $\Phi^{(n)}$ associated to $Y$. This may or may not correspond to asymptotic properties in time and/or space depending on the form of the vector field $Y$ we are considering. $\odot$

4.2.   Examples

Example 4.1. Consider the DS in ${{\bf R}}^2$

this is symmetric under both time translations and spatial rotations. Thus if we consider the two-parameters family of vector fields

all of these are symmetries for our DS. Note that for ${\alpha} > 0$ the action of $X$ gives translations forward in time, while for ${\alpha} < 0$ we have translations backward in time; similarly, the sign of ${\beta}$ controls the direction of rotation in the $(x, y)$ plane. We assume ${\alpha} > 0$ for definiteness and ease of discussion.

The action of $X$ on the space ${\Gamma} = \{ x_{\lambda} (t), y_{\lambda} (t) \}$ of sections of $(R^2 \times R, \pi_0, R)$ – which we denote by $\widetilde {X}$ – is described by

Thus there is no fixed point for the action of $\widetilde {X}$ unless we choose ${\beta} = {\alpha} A$, i.e.,

With this choice instead, we get

thus all solutions admit the vector field $X_0$ as an ${\alpha}$-asymptotic symmetry, and the vector field $X_0^* = -X_0$ as an ${\omega}$-asymptotic one (their role is reversed for ${\alpha} < 0$ in the definition of $X_0$). Note that the solutions with $x_0^2 +y_0^2$ equal to either zero or one also admit $X_0$ as a full symmetry. $\diamondsuit$

Example 4.2. Consider the FKPP (Fisher-Kolmogorov-Petrovskii-Piskunov) equation ^[64,65]

for $u = u(x, t)$, with $A$ and ${\varepsilon}$ positive real constants; in biological applications, it is required that $u(x, t) \ge 0$ for all $x$ and $t$. This obviously has some stationary homogeneous (i.e., $u_x = 0$) states $u(x, t) = 0$ and $u(x, t) = 1$, with the former being unstable and the latter stable against small perturbations. It is known that if the initial datum is suitably concentrated, in particular if it decays exponentially fast for $|x| \to \infty$, then for $t \to \infty$ and $x \to \infty$ the solution is in the form of a traveling front

It is convenient to rescale coordinates via

Using these coordinates – and omitting the tildes for ease of writing – the equation reads

and the asymptotic solution is

As already mentioned, the asymptotic above describes the behavior of the solution for large $t$ and $x$, which also means for small $u$. But for small $u$ the equation is approximated by its linearization around $u = 0$, i.e., simply by

the ansatz $u(x, t) = w(z) = w(x -2 t)$ transforms this into

with solutions

We denote by $\mathcal{W}$ the space of such solutions; the constants $(c_+, c_-)$ provide coordinates in this space. The requirement that $w(z) \to 0$ for $z \to \infty$ (i.e. for $x \to \infty$ at any finite $t$) yields $c_- = 0$ and hence $w(x) = c_+ e^{- z}$, which is just the $f_0 (x, t)$ given above.

The symmetry algebra of the linearized Eq (33) is generated by

We also consider

note that $X_+ = {\partial}_z$, $X_- (z) = 0$. Moreover, $[ X_0, X_\pm] = 0$.

The quotient (i.e., symmetry-reduced) Eq (34) admits $X_0$ (now written as $X_0 = w {\partial}_w$) as a scaling symmetry, and is also invariant under $X_+$. It is easily checked that the propagating front solutions correspond to a subspace of $\mathcal{W}$ invariant under the action of the group generated by the vector fields $X_0$ and $X_+$.

As for asymptotic symmetries, the situation is less simple; one can show the following, which we quote from ^[63]: if we consider a scaling vector field $X$ such that the limit for ${\lambda}\to \infty$ of $\exp [ {\lambda} X]$ extracts the behavior for large $x$ and $t$, and ${\Delta}_0$ is the FKPP equation, we set ${\Delta}_{\lambda}: = e^{{\lambda} X} {\Delta}_0$. Then

is the heat equation $u_t = u_{xx}$.

In fact, consider the most general scaling vector field,

one of the constants $a, b, c$ can always be set to unity (provided it is nonzero), which amounts to a redefinition of the group parameter. Then choosing ${\Delta}_0$ the FKPP equation in its standardized form (32) we get for ${\Delta}_{\lambda} = \exp[X_s {\lambda}] {\Delta}_0$ the expression

In order to have a meaningful (and nonzero) limit for ${\lambda} \to \infty$ we set $b = c$, and we will choose $b = c < 0$; in order to have a nontrivial limit for ${\lambda} \to \infty$ we also set $a = b/2$. Moreover, we can reparametrize the flow setting $|b| = 1$, i.e., $b = -1$. This yields

which reduces to the heat equation for ${\lambda} \to \infty$. $\diamondsuit$

5.   Symmetry of stochastic differential equations

We have so far discussed symmetries of deterministic differential equations. The symmetry approach has also been applied to stochastic differential equations ^{[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]}.

Here we will be specially concerned with equations (as in the case of deterministic equation, here an equation might be of vector type, i.e., a system of scalar equations) of Ito type, i.e., in the form

where$f^i$ and ${\sigma}^i_{\ k}$ are smooth functions. We will also consider Stratonovich type equations

As it is well known, to each Ito equation there is an associated Stratonovich equation (and conversely) which carries the same statistical information (the correspondence between the two presents some subtleties, see e.g., ^[34,37] for a discussion of these), in particular Eqs (35) and (36) are in such a relation if their drifts satisfy

We will consider in particular the approach which parallels the usual treatment of deterministic equations in the stochastic case (to which we gave several contributions in recent years); for a discussion of other approaches, including earlier attempts, see e.g., the review paper ^[22]. See also ^[66,67] for an approach relating symmetries to Girsanov theory, and more generally ^{[68,69,70,71,72]} for a related concurrent approach.

While in the case of deterministic differential equations there exists a well developed geometric theory, which allows to set the symmetry theory on firm geometrical basis, this does not hold for stochastic equations. Actually, the main characteristic of Ito differentials is that under diffeomorphisms they do not transform by the familiar chain rule, but according to Ito rule. This shows that the lack of a geometrical setting for symmetries of Ito equations is not due to insufficient development of the theory, but to its intrinsic features. The situation is slightly different for Stratonovich equations. In this case the chain rule is preserved, so that a geometrical setting would be possible; but, as it is well known, the Stratonovich formulation presents several delicate points from the point of view of stochastic processes (many of them being actually associated to its more relevant advantage from the point of view of a physicist, i.e., the preservation of the time inversion symmetry), so that a formulation of the basic theory in terms of the Ito theory is preferable. The relation between symmetry theory for Ito and Stratonovich equations is thoroughly discussed e.g., in ^[27]; see also ^[19] for earlier results.

5.1.   Symmetry of Ito equations

Let us consider an Ito Eq (35); this involves the time variable $t$, the driving Wiener processes $w^k (t)$, and the spatial variables $x^i$, which evolve through a stochastic process depending on the realization of the $w^k (t)$ and described by the Ito equation itself.

Thus our problem lives in the space $\mathcal{E}$ of the variables $\{ t; x; w \}$ (note that $x$ and $w$ are vector variables); a blind application of the Lie approach would consider general vector fields in this space, i.e., vector fields of the form

It is clear that the action of such a vector field mixes all the variables, discarding the difference between driving and driven stochastic processes, and, even more, introducing a stochastic time. This kind of transformation is well known and widely used by probabilists, and also in the study of stochastic differential equations (including e.g., in the study of normal forms for SDEs, which has close relations with symmetry considerations ^[73]); but here we are interested in transformations which leave the equation invariant. Transformations which even change the nature of the variables are surely not leaving the equation invariant, and hence will not be considered.

If we want to leave the equation invariant, a necessary (but by no means sufficient) condition is that the nature of the involved variables is not changed. We should also ask the transformed $w^k$ variables to be still independent Wiener processes. These requirements (see ^[30] for how the latter one restricts the form of the $h^k$ functions) lead to consider vector fields in $\mathcal{E}$ of the form

with $R$ a matrix in the Lie algebra of the conformal linear group $\mathrm{CL (n)}$. These vector fields are called admissible in $\mathcal{E}$, and in the following we only consider such fields.

