In this work, by stochastic analyses, we study stochastic representation, well-posedness, and regularity of generalized time fractional Schrödinger equation
{∂wtu(t,x)=Lu(t,x)−κ(x)u(t,x),t∈(0,∞),x∈X,u(0,x)=g(x),x∈X,
where the potential κ is signed, X is a Lusin space, ∂wt is a generalized time fractional derivative, and L is infinitesimal generator in terms of semigroup induced by a symmetric Markov process X. Our results are applicable to some typical physical models.
Citation: Rui Sun, Weihua Deng. A generalized time fractional Schrödinger equation with signed potential[J]. Communications in Analysis and Mechanics, 2024, 16(2): 262-277. doi: 10.3934/cam.2024012
[1] | Ekaterina Kldiashvili, Archil Burduli, Gocha Ghortlishvili . Application of Digital Imaging for Cytopathology under Conditions of Georgia. AIMS Medical Science, 2015, 2(3): 186-199. doi: 10.3934/medsci.2015.3.186 |
[2] | Tamás Micsik, Göran Elmberger, Anders Mikael Bergquist, László Fónyad . Experiences with an International Digital Slide Based Telepathology System for Routine Sign-out between Sweden and Hungary. AIMS Medical Science, 2015, 2(2): 79-89. doi: 10.3934/medsci.2015.2.79 |
[3] | Sanjay Konakondla, Steven A. Toms . Cerebral Connectivity and High-grade Gliomas: Evolving Concepts of Eloquent Brain in Surgery for Glioma. AIMS Medical Science, 2017, 4(1): 52-70. doi: 10.3934/medsci.2017.1.52 |
[4] | Joy Qi En Chia, Li Lian Wong, Kevin Yi-Lwern Yap . Quality evaluation of digital voice assistants for diabetes management. AIMS Medical Science, 2023, 10(1): 80-106. doi: 10.3934/medsci.2023008 |
[5] | Adel Razek . Augmented therapeutic tutoring in diligent image-assisted robotic interventions. AIMS Medical Science, 2024, 11(2): 210-219. doi: 10.3934/medsci.2024016 |
[6] | Belgüzar Kara, Elif Gökçe Tenekeci, Şeref Demirkaya . Factors Associated With Sleep Quality in Patients With Multiple Sclerosis. AIMS Medical Science, 2016, 3(2): 203-212. doi: 10.3934/medsci.2016.2.203 |
[7] | Jonathan Kissi, Daniel Kwame Kwansah Quansah, Jonathan Aseye Nutakor, Alex Boadi Dankyi, Yvette Adu-Gyamfi . Telehealth during COVID-19 pandemic era: a systematic review. AIMS Medical Science, 2022, 9(1): 81-97. doi: 10.3934/medsci.2022008 |
[8] | Vanessa Kai Lin Chua, Li Lian Wong, Kevin Yi-Lwern Yap . Quality evaluation of digital voice assistants for the management of mental health conditions. AIMS Medical Science, 2022, 9(4): 512-530. doi: 10.3934/medsci.2022028 |
[9] | Van Tuan Nguyen, Anh Tuan Tran, Nguyen Quyen Le, Thi Huong Nguyen . The features of computed tomography and digital subtraction angiography images of ruptured cerebral arteriovenous malformation. AIMS Medical Science, 2021, 8(2): 105-115. doi: 10.3934/medsci.2021011 |
[10] | Ilige Hage, Ramsey Hamade . Automatic Detection of Cortical Bones Haversian Osteonal Boundaries. AIMS Medical Science, 2015, 2(4): 328-346. doi: 10.3934/medsci.2015.4.328 |
In this work, by stochastic analyses, we study stochastic representation, well-posedness, and regularity of generalized time fractional Schrödinger equation
{∂wtu(t,x)=Lu(t,x)−κ(x)u(t,x),t∈(0,∞),x∈X,u(0,x)=g(x),x∈X,
where the potential κ is signed, X is a Lusin space, ∂wt is a generalized time fractional derivative, and L is infinitesimal generator in terms of semigroup induced by a symmetric Markov process X. Our results are applicable to some typical physical models.