Remark 5.1. We recall $\mathrm{CL (n)} \approx {\bf R}_+ \times \mathrm{O(n)}$, where $\mathrm{O(n)}$ is the ($n$-dimensional) orthogonal group, and ${\bf R}_+$ represents here the dilation group. An alternative characterization of matrices $A \in \mathrm{CL (n)}$ is simply that they satisfy $A^+ A = {\lambda}^2 I$ for some ${\lambda} \in {\bf R}$. We are interested in matrices $R$ in the Lie algebra of $\mathrm{CL (n)}$; in practice, such an $R$ is the sum of a diagonal matrix which is a multiple of the identity (corresponding to the linear space spanned by the generator of the dilation group), and a skew-symmetric one (these correspond to the linear space spanned by the generators of the orthogonal part of the group). Thus $R$ has $1+n (n-1)/2$ independent elements. $\odot$

It is then straightforward to consider the change of variables induced by an infinitesimal action of $X$, i.e.,

and hence on the Eq (35). Our restrictions on the form of the vector field guarantees that the transformed equation is still an Ito one, which can hence be written as (recall the variables $t, x, w$ are now the transformed ones)

where the functions $\widehat{f^i}$ and $\widehat{\sigma}^i_{\ k}$ – which can be readily computed in explicit terms – are in general different from the original ones, $f^i$ and ${\sigma}^i_{\ k}$.

Requiring that the equation is left invariant under the action of $X$ means simply requiring that $\widehat{f^i} (x, t) = f^i (x, t)$ and that $\widehat{\sigma}^i_{\ k} (x, t) = {\sigma}^i_{\ k} (x, t)$. We have thus $n + n^2$ equations involving the unknown functions $\tau, {\varphi}^i, {\sigma}^i_{\ k}$, which determine the vector fields which leave the Eq (35) invariant, i.e., identify its (Lie-point) symmetries; these are thus called the determining equations.

An important point is that the form (39) for symmetry vector fields is actually still too general: albeit all vector fields of this form are admissible, not all of them are useful when it comes to applying symmetry theory. In particular, as in the deterministic case, the main purpose of computing symmetries is to then apply them for reducing – or even solving – the stochastic equations. But to this purpose, only so called simple symmetries are useful under the present theory ^{[16,17,18,28]}. Simple symmetries are those not acting (but generally depending) on the time variable; i.e., those, in the notation of (39), with $\tau = 0$). Thus admissible simple symmetries have generators of the form

with $R$ as discussed above.

We stress that this restriction to simple (admissible) symmetries will be given for granted in the following; it will also simplify a number of matters.

Some further nomenclature within simple symmetries may also be useful: when $R = 0$ and the ${\varphi}^i$ do not depend on the $w$ variables, we have deterministic symmetries; when $R = 0$ and (at least some of) the ${\varphi}^i$ do actually depend on the $w$ variables, we have random symmetries; and finally when $R\not = 0$ (and hence the $X$ has a component along the $w$ variables) we have W-symmetries.

Remark 5.2. There is a substantial difference in computing the action of a vector field $X$ on a stochastic equation with respect to the case of a standard (i.e., deterministic) ordinary differential equation. In the latter case the ODE defines a vector field $Z$ in a suitable jet bundle $J$, and we compute how the vector field $X$ and its prolongations act on the vector field $Z$; in the case of stochastic Ito equations, we have to compute how $X$ acts on the functions $f^i (x, t)$, ${\sigma}^i_{\ k} (x, t)$ and on the differentials $d x^i$, $d t$, $d w^k$. Thus, roughly speaking, we are dealing with one-forms rather than with vector fields. This might be compared with recent work by Anco and Wang ^[74] taking the same point of view for deterministic ODEs. $\odot$

When we write down the explicit expressions for $\widehat{f^i}$ and $\widehat{\sigma}^i_{\ k}$, we also obtain the explicit expressions for the determining equations. We set this in the form of a Proposition; this is quoted from ^[30], and we refer to there for a proof.

Here and below the symbol $\Delta$ denotes the Ito Laplacian,

Proposition 5.1. The determining equations for simple symmetries of the Ito Eq (35) are

Obviously, several special cases are obtained by considering special forms of the functions $\tau$, ${\varphi}^i$ and $R$. We will not enter into this discussion, and just refer the reader to ^[22,30].

We will however stress that solutions with $\tau = 0$ (i.e., such that the time variable is not acted upon by the vector field $X$) are known as simple symmetries, and are specially useful in that the symmetry reduction based on these is specially simple.

Remark 5.3. The determining Eqs (43), (44) are a set of $n+n^2$ equations for the $n$ functions $\{ {\varphi}^1, ..., {\varphi}^n \}$ and the $n \times n$ matrix $R$, having (as noted above) $n (n-1)/2 +1$ independent elements. In order to concretely use the information about the symmetry, it does not suffice to prove that a solution exists, but we need of course to determine it in an explicit way. $\odot$

5.2.   Ito versus Stratonovich equations

As it is well known – and recalled above – the Ito Eq (35) is associated with the Stratonovich Eq (36), the coefficients of the two being related by (37). With this condition, (36) carries the same statistical information as (35); see e.g., ^[37] for more detail. From the point of view of symmetry, there is a substantial difference: while the Ito equations transforms, under the action of a diffeomorphism and thus in particular under the flow generated by a vector field, as described by the Ito rule, a Stratonovich equation transforms under the usual chain rule and thus has a geometrical meaning.

It is quite natural to wonder if symmetries of an Ito equation and of the associated Stratonovich equation are related or coincide. The question was first tackled by Unal ^[15], and then fully solved by Kozlov ^[27], who showed that deterministic and random symmetries of an Ito and of the associated Stratonovich stochastic equations are just the same – as indeed natural given the equivalence of the two formulations.

The situation is slightly different for W-symmetries; the relations of these for an Ito and the associated Stratonovich equation are discussed in ^[30], see Section VII (and in particular Section VII.C) therein.

One could of course also analyze symmetries of a Stratonovich Eq (36) along the lines of Section 5.1, with the same limitation on the form if the admitted vector fields; in this way one would obtain the determining equations for (Lie-point) symmetries of a Stratonovich equation. See again ^[30] for a proof to the following statement.

Proposition 5.2. The determining equations for simple symmetries of the Stratonovich Eq (36) are

Note that (46) coincides with (44); as for (45), this coincides with (43) when we impose the correspondence (37). See also ^{[24,25,26,27]} for further detail.

5.3.   Adapted variables and integration of stochastic equations

In the view of Sophus Lie, the symmetry analysis of differential equations was not a mathematical curiosity, but a concrete method to obtain order/dimensional reduction, and when possible exact solutions, of the differential equations.

The same applies for symmetry analysis of stochastic equations: knowledge of their symmetries allows to obtain a reduction, and when we have enough symmetries (e.g., one symmetry for a scalar equation) there is a concrete method to obtain a complete solution of the equation.

Remark 5.4. It is maybe worth specifying what we mean by this, i.e., by a complete solution of an Ito equation. We mean an explicit correspondence between a concrete realization of the driving ($n$-dimensional) Wiener process and the corresponding realization of the ($n$-dimensional) stochastic process $x(t)$. Example of this correspondence are provided below in this subsection. $\odot$

As for deterministic equations, the key to integration of a symmetric equation lies in passing to symmetry-adapted variables (an alternative method uses invariants, and will be discussed in the next Section). In these, the equation takes a specially simple form and is integrated in an elementary way; that is, the difficulty of integrating the equation is moved to the difficulty of determining the symmetry and the symmetry-adapted variables.

In fact, once we have determined (simple) symmetries of a given Ito equation, these can be used pretty much in the same way as in the case of deterministic equations to integrate or reduce the equation at hand.

Proposition 5.3. Let the Eq (35) for $x \in M \approx {\bf R}^n$ admit a $r$-dimensional solvable algebra $\mathcal{G}$ of simple deterministic Lie symmetries. Then, passing to $\mathcal{G}$-adapted variables, the equation can be reduced to a stochastic equation for $y \in M/\mathcal{G} \approx {\bf R}^{n-r}$ plus a set of $r$ reconstruction equations; the latter amount to simple Ito integrations. In the case $r = n$, the equation is explicitly integrated.

Proof. This is given and proved e.g., in ^[25]; it is however worth sketching the proof. The first step, i.e., the determination of (simple) Lie symmetries, has already been mapped in the previous part of this section to the problem of solving the determining equations. Assume that a symmetry vector field $X$ – written in the form (41) – has been found. We want then to pass from $(x, t;w)$ to new variables $(y, \theta; z)$, where the $z^k$ are still independent Wiener processes, such that in the new variables $X$ reads simply as $X = {\partial} / {\partial} y^n$. In fact, if this is the case the Ito equation will be written, in the new variables, as

with both the $F^i$ and the $S^i_{\ k}$ being independent of $y^n$. The equation decouples then into an $(n-1)$-dimensional Ito equation plus a "reconstruction equation", which is written in terms of an Ito integral as

In the case of a scalar equation, this gives directly a full solution of the equation.

When we have a higher dimensional equations and several symmetries spanning a solvable Lie algebra, we can proceed step by step by reducing the dimension of the system provided we operate according to the structure of the Lie algebra, i.e., according to the derived series for $\mathcal{G}$, as in the case of deterministic equations ^{[3,4,5,6,7,8,9]}. $\diamondsuit$

We stress that for simple Lie-point symmetries, i.e., vector fields of the form

there is a simple way to relate the sought-for change of variables to the symmetry $X$ ^[16,17,18]. First of all we note that in this case the time variable can be left unaffected by the change of variables. It suffices then to require that

this requires to solve the linear equations

which are solved by the classical method of characteristics.

In particular, for $n = 1$ we have to solve only the last equation, and its solution is promptly written as

Remark 5.5. The reader can wonder why this Proposition considers only deterministic symmetries. The reason is that in the case of random or $W$-symmetries, one is not guaranteed that the transformed equation is still of Ito type (this depends on a certain compatibility condition being satisfied, see ^[25,30]). Note that the problem is specially present when attempting multiple reduction. In fact, the first reduced equation could be not of Ito type, and in this case our theory can not say anything about successive reductions.

We also stress that in the scalar case the reduced equation, albeit not of Ito type, could be integrable; in this case its solution provides – inverting the transformation – the solution to the original equation.

Finally, we note that in ^[30], see in particular Sect. V therein, condition ensuring the symmetry reduced equations are still of Ito type are discussed in some detail (in any case, one can perform the reduction and verify if the reduced equation is Ito or otherwise). When all the reduced equations are of Ito type, Proposition 3 extends also to random or W-symmetries. $\odot$

5.4.   The Lie algebraic structure of symmetries for a stochastic equation

It is well known that symmetries of a deterministic differential equations form a Lie algebra ^{[3,4,5,6,7,8,9]} (for an explicit proof, see e.g., Sect. 2.3 (and Sect. 5.1 for the case of generalized symmetries) in ref.^[6]). The same holds for stochastic equations, at least for deterministic or random symmetries; note this includes the class of symmetries (deterministic ones) which are guaranteed to allow multiple reductions (see Remark 5.5 above):

Proposition 5.4. The set of simple deterministic or random symmetries of a given Ito equation has the standard Lie algebra structure, i.e., if $X, Y$ are symmetries of a given equation, their commutator $Z = [X, Y]$ is also a symmetry for the same equation.

Proof. This is Theorem 4.3 in ^[26]; see there for the proof. Alternatively, note that symmetries of an Ito equation correspond to symmetries of the associated Stratonovich equation; for the latter the standard chain rule holds, so it is obvious that symmetries have the standard Lie algebra structure, proceeding just as in the case of deterministic equations. $\diamondsuit$

Remark 5.6. It should be noted that ^[20] contains a statement in contrast with Proposition 5.4 (see Sect. IX.A therein). That statement concerns transformations of a rather more general form than the one considered here^† and it is thus wrong if seen in the present context, while Proposition 5.4 above is correct. $\odot$

^†In fact, far too general transformations, in particular functions $h(x, t, w)$, are allowed at that point of the discussion in ^[20]; restricting to $h(x, t, w) = R w$, as we do here, produces the right result.

Remark 5.7. Our Proposition 5.4 does not say anything concerning W-symmetries. In fact, the following Examples will show that in some cases these also form a Lie algebra (or possibly even a Lie module), while in other cases this is not the case. In other words, no general statement is possible about the Lie algebraic structure of W-symmetries of a given equation. $\odot$

5.5.   Stochastic versus diffusion equations

Finally, we recall that for any Ito Eq (35) there is an associated diffusion (Fokker-Planck) equation

describing the evolution of the density $u (x, t)$ in time; here $u_i : = ({\partial} u / {\partial} x^i)$ and similarly $u_{ij} : = ({\partial}^2 u / {\partial} x^i {\partial} x^j)$, and the coefficients in the Fokker-Planck Eq (50) are related to those in the Ito Eq (35) by

Remark 5.7. The Fokker-Planck equation is often also written, more synthetically, as

Combining this equation with those for the transition from an Ito equation to the associated Stratonovich one, see Eq (37), it is immediate to get the Fokker-Planck equation associated to a Stratonovich equation. $\odot$

Remark 5.8. We also recall that, as discussed in ^[13,19,22], any symmetry of the Ito Eq (35) leads (adding a standard term proportional to ${\partial}_u$) to a symmetry of the associated Fokker-Planck Eq (50), but the converse is not true. This is rather natural, given that different Ito equations can have the same associated Fokker-Planck equation; in fact, symmetries of a Fokker Planck equation will in general map these different Ito equations one into the other. We refer to ^{[13,14,19,22]} for further detail and examples. $\odot$

5.6.   Examples

We consider here some examples illustrating the results discussed in this Section. We recall that we are considering simple admissible symmetries or W-symmetries; that is, vector fields of the form (41) with $R$ a constant matrix of the form discussed in Sect. 5.1.

Example 5.1. Consider the equation

The determining equations are then

their general solution is

with $k$ a constant. Thus we have two symmetries (a deterministic one and a W-symmetry), i.e.,

It is immediate to check that

is still a symmetry: thus in this case the set of (simple) symmetries, involving both "ordinary" and W-symmetries, is indeed a Lie algebra. $\diamondsuit$

Example 5.2. Consider the scalar Ito equation

where ${\alpha}$ and ${\beta}$ are real constants. This describes the geometric Brownian motion in one dimension, and it is well known that this equation can be integrated (see e.g., ^[36], Chapter 5). Note that if we start with $x(0) > 0$, we are guaranteed to have $x(t) \ge 0$ for all times. The determining equations (43), (44) for simple symmetries^‡ of (53) are

^‡Recall this means $\tau = 0$ in terms of (39), i.e., that $X$ should be of the form (41).

Solving (55) yields

Inserting this into (54), the latter reads

the solution to this equation is

having defined

Going back to evaluate ${\varphi}$, we thus we have symmetries depending on the arbitrary function $Q$ and on the arbitrary constant $R$

The simplest choice is of course $R = 0$, $Q(\zeta) = 1$, which gives ${\varphi} = x$, i.e. the scaling symmetry

The associated integrating change of variables is simply

In fact, Ito formula for the change of variables gives immediately (using (53) for the time evolution)

which yields at once (setting for ease of writing $t_0 = 0$ and $w(0) = 0$)

and hence

We can use this Example also to illustrate the Lie algebraic structure of the set of symmetries. Let us first set $R = 0$, i.e., consider random symmetries, and two symmetries of the general form (56) with $R = 0$, say

An explicit computation shows that

where the function $H$ is given explicitly by

This is still of the functional form (56) with $R = 0$, and hence is immediately known to be still a (random) symmetry for our Eq (53).

On the other hand, let us consider the general case, i.e., let us not enforce $R = 0$. We set a simplified notation writing

and consider vector fields

by our discussion above, they are symmetries of (53); we could actually also take $R_1 = R_2 = R \not = 0$. Then an explicit computation yields

where the function $h$ is given by

The vector field $Z$ thus defined is not of the form (56) and hence is not a symmetry of the Eq (53).

This shows that W-symmetries of a given equation, contrary to simple deterministic and random symmetries, in general do not form a Lie algebra. $\diamondsuit$

Example 5.3. Consider the stochastic logistic equation with environmental noise

Here ${\alpha}, {\beta}, {\gamma}$ are nonzero real constants; in the biological applications they have positive sign, but for our present purposes this is not relevant.

We will again look at the case with $R = 0$, i.e., search for symmetries of the form $X = {\varphi} (x, t;w) {\partial}_x$. In this case the second determining equation (44) reads

which yields immediately

Then the first determining Eq (43) splits into two equations (for the coefficients of $x$ and $x^2$ in the resulting expression):

These are easily solved, and we get in the end

Note this yields a random symmetry

The associated change of variables and its inverse are given by

When we express the evolution of $y$ using Ito's rule and (57), we get

note this is not an Ito equation. It can however be directly integrated to give

Transforming back to the original variable $x$ via (59) we obtain the solution to (57) in closed – albeit involved – form. The reader can see ^[75] for numerical experiments confirming the analytical procedure. $\diamondsuit$

Example 5.4. Consider the equation

(note that $x(0) > 0$ entails $x(t) \ge 0$ for all $t \ge 0$.) This admits as symmetry

where $F$ is an arbitrary function of

The simplest choice is of course $F (\zeta) = 1$, i.e.,

note this is a deterministic symmetry. The associated change of variables is just $y = \log (x)$, and with this our Eq (60) is mapped into

which is directly integrated to

This yields the solution to our Eq (60) upon inversion of the change of variables, i.e., setting $x = e^y$.