Diophantine equation is a classical problem in number theory. Let [α] denote the integer part of the real number α and N be a sufficiently large integer. In 1933, Segal [27,28] firstly studied additive problems with non-integer degrees, and proved that there exists a k0(c)>0 such that the Diophantine equation
[xc1]+[xc2]+⋯+[xck]=N | (1.1) |
is solvable for k>k0(c), where c>1 is not an integer. Later, Deshouillers [5] improved Segal's bound of k0(c) to 6c3(logc+14) with c>12. Further Arkhilov and Zhitkov [1] refined Deshouillers's result to 22c2(logc+4) with c>12. Afterwards, many results of various Diophantine equations were established (e.g., see [7,10,14,16,17,18,19,21,25,41,42]). In particular, Laporta [17] in 1999 showed that the equation
[pc1]+[pc2]=N | (1.2) |
is solvable in primes p1, p2 provided that 1<c<1716 and N is sufficiently large. Recently, the range of c in (1.2) was enlarged to 1<c<1411 by Zhu [40]. Kumchev [15] showed that the equation
[mc]+[pc]=N | (1.3) |
is solved for almost all N provided that 1<c<1615, where m is an integer and p is a prime. Afterwards, the range of c in (1.3) was enlarged to 1<c<1711 by Balanzario, Garaev and Zuazua [3].
In 1995, Laporta and Tolev [18] considered the equation
[pc1]+[pc2]+[pc3]=N | (1.4) |
with prime variables p1,p2,p3. Denote the weighted number of solutions of Eq (1.4) by
R(N)=∑[pc1]+[pc2]+[pc3]=N(logp1)(logp2)(logp3). | (1.5) |
They established the following asymptotic formula
R(N)=Γ3(1+1c)Γ(3c)N3c−1+O(N3c−1exp(−log13−δN)) |
for any 0<δ<13 and 1<c<1716. Afterwards, the range of c was enlarged to 1<c<1211 by Kumchev and Nedeva [16], to 1<c<258235 by Zhai and Cao [39], and to 1<c<137119 by Cai [4].
In this paper, we first show a more general result related to (1.5) by proving the following theorem.
Theorem 1.1. Let N be a sufficiently large integer. Then for 1<c<3+3κ−λ3κ+2, we have
R(N)=Γ3(1+1c)Γ(3c)N3c−1+O(N3c−1exp(−log13−δN)) | (1.6) |
for any 0<δ<13, where (κ,λ) is an exponent pair, and the implied constant in the O−symbol depends only on c.
Choosing (κ,λ)=BA2BABABABAB(0,1)=(81242,132242) in Theorem 1.1, we can immediately get the following corollary, which further improves the result of Cai [4].
Corollary 1.2. Under the notations of Theorem 1.1, for 1<c<837727 the asymptotic formula (1.6) follows.
It is easy to verify that the range of c in Corollary 1.2 is larger than one of Cai's result. Our improvement mainly derives from more accurate estimates of exponential sums by combining Van der Corput's method, exponent pairs and some elementary methods. Also the estimates of exponential sums has lots of applications in problems including automorphic forms (e.g., see [8,11,12,20,22,24,29,30,31,32,33,34,35,36,37,38]).
Notation. Throughout the paper, N always denotes a sufficiently large integer. The letter p, with or without subscripts, is always reserved for primes. Let ε∈(0,10−10(3+3κ−λ3κ+2−c)). We denote by {x} and ‖x‖ the fraction part of x and the distance from x to the nearest integer, respectively. Let 1<c<3+3κ−λ3κ+2 and
P=N1c, τ=P1−c−ε, e(x)=e2πix, S(α)=∑p≤P(logp)e(α[pc]). |
To prove Theorem 1.1, we need the following lemmas.
Lemma 2.1 ([9,Lemma 5]). Suppose that zn is a sequence of complex numbers, then we have
|∑N≤n≤2Nzn|2≤(1+NQ)Q∑q=0(1−qQ)Re(∑N≤n≤2N−q¯znzn+q), |
where Re(t) and ¯t denote the real part and the conjugate of the complex number t, respectively.