Suppose we do not want to make this simplest choice, and go for the next simpler one; this will produce a random symmetry. So let us choose $F(\zeta) = \zeta$, and hence

Then the associated change of variables is

the equation for $y$ is then

Thus we should choose ${\beta} (t, w) = b (t) + c w$ in order to get an Ito equation. With this choice, the initial equation is mapped into

which is readily integrated to yield

Inverting the change of coordinates – which is now a more complex operation than in the previous case – we obtain the solution for $x(t)$.

We can again check that the symmetries, i.e., vector fields of the form (61) (which are random symmetries), form a Lie algebra. In this case, it is in fact immediate to check that if

then we have

where we have defined

Thus $Z$ is of the form (61), and hence a symmetry for Eq (60), as claimed. $\diamondsuit$

Example 5.5. Consider the two-dimensional equation (here and elsewhere in concrete examples, we write vector indices as lower ones, to avoid any possible confusion with squares)

It is easily checked that this admits the symmetry vector field

symmetry-adapted variables are

in these variables, the vector field is

and the Eq (62) reads simply

That is, we have an autonomous Ito equation for $y_1$; if this is solved, say with $y_1 = \Phi [ t, w_1 (t) ]$, then the equation for $y_2$ is immediately integrated to give

The solution to the initial problem (62) is then obtained by inverting the change of variables (64). $\diamondsuit$

Example 5.6. Consider, as in ^[30], the two-dimensional isotropic linear stochastic equation

This was considered, for what concerns W-symmetries in ^[30] (this was Example 7 therein), but only W-symmetries were considered in that work; we will thus have to derive the full symmetry algebra.

We write, for ease of notation, $R$ in the form

Solving the ${\sigma}$-determining equations for ${\varphi}^i ({{\bf x}}, t; {{\bf w}})$ yields

With this, the $f$-determining equations read

Note that the $\psi$ are functions of the $z_i$ variables alone; thus coefficients of the $x_i$ must vanish separately, and we get

where $c_i$ are arbitrary constants.

In conclusion, the symmetry algebra $\mathcal{G}$ for (66) is spanned by the vector fields

Note that the $X_i$ are W-symmetries, while the $Y_i$ are deterministic symmetries. A trivial computation shows that the commutation relations are given by

This shows that $\mathcal{G}$ is indeed a Lie algebra; moreover ${{\mathcal G}}_X = \{ X_1, X_2\}$ and ${{\mathcal G}}_Y = \{ Y_1, Y_2 \}$ are abelian subalgebras, and ${{\mathcal G}}_Y$ an abelian ideal in ${{\mathcal G}}$. $\diamondsuit$

6.   The method of invariants

Consider a general function

its Ito differential is

When we evaluate this on the dynamics described by the Ito Eq (35), we get

It follows immediately that if (and only if) $F$ satisfies the equations

then $F$ is an invariant for the SDEs (35).

Note that in general the invariants will be arbitrary functions of some "basic invariants" functions.

Example 6.1. Consider the equation

then the Eq (70) yield immediately that

A straightforward computation shows that in this case $\Delta F = 0$ (note in particular $\Delta z^i = 0$). The Eq (69) reads then

and it follows immediately that

having defined

where $\Phi$ is the primitive of $f$,

Obviously, the $\zeta^i$ are in this case the basic invariants mentioned above. $\diamondsuit$

The existence of invariants allows to express the solutions to a SDE system in terms of a reduced system of Ito equations; if there are as many independent invariants as degrees of freedom, then the solution can be expressed purely in terms of invariants.

In other words, the presence of a sufficient number of invariants guarantees the integrability of the equation, pretty much as in the deterministic case the presence of a sufficient number of conserved quantities guarantees integrability.

Example 6.2. In the case of Eq (71), we have the basic invariants defined in (72); at the initial time $t = 0$ these will satisfy $\zeta^i = \zeta^i (0)$. If $w^i (0) = 0$, and choosing the constant in the primitives of (73) so that $\Phi^i (0) = 0$, at $t = 0$ we have

Thus we have at all times and for each realization of the Wiener processes $w^k (t)$, and with $\Phi^i$ as in (73) above,

This yields directly the solution to (71). $\diamondsuit$

It is natural to wonder if and how invariants are related to symmetries of the stochastic equation.

Comparing Eqs (43), (44) one the one hand, and (69), (70) on the other, we immediately observe a rather trivial (but useful) relation: if both $f^i$ and ${\sigma}^i_{\ k}$ do not actually depend on the $x$ variables (but possibly depend on $t$) and set $R = 0$, the symmetry coefficients ${\varphi}^i$ are also invariants for the Ito equation. (Note that such relation does not exist for $R \not = 0$, i.e., for proper W-symmetries ^[30].)

We would be interested in knowing if other relations exist. This matter has been studied in recent papers by one of us ^[26,27,28] (to which we refer for details and examples), and we briefly report the relevant results here.

Proposition 6.1. The set of invariants for a given Ito equation is a ring over ${\bf R}$, i.e., if $F, G$ are invariants for the given equations and ${\alpha}, {\beta}$ are real constants, so are

Proposition 6.2. If a given Ito Eq (35) admits the symmetry $X$ and the invariant $F$, then $G = X(F)$ is also an invariant for the same equation.

Proposition 6.3. The set of symmetries for a given Ito equation has, beside the structure of Lie algebra, the structure of a Lie module over the ring of invariants; i.e., if $X, Y$ are symmetries for the given equation, and $F, G$ invariants for the same equation, then

is still a symmetry for the equation.

Proof. Proposition 6.1 is trivial; Propositions 6.2 and 6.3 are respectively Theorems 4.4 and 4.6 in ^[26]; see there for proofs. $\diamondsuit$

We stress that, obviously, a generic stochastic equation has no invariants, even in the case it has symmetries.

Example 6.3. Consider again the two-dimensional isotropic linear stochastic Eq (66) considered in Example 5.6 above (where it was found it has a nontrivial symmetry algebra). Now a direct computation shows that there are no invariants. In fact, invariants should satisfy the equations

Note in the last equation there should also be a term $(1/2) \Delta J$ with $\Delta$ the Ito Laplacian (42), but in this case it results $\Delta J = 0$.) Solving the first two equations yields

Plugging this into the last Eq (74) yields $H = const$, thus showing that only trivial invariants are present. $\diamondsuit$

7.   Conditional and asymptotic symmetries for stochastic equations

We now want to argue that – pretty much in the same way as a great extent of the standard theory of symmetry of deterministic differential equations can be extended to the framework of stochastic differential equations – the concepts of conditional, partial and asymptotic symmetry also apply for stochastic differential equations. What is more, these are also useful to determine solutions to stochastic equations.

It will happen that these are intimately related – much more than in the deterministic setting – to similar concepts for invariants, i.e., to conditional, partial or asymptotic invariants. These, and the relation with symmetries, will be discussed in the next Sect. 8. We will first discuss the matter in abstract terms – actually with a more formal writing than in previous Sections, as now we develop new material and results; and then provide, in Sect. 9 below, a number of concrete Examples.

7.1.   Conditional and partial symmetries

Our first observation is that conditional and partial symmetries can be, in principle, defined for stochastic equations exactly as in the case of deterministic ones.

The difference – and the reason to write "in principle" in the lines above – is that while a solution of a deterministic equation is determined by the initial condition, in the case of stochastic equation it is determined by the initial condition and by the realization of the driving Wiener processes. But, of course, we do not know a priori what the realization of the Wiener process will be, so we do not want to discuss properties which depend on such realization. (A partial exception to this is that often we are satisfied with properties which hold for generic realizations of the Wiener processes, i.e., for a full measure set of the driving Wiener paths.)

Thus, in practice, it is quite difficult – and makes little sense in view of applications – to study symmetries of a given solution, or of a given set of solutions.

The exception is provided by the case where the solutions in the set which is studied are identified by their living on an invariant manifold for the equation. By this we mean that the Eq (35) describes a variable $(x, t) \in M$ ($M$ is the phase manifold), and there is a proper submanifold $M_0 {\subset} M$ such that if $(x(0), t_0) \in M_0$, then for any realization of the driving Wiener processes, and for any $t \ge t_0$, $(x(t), t) \in M_0$.

We will actually confine ourselves to the case of autonomous equations, i.e., the case where the coefficients $f^i (x, t)$ and ${\sigma}^i_{\ k} (x, t)$ in (35) are actually independent of $t$. In this case we can work with the reduced phase manifold^§, and in the following $M$ should be interpreted as the reduced phase manifold. Extension of our discussion and results to the non-autonomous is rather simple, and in any case can be obtained – remaining within the autonomous formalism – by adding a new variable $x^0$ with evolution governed by $d x^0 = d t$.

^§This was denoted as $M_0$ in Sect. 2; but here we reserve this notation to the dynamically invariant submanifold. We trust no confusion will arise.

We will see in the following that the existence of these invariant manifolds is naturally related to "conditional" (or "partial") invariants for the equation.^¶

^¶The simplest – but rather trivial – case in which there is an invariant manifold is that of a fixed point; for example the stochastic Eq (35) with coefficients satisfying $f^i (0, t) = 0$, ${\sigma}^i_{\ j} (0, t) = 0$, automatically admit ${{\bf x}} = 0$ as a fixed point, and hence a (highly symmetric) solution $x(t) \equiv 0$. This is of little interest.

The simplest nontrivial case is that in which the dynamics admits a nontrivial invariant submanifold $M_0 {\subset} M$ (this is of course more significant if $M_0$ is stable or attracting, as will be discussed later on). In this case we can consider the restriction of our system to $M_0$, and this may have symmetries which are not present for the full system. These in turn may allow for the integration of the restricted equation; solutions obtained in this way are also special solutions for the full system.

We start by stating some very simple – but useful – facts.

Lemma 7.1. Let the SDE $\Delta$ (35) in the $n$-dimensional manifold $M$ admit an invariant submanifold $M_0 {\subset} M$ of dimension $m < n$. Consider the restriction ${\Delta}_0$ of ${\Delta}$ to $M_0$. Any solution to ${\Delta}_0$ is also a solution to ${\Delta}$, and the most general solution to ${\Delta}$ with $x (0) \in M_0$ is obtained as a solution to ${\Delta}_0$.

Proof. Obvious.

Lemma 7.2. In the setting of Lemma 7.1, let ${\Delta}_0$ admit $m$ simple symmetries spanning a solvable algebra. Then it can be integrated, and hence all the solutions to the initial equation ${\Delta}$ with initial conditions on $M_0$ can be obtained.

Proof. Integrability of ${\Delta}_0$ follows from Proposition 5.3 above. Once ${\Delta}_0$ is integrable, the rest of the statement follows immediately from Lemma 7.1. $\diamondsuit$

This construction is also generalized to the case where the symmetry algebra has smaller dimension, as stated in the following Lemma.

Lemma 7.3. Let the Ito SDE $\Delta$ in the $n$-dimensional manifold $M$ admit an invariant submanifold $M_0 {\subset} M$ of dimension $m < n$. Let ${\Delta}_0$ be the restriction of ${\Delta}$ to $M_0$. Then

$(a)$ If ${\Delta}_0$ admits $s < m$ simple symmetries spanning a solvable algebra ${{\mathcal G}}$ which generates a local Lie group $G$, then ${\Delta}_0$ can be reduced to an equation $\widehat{\Delta}_0$ on $M_0/G$.

$(b)$ Let $\widehat x (t)$ be any solution to $\widehat{\Delta}_0$; then $\widehat x (t)$ extends to a solution $x(t)$ of ${\Delta}_0$, and this is turn describes the most general solution to ${\Delta}$ with $x(0) \in M_0$.

Proof. Point $(a)$ follows by Proposition 5.3 above. Solutions to $\widehat{\Delta}_0$ are extended to solutions to ${\Delta}_0$ again by means of Proposition 5.3, i.e., via the reconstruction equations. In view of Lemma 7.1, the exact solutions obtained in this way are also all the solutions to the initial equation ${\Delta}$ with initial conditions on $M_0$. $\diamondsuit$

Remark 7.1. The statement in these Lemmas are rather trivial, as shown by their very simple proofs. Actually these Lemma are more the description of a strategy to obtain a set of solutions (identified by living on $M_0$) to ${\Delta}$. $\odot$

Remark 7.2. If the equation admits a standard (global) invariant, then all the level manifolds for this invariant are invariant manifolds of the differential equation; see Example 9.1 below. Thus the Lemmas 7.1, 7.2 and 7.3 given above apply in general, in the sense that any initial datum lies on such an invariant manifold. But they apply also to the case where no global invariant exists (we will see in a moment a weaker type of invariant is involved here), i.e., when we have isolated invariant manifolds. $\odot$

Remark 7.3. The presence of an isolated invariant manifold can be described in terms of a conditional invariant for the stochastic equation. That is, if $M_0$ is the submanifold described by $F (x) = m_0$ (here $m_0 \in {\bf R}$) and $M_0$ is an invariant manifold while there are no nearby invariant manifolds, then $F (x)$ is a conditional invariant for our equation. Note that this means that on the dynamics we have

with nonzero ${\alpha}, {\beta}_k$. In this case only the $m_0$ level set is invariant, while the other ones are in general not invariant, as in our Examples 9.2 and 9.3 below (in both cases the conditional invariant is just $\rho$ and the invariant level set is $\rho = 1$). We will discuss this point in more detail in Sect. 8 below. $\odot$

Remark 7.4. This situation is similar to the setting considered by Misawa and by Albeverio & Fei ^{[76,77,78,79]} in their early study of symmetry of stochastic equations (see the discussion in ^[22]). In their case, however, we had a global (proper) invariant, hence a global constant of motion – which allowed for a dimensional reduction of the stochastic system – as in Example 9.1 below, while in the setting discussed here we only have a conditional one, hence reduction can take place only on the specific level set of the conditional constant of motion, viz. the manifold $M_0$. $\odot$

Remark 7.5. The situation discussed in this Section is related to conditional constants of motion ^{[80,81,82,83,84]} and conditional invariants, so that it would be natural to speak of conditional symmetries. On the other hand, in this case the solutions (to the restricted problem) are not invariant under the conditional symmetry: they are actually mapped one into another by the action of the latter. In this sense, the situation is more similar to that encountered when dealing, in the deterministic framework, with partial symmetries; from this point of view it might be more appropriate to speak of partial invariants; but we will prefer to use the term conditional invariant in view of the first mentioned parallel with the deterministic case, and the established use in the latter. $\odot$

7.2.   Invariant manifolds and (conditional) configurational invariants

In the previous discussion, in particular Lemmas 7.1 through 7.3, we have assumed the existence of an invariant manifold $M_0 {\subset} M$ for the dynamics described by our Ito Eq (35). Needless to say, this is a highly non-generic situation. In fact, as anticipated in Remark 7.3, the existence of invariant manifolds is related to the existence of (conditional) invariants for the dynamics.

In order to identify invariant manifolds in $M$, or in $M \times {\bf R}$, these invariants should depend only on the $x$, or the $(x, t)$, variables. We have thus the following:

Definition 7.1. When the invariant $J = J(x, t;w)$ for an Ito equation does not depend on the $w$ variables, i.e., $J(x, t;w) = \Psi (x, t)$, we say that $J$ is a phase invariant; if moreover $\Psi$ is independent of $t$, we say $J$ is a configurational invariant.

Remark 7.6. We stress that while configurational invariants can identify invariant manifolds, and phase invariant can identify time-varying invariant manifolds, in order to identify solutions in terms of invariants we need invariants which also depend on the $w^i$ variables, and hence on the realization of the Wiener processes. See also Example 9.1 in this respect. $\odot$

We recall that if $J : M \to R$ is a smooth function defined on the phase manifold $M$, or more generally $J : M \times \mathcal{W} \to R$ where $\mathcal{W} \approx {\bf R}^n$ is the space in which the driving Wiener processes $w^i = w^i (t)$ take values, then its evolution on the dynamics described by the Ito equation ${\Delta}$ – written as (35) – is given by

We are specially interested in the case where $J$ is a function of $x$ and $t$ alone, $J(x, t;w) = \Psi (x, t)$; we say then that $J$ is a phase space function, and if $\Psi$ does not depend on $t$ we say it is a configurational function. In both these cases the Ito Laplacian reduces to

Lemma 7.4. The evolution of a phase space function under the dynamics described by an Ito Eq (35) is itself described by an Ito equation.

Proof. Using (75) and specializing it to the case $J = \Psi (x, t)$, we have

Recalling also (76), we note that now all terms within the square brackets depend on $(x, t)$ alone, i.e., we have again an equation of Ito type. $\diamondsuit$

It is now natural to introduce the following Definition, which leads immediately to the foregoing Lemma 7.5.

Definition 7.2. (a) If the function $J$ is such that $d J = 0$ on solutions to the Ito equation (35), i.e., (75) is identically zero, we say that it is an invariant for the Ito equation (a phase space invariant if $J = \Psi (x, t)$, and (78) is identically satisfied, a configurational invariant if $J = \Psi (x)$).

(b) If this relation is valid only on the level set $J = c$, we say that $J$ is a conditional invariant (a conditional phase space invariant if $J = \Psi (x, t)$, a conditional configurational invariant if $J = \Psi (x)$) for the Ito equation.