Lemma 2.2. Suppose that |x|>0 and c>1. Then for any exponent pair (κ,λ) and M≤a<b≤2M, we have
∑a≤n≤be(xnc)≪(|x|Mc)κMλ−κ+M1−c|x|. |
Proof. We can get this lemma from [6,(3.3.4)].
Lemma 2.3 ([2,Lemma 12]). Suppose that t is not an integer and H≥3. Then for any α∈(0,1), we have
e(−α{t})=∑|h|≤Hch(α)e(ht)+O(min(1,1H‖t‖)), |
where
ch(α)=1−e(−α)2πi(h+α). |
Lemma 2.4 ([9,Lemma 3]). Suppose that 3<U<V<Z<X, and {Z}=12, X≥64Z2U, Z≥4U2, V3≥32X. Further suppose that F(n) is a complex valued function such that |F(n)|≤1. Then the sum
∑X≤n≤2XΛ(n)F(n) |
can be decomposed into O(log10X) sums, each of which either of type {I}:
∑M≤m≤2Ma(m)∑N≤n≤2NF(mn) |
with N>Z, where a(m)≪mε and X≪MN≪X, or of type {II}:
∑M≤m≤2Ma(m)∑N≤n≤2Nb(n)F(mn) |
with U≪M≪V, where a(m)≪mε,b(n)≪nε and X≪MN≪X.
Lemma 2.5. Let f(t) be a real value function and continuous differentiable at least three times on [a,b](1≤a<b≤2a), |f‴(x)|∼Δ>0, then we have
∑a<n≤be(f(n))≪aΔ16+Δ−13. |
Moreover, if 0<c1λ1≤c2λ1, |f″(x)|∼λ1a−1, then we have
∑a<n≤be(f(n))≪a12λ121+λ−11; |
if c2λ1≤12, then we have
∑a<n≤be(f(n))≪λ−11. |
Proof. The first result was proved by Sargos [26]. And the remaining two results were due to Jia [13].
Lemma 2.6 ([23,Lemma 2]). Let M>0, N>0, um>0, υn>0, Am>0, Bn>0 (1≤m≤M,1≤n≤N). Let also Q1 and Q2 be given non-negative numbers, Q1≤Q2. Then there is one q such that Q1≤q≤Q2 and
M∑m=1Amqum+N∑n=1Bnq−υn≪M∑m=1N∑n=1(AυnmBumn)1um+υn+M∑m=1AmQum1+N∑n=1BnQ−υn2. |
Lemma 2.7 ([36,Lemma 5]). Let f(x), g(x) be algebraic functions in [a,b], |f″(x)|∼1R, f‴(x)≪1RU, U≥1, |g(x)|≪G, |g′(x)|≪GU−1. [α,β] is the image of [a,b] under the mapping y=f′(x). nu is the solution of f′(n)=u.
bu={1,α<u<β,12,u=α∈Noru=β∈N. |
Then we have
∑a<n≤bg(n)e(f(n))=∑α<u≤βbug(nu)√|f″(nu)|e(f(nu)−unu+18)+O(Glog(β−α+2)+G(b−a+R)u−1)+O(Gmin(√R,1‖α‖)+Gmin(√R,1‖β‖)). |
Lemma 2.8 ([13,Lemma 3]). Suppose that x∼N, f(x)≪P, and f′(x)≫Δ. Then we have
∑n∼Nmin(D,1‖f(n)‖)≪(P+1)(D+Δ−1)log(2+Δ−1). |
Lemma 2.9. For 0<α<1 and any exponent pair (κ,λ), we have
T(α,X)=∑X<n≤2Xe(α[nc])≪Xκc+λ1+κlogX+XαXc. |
Proof. Throughout the proof of this lemma, we write H=X−κc+1−λ+κ1+κ for convenience. Using Lemma 2.3 we can get
T(α,X)=∑|h|≤Hch