Lemma 7.5. Phase space invariants are identified by the $n+1$ equations

Proof. This follows at once from (77). $\diamondsuit$

Remark 7.7. If the solutions to the (78) identify one or more level sets of $J = \Psi$ (in practice, if the l.h.s. of the above equations can be factorized with a factor being just a function of $\Psi$), then this or these level sets correspond to invariant manifolds for the dynamics. We will see in concrete Examples that this may happen. $\odot$

Remark 7.8. As configurational invariants (and functions) are special cases of phase space invariants (and functions), the Lemmas 7.4 and 7.5 above also apply to them. Note that the Eq (78) read in this case

where we have taken into account the fact that configurational invariants naturally arise in autonomous systems. $\odot$

7.3.   Asymptotic symmetries

We have so far discussed the case where an invariant submanifold $M_0 {\subset} M$ exists, and considered dynamics on it. We have not discussed neither what happens nearby the invariant manifold, nor the stability of such manifold.

As we are considering stochastic systems, we should consider at least two types of (local or global) stability ^[85,86].

(A) On the one hand, it is possible that for any initial datum $x_0$ in a neighborhood $\mathcal{B}_0 \supset M_0$ and any realization of the Wiener processes $w^i (t)$, the dynamics is attracted towards $M_0$.

(B) On the other hand, it is possible that for any $x_0 \in \mathcal{B}_0$, considering the Fokker-Planck equation with initial condition $u (x, 0) = \delta (x - x_0)$ the probability density function $u(x, t)$ remains concentrated around $M_0$.

Definition 7.3. When the situation described in case (A) above occur, we say that $M_0$ is locally strongly attractive; if this holds for any $x_0 \in M$, we say it is globally strongly attractive.

Definition 7.4. When the situation described in case (B) above occur, we say that $M_0$ is locally weakly attractive; if this holds for any $x_0 \in M$, we say it is globally weakly attractive.

Remark 7.9. Note that in case (B) above typically there will be a parameter ${\sigma}_0 \ge 0$ – related in physical terms to the intensity of the noise described by the Wiener processes – which controls the spreading of the asymptotic measure around $M_0$, and such that in the limit ${\sigma}_0 \to 0$ the asymptotic measure $u_* (x) = \lim_{t \to \infty} u(x, t)$ collapses to a measure on $M_0$. $\odot$

Lemma 7.6. In the case (A), the most general asymptotic solution to ${\Delta}$ will be described by the most general solution to ${\Delta}_0$.

Proof. In fact, under these hypotheses any dynamics starting in $\mathcal{B}$ will be attracted to the invariant manifold $M_0$, and on this the dynamics is described by ${\Delta}_0$. The conclusion follows then by Lemma 7.1 above. $\diamondsuit$

Remark 7.10. Needless to say, even in the case ${\Delta}_0$ is fully integrable we will in general not be able to describe which ones of the solutions to ${\Delta}_0$ will describe the asymptotic behavior for a given solution to ${\Delta}$, i.e., for given initial datum $x_0 \in \mathcal{B}$ and given realization of the Wiener processes, as we will not be able to integrate the transient dynamics. $\odot$

Remark 7.11. The Remark above refers to case (A). In case (B), on the other hand, the dynamics can not be reduced, even asymptotically, exactly to the dynamics on $M_0$ and hence to ${\Delta}_0$. Thus even in the case where ${\Delta}_0$ is exactly integrable – and even for $x_0 \in \mathcal{B}_0$ – this does not give a description of the asymptotic dynamics for ${\Delta}$. In this case we have a weaker result. $\odot$

Lemma 7.7. In the case (B), for small enough ${\sigma}_0$, the asymptotic dynamics will be described for most of the time, by the linearization $\widetilde {{\Delta}}_0$ of ${\Delta}$ around $M_0$.

Proof. Under the hypotheses holding in case (B) and for small enough ${\sigma}_0$, the measure will be asymptotically concentrated around $M_0$, so that the linearization $\widetilde {{\Delta}}_0$ (which, in turn, will be a linearization around ${\Delta}_0$) will describe "most" of the asymptotic solutions; by this we mean that there could be from time to time large fluctuations driving the dynamics for a short time outside of the immediate neighborhood of $M_0$, i.e., outside the region where ${\Delta}$ reduces to $\widetilde {{\Delta}}_0$. These large fluctuations will occur with a frequency exponentially small in ${\sigma}_0$ and will decay rapidly so that the dynamics is quickly driven back to the region around $M_0$. $\diamondsuit$

Remark 7.12. This Lemma does not give a result of the same strength as the previous one; on the other hand, it says that one will have a reliable description in terms of $\widetilde {{\Delta}}_0$, except for rare and short fluctuations. $\odot$

As far as symmetries are concerned, our discussion suggests naturally the following definitions.

Definition 7.5. Let $M_0 {\subset} M$ be a strongly attractive submanifold for the equation ${\Delta}$, and let ${\Delta}_0$ be the restriction of ${\Delta}$ to $M_0$; let $X$ be a symmetry for ${\Delta}_0$, hence a conditional symmetry for ${\Delta}$. We say that $X$ is also a strong asymptotic symmetry for ${\Delta}$.

Definition 7.6. Let $M_0 {\subset} M$ be a weakly attractive submanifold for the equation ${\Delta}$, let ${\Delta}_0$ be the restriction of ${\Delta}$ to $M_0$, and let $\widetilde {{\Delta}}_0$ be the linearization of ${\Delta}$ around $M_0$; let $X$ be a symmetry for ${\Delta}_0$, hence a conditional symmetry for ${\Delta}$. If $X$ is also a symmetry for $\widetilde {{\Delta}}_0$, then we say that $X$ is also a weak asymptotic symmetry for ${\Delta}$.

Both these cases are related to the presence of asymptotic conditional invariants, as we are now going to discuss in the next Section 8.

We stress that we are again interested in the case of phase space or configurational invariants, i.e., where the would-be invariants depend on $(x, t)$ or on the $x$ variables alone; thus we are in the setting discussed above, and the evolution is described by (77).

8.   Asymptotic invariants and asymptotic symmetries for stochastic equations

The discussion of Section 7.2, referring to conditional symmetries, can be extended to the case of asymptotic symmetries.

The key observation is that the evolution of a phase space function under the dynamics of an Ito Eq (35) is described by (77), i.e., by an equation of Ito type; this will be denoted, for ease of reference, as ${\Delta}_J$. So our discussion of (stochastic) stability for submanifolds $M_0$ of $M$ under ${\Delta}$ can be repeated for the stability under ${\Delta}_J$.

Definition 8.1. If $J = \Psi (x)$ is a conditional configurational invariant for the Ito equation ${\Delta}$ on the invariant manifold $M_0$, and $M_0$ is a strongly attractive manifold for ${\Delta}$, we say that $J$ is a strong asymptotic invariant for ${\Delta}$.

Definition 8.2. If $J = \Psi (x)$ is a conditional configurational invariant for the Ito equation ${\Delta}$ on the invariant manifold $M_0$, and $M_0$ is a weakly attractive manifold for ${\Delta}$, se say that $J$ is a weak asymptotic invariant for ${\Delta}$.

Lemma 8.1. Let $J$ be a strongly asymptotic configurational invariant for the Ito equation ${\Delta}$, associated to the invariant manifold $M_0$, and let ${\Delta}_0$ be the reduction of ${\Delta}$ to $M_0$. Then the asymptotic solutions to ${\Delta}$ are described by the solutions to ${\Delta}_0$.

Proof. By definition, under the hypotheses of the Lemma $M_0$ is a strongly attractive invariant manifold for ${\Delta}$, hence the statement follows immediately from Lemma 7.6, which itself uses Lemma 7.1. $\diamondsuit$

Lemma 8.2. Let $J$ be a weakly asymptotic configurational invariant for the Ito equation ${\Delta}$, associated to the invariant manifold $M_0$, and let $\widetilde {{\Delta}}_0$ be the linearization of ${\Delta}$ around $M_0$. Then the asymptotic solutions to ${\Delta}$ are described by the solutions to $\widetilde {{\Delta}}_0$, up to large fluctuations.

Proof. By definition, under the hypotheses of the Lemma $M_0$ is a weakly attractive manifold for ${\Delta}$. Thus, according to Lemma 7.7, the dynamics of ${\Delta}$ lives, for most of the time and up to rare and short-lived large fluctuations, in a neighborhood of $M_0$; thus it is described by $\widetilde {{\Delta}}_0$, and the Lemma is just a restatement of these facts in terms of (asymptotic) invariants. $\diamondsuit$

Remark 8.1. The approach discussed in this subsection is strongly related to stochastic Lyapounov functions, see e.g., ^[85,86]. Some of our ideas are also related to ideas developed in the context of bifurcation for stochastic systems ^[87]. $\odot$

Remark 8.2. Our discussion used configurational invariants to relate invariants to attractive invariant submanifolds. As stressed above, see Remark 7.6, configurational or phase space (asymptotic) invariants can only do this, i.e., they cannot be used to obtain solutions to Ito equations, at difference with general (asymptotic) invariants, i.e., invariants depending also on the $w$ variables. See also the discussion in Sect. 6 for the case of full invariants. $\odot$

9.   Examples

We will now consider concrete Examples related to the matters discussed in Sections 7 and 8 above. These matters are of course strongly interconnected, but we have separated the Examples in different types – and correspondingly separated the present Section in different subsections – for ease of reading.

In most cases, our Examples are built so that the compact invariant manifold is just a circle (usually the unit one), as we want to focus on conceptual issues rather than on computational difficulties. Thus it is not surprising that things will be simpler in polar or spherical coordinates, and indeed we will mostly work in polar coordinates, albeit in some cases we will also start from Cartesian ones.

9.1.   Full symmetry, full invariants

We start by considering an Example due to Misawa ^[79]; this is Example 1 in there, where it is considered in the equivalent Stratonovich form, and only up to checking $J$ is an invariant.

Example 9.1. Consider the Ito equation

(note that only one Wiener process is involved). In this case the function

is an invariant. In fact one has

and then $d J = 0$ on the Eq (80) follows immediately from a direct computation. Thus any sphere of radius $\rho$ is invariant under the evolution described by (80).

It is also easily checked, by the same procedure or by just looking at (80), that

is another invariant.

By considering the Jacobian matrix for the change of coordinates $(x_1, x_2, x_3) \to (J, H, z)$ with $z = x_3$, we observe that its determinant is $2 (x_1 - x_2)$, i.e., this is singular on the plane $x_1 = x_2$; we will thus have to consider two regions.

Implementing the change of variables $(x_1, x_2, x_3) \to (J, H, z)$ we have that the inverse change is given (in the two regions $x_1 < x_2$ and $x_1 > x_2$) by

Note that by elementary algebra, and in view of the definitions of $J, H$, and $z$, we have

thus the argument of the square root in the formula above is always positive. We introduce, for ease of writing, the new real function

so that the above inverse change of variables reads

On the one-dimensional manifolds (corresponding to the $\pm$ sign in the formulas above) identified by given values for $J = \rho^2$ and $H$ we have the reduced equation

Note that here $z \in [-\rho, \rho]$ (recall $J = \rho^2$), and that for $z = \pm \rho$ we have $J = z^2$, $H = z$ and $x_1 = x_2 = 0$.

We can further look for symmetries of the reduced equation. We disregard W-symmetries, i.e., set $R = 0$ in (44)^||; with straightforward computations we get

^||The case with $R\not = 0$ can also be analyzed completely, yielding of course more complex formulas.

where the $\pm$ refers to the two determinations given above for $x_1$ and $x_2$ in terms of $z$, $\psi_\pm$ are arbitrary function of their argument, and

The simplest case does of course correspond to $\psi_\pm (u_\pm) = c_\pm$, say

for ease of discussion (note the two $\pm$ signs are independent), i.e., to^**

^**This notation might be confusing: a more precise notation would be ${\varphi}_\pm = s_\pm \chi$, with $s_\pm$ a sign.

With this, the integrating change of variables is given by

With a direct application of Ito rule, the evolution equation for the random variable $\xi$ turns out to be

This is immediately integrated, yielding

Going back to the $z$ variable we obtain (note that $3 J - H^2 \ge 0$)

Having obtained $z(t)$, i.e., $x_3 (t)$, now $x_1 (t)$ and $x_2 (t)$ are also obtained via (83). $\diamondsuit$

9.2.   Full or conditional symmetries, conditional invariants

We now consider several examples with conditional invariants; at this stage we will not yet look at asymptotic symmetries or invariants.

Example 9.2. Let us work in $R^2$ with polar coordinates $(\rho, \vartheta)$, and consider the Ito equations

with $a, b, {\omega}, {\sigma}$ nonzero real constants.

It is clear that the unit circle $\rho = 1$ is an invariant manifold for these equations; the dynamics on the unit circle is described by

i.e., by a uniform rotation with a superimposed Wiener process. This yields immediately

Thus albeit we have no information about the dynamics for generic initial conditions, we have a full description of the dynamics with initial condition on the unit circle.

The discussion is immediately generalized – in its entirety – to equations of the form

with ${\alpha}, {\beta}, {\sigma}_i$ smooth functions and

for some $\rho_0$.

If we drop the last condition, i.e., require only ${\alpha} (\rho_0, \vartheta) = 0 = {\sigma}_1 (\rho_0, \vartheta)$, we still have reduction to a one dimensional equation on the circle of radius $\rho_0$, albeit we have no information on the solutions living on this circle.

Note that in this case the full equations are rotationally invariant, as immediately seen from (86); but the associated function $J = \rho$ is a conditional invariant. $\diamondsuit$

Example 9.3. Consider $R^3$ with cylindrical coordinates $(\rho, \vartheta, z)$ and the Ito equations

with $\{ a, b, c, \kappa, {\omega} \}$ nonzero real constants and $\{{\gamma}, {\sigma}_1, {\sigma}_2 \}$ smooth functions. By construction, the cylinder $M_0$ of equation $\rho = 1$ is an invariant submanifold (the function $J_1 = \rho$ is a conditional invariant), and on it the system reduces to

Suppose now that in general $\{{\gamma}, {\sigma}_1, {\sigma}_2\}$ depend effectively on $\vartheta$, but that

Then the restricted system admits rotations in $\vartheta$ as symmetries, and hence the system can be further reduced to an equation in $z$. Albeit we are in general not able to solve this (one-dimensional) equation, we know that given a solution we can reconstruct a solution of the two-dimensional restricted system, and this is a special solution to the full three-dimensional problem.

Similarly it is easy to produce examples in which not only the cylinder $\rho = 1$ is invariant, but within this we have the circle $S^1_0 = \{ \rho = 1 \, \ z = 0 \}$ which is also invariant. In this case the equation is fully integrated for initial data on $S^1_0$, partially integrated for initial data on the cylinder $\rho = 1$, and in general nothing can be said for general initial data (but Eq (87) show that an asymptotic analysis will be possible). $\diamondsuit$

Example 9.4. In the previous examples, everything was rather obvious by construction, thanks to the use of polar coordinates and the fact the invariant manifold was just a circle of radius one. Things are slightly less evident if we use Cartesian coordinates, and we want to show now that one can indeed work also with a less "ready to use" setting.

Consider, with $r^2 = x_1^2 + x_2^2$ for short, the equation

We consider now the function $J = r^2$, and look at its variation on the dynamics (88). This yields

Thus we obtain that $J$ is not an invariant; but it is a conditional invariant: on the level sets $J = 0$ (corresponding to the origin in the Cartesian plane) and $J = 1$ (the unit circle in the Cartesian plane) we have $d J = 0$, and hence these are invariant manifolds for the dynamics (88).

Actually, passing to polar coordinates $(\rho, \theta)$ (on $\rho = 0$ the change of coordinates is singular, and $\theta$ ill-defined) the Eq (88) reads simply

It is thus evident that on $J = \rho^2 = 1$ the dynamics reduces to

It should be noted that, as immediately apparent from (90), in this case rotations in the $(x_1, x_2)$ plane are a symmetry of the full equation, and not only of the reduced one. We thus see again that one can have a conditional invariant associated to a full symmetry. $\diamondsuit$

Example 9.5. We generalize the previous Example; we use Cartesian coordinates $(x, y)$ and write again $r = \sqrt{x^2 + y^2}$. Consider the equations

Here $f_i = f_i (x, y)$ and $s_{ij} = s_{ij} (x, y)$ are arbitrary smooth functions, and of course the system is written in this rather involved way so that it produces a simple result.

We consider again the function $J = r^2$; it results that on solutions to (91) we have

It is thus immediately apparent that $J$ is in general not an invariant for (91), but that it is always a conditional invariant on the circle $r = 1$. Thus the latter is an invariant manifold for all the equations of the form (91).

In fact, passing to polar coordinates $(r, \theta)$, and writing

the Eq (91) reads simply

Now, unless all the functions $F_i$ and $S_{ij}$ are actually independent of $\theta$, rotations are not a symmetry of the full Eq (92), and hence of the original Eq (91).

On the other hand, the equation reduced to the invariant manifold $r = 1$ reads just

(note this is a deterministic equation), and if $F_2 (1, \theta)$ is independent of $\theta$, it has a rotational symmetry. On the other hand, if $F_2 (1, \theta)$ does actually depend on $\theta$, we do not have a rotational conditional symmetry, despite the presence of the conditional invariant $J$ on $r = 1$. $\diamondsuit$

9.3.   Asymptotic symmetries

The reader has surely noted that, albeit we only discussed conditional symmetries and/or invariants in our previous Examples, these are actually tailored so to have asymptotic symmetries and/or invariants. We will look again at them under this point of view, but we will first consider an even simpler one dimensional Example.

Example 9.6. Consider the one-dimensional equation

in this case the invariant manifold is just the single point $x = 0$; this is of course of little interest in itself.

This equation does not have continuous symmetries (an obvious reflection symmetry $(x, t, w) \to (- x, t, w)$ is present), but as the dynamics is attracted towards $x = 0$ we know that asymptotically the system dynamics will be described by its linearization

This has a scaling symmetry,

thus we can just use the prescription discussed in Sect. 5 and pass to the new variable

Note here that we should actually write $y = \log (|x|)$ and distinguish the cases $x > 0$ and $x < 0$. This is just fine as $x = 0$ is a barrier for this dynamics. We will then use $x > 0$ (so that in particular ${\partial} y / {\partial} x = 1/x$), and $x = \exp[y]$. A similar discussion would apply for $x < 0$.

If we apply this transformation to the full equation, we get

Now we should observe that we are interested in the region $x \approx 0$, i.e., $y \approx - \infty$; in this limit the term $e^{2 y}$ can be discarded. Actually, the $y$ dynamics will drive the system towards this limit. Thus in this asymptotic regime the dynamics is just governed by

it is clear that this equation is readily integrated to

and also that it admits ${\partial}_y$ as a symmetry. $\diamondsuit$

Example 9.7. Consider again the system of Example 9.2, i.e., (86), which we rewrite here for ease of reference:

The equation for $\rho$ is autonomous, and we readily note that $\rho = 1$ is an invariant manifold, and actually for $a > 0$ a weakly attractive one.

It appears from (97) that – at least for $|{\sigma}| \ll 1$, i.e. when the drift overcomes the diffusion term – the dynamics converges to the circle $\rho = 1$. Note this will be a weakly attracting manifold. We will thus linearize (97) around $\rho = 1$; we do this by writing

and taking only first order terms in $\eta$. This yields

The dynamics is attracted to $\eta = 0$ (corresponding to $\rho = 1$) and on this it is described by $d \vartheta = {\omega} dt + {\sigma} dw_2$, while in the region around the attracting manifold it is described by (98).

The original Eq (97) is rotationally invariant; this symmetry is also present in the reduced Eq (98).

Note also that the first equation in (98) admits $X = \eta {\partial}_\eta$ as a (scaling) symmetry; thus it can be integrated (passing to the variable $y = \log (\rho) = \log (1 + \eta)$), which yields $\eta = \eta (t)$ as an explicit function, depending on the realization of the Wiener process $w_1 (t)$. With this, the second equation in (98) is also readily integrated.

It may be interesting to see what happens if we proceed in a slightly different way. We can perform the change of variables $y = \log (\rho)$ directly on the original equation (97); this yields in mixed notation (obviously we have a singularity at $\rho = 0$) for the first equation

Passing now to write $\rho = 1 + \eta$ as above, the full equation reads

expanding in series at first order in $\eta$, we get $y = \log (1 + \eta) \simeq \eta$, hence $dy \simeq d \eta$, and

where $|{\sigma} | \ll 1$ has been taken into account, exactly as before. $\diamondsuit$

Example 9.8. Consider the equation

Here $a, b$ are real constants, ${\alpha}, {\beta}, {\omega}$ are arbitrary smooth functions of $x_1, x_2$. We assume $a > 0$ for definiteness.

In this case the evolution of $J = r^2$ is given by

note this is a deterministic equation, so that any initial datum with $r \not = 0$ is attracted by the circle $r = 1$. In this case, we have an invariant and strongly attractive submanifold. Correspondingly, $J$ is a strongly asymptotic invariant.

Passing to polar coordinates, we have

Needless to say, the functions ${\alpha}, {\beta}, {\omega}$ should now be thought as functions of $(r, \theta)$; we will denote by $\widehat{\alpha}, \widehat{\beta}, \widehat{\omega}$ the restrictions of these to the circle $r = 1$. On the invariant strongly attractive submanifold $r = 1$ the evolution is described by

It is rather clear that choosing suitably the functions ${\alpha}, {\beta}, {\omega}$ we can have different situations:

● If these do not depend on $\theta$, the full Eq (99) is rotationally invariant, i.e., $X = - x_2 {\partial}_1 + x_1 {\partial}_2$ is a full symmetry of (99);

● If these do depend on $\theta$, but their restrictions $\widehat{\alpha}, \widehat{\beta}, \widehat{\omega}$ do not depend on $\theta$, then (99) does not admit $X$ as a full symmetry, but admits it as a conditional symmetry on the invariant (and attractive) manifold $M_0$ identified by $r = 1$;

● If (at least one of) the $\widehat{\alpha}, \widehat{\beta}, \widehat{\omega}$ depend on $\theta$, then (99) does not even admit $X$ as a conditional symmetry.

It is maybe worth stressing again that in all these cases $J$ is a strongly asymptotic invariant. $\diamondsuit$

10.   Discussion and conclusions

In many physically interesting situations, one observes that equations not enjoying a given symmetry can have special solutions which are symmetric. Even more frequently, the asymptotic behavior of the system under study is characterized in terms of symmetry, independently of the symmetry of initial conditions; typically this involves a simple, i.e., a rotational or a scaling symmetry. A theory of conditional and asymptotic symmetries for deterministic equations has been developed in the literature.

In the present paper, we have extended this theory to the case of stochastic differential equations, considering also the interplay between (asymptotic) symmetry and (asymptotic) integrability for stochastic equations. A special role in this frame is played by (full or conditional or asymptotic) invariants for stochastic equations. We have also shown by concrete (simple) examples how the theory developed here can be applied in practice.

Our examples show that the interrelation between the presence of (conditional and/or asymptotic) symmetry and invariants is not unique: the presence of conditional of asymptotic configurational invariants is needed to have a conditional or asymptotic symmetries, but the converse is not true.

Our discussion shows that the pre-existing theory for deterministic equations can be fully extended to stochastic ones, and the examples show that it can be applied with a well definite algorithm and standard computations; these were specially simple in the considered examples – which were built precisely in order to show the conceptual issues without blurring them by computational details – but our discussion shows that even in more involved cases (apart from computational complications) the procedure is rather straightforward.

Needless to say, its success depends on the equations under study possessing the required (full, conditional or asymptotic) symmetry properties. Symmetry is a non-generic property; but conditional or even more asymptotic symmetry is more common than full symmetry, so we are confident that our method can be widely applicable, as it already happens in the context of deterministic equations.

Our approach was focusing – again, as in the deterministic case – on continuous (Lie) symmetries. At difference with the deterministic case, we have dealt only with ordinary stochastic differential equations, and not discussed partial stochastic differential equations. The reason for this is that a symmetry theory for these equations is not as well developed as for ordinary ones; we expect that once this (substantial) obstacle is removed, the theory can be extended from full to conditional and asymptotic symmetries as in this work. In view of the relevance of conditional and asymptotic symmetries for the study of deterministic PDEs, we expect this extension to stochastic PDEs to be specially relevant for applications, and we hope it may soon be obtained.

Finally, we would like to note that in a way the approach pursued in the literature dealing with conditional or asymptotic symmetries is inverse to that of the well known symmetry breaking approach. In that case, attention is focused on the fact that equations possessing a given symmetry $G$ can have solutions which are invariant only under a subgroup $G_0 {\subset} G$ (possibly with $G_0 = \{ I \}$); here instead one is focusing on the fact that an equation possessing a symmetry $G_0$ (possibly with $G_0 = \{ I \}$) can have solutions with symmetry $G \supset G_0$. Similarly, for an equation with symmetry $G_0$ it may happen that the asymptotic dynamics is described by an effective equation with symmetry $G \supset G_0$, and this is turn may have solutions with symmetry $G$ or even $\widetilde {G} \supset G \supset G_0$.

Acknowledgements

The work of GG is also supported by SMRI, which he was visiting while the first draft of this paper was prepared and again at the time of final writing. The work of FS is supported by ERC SG CONSTAMIS grant. GG and FS are also members of GNFM-INdAM. We thank these Institutions for their support.

Conflict of interest

The authors declare no conflict of interest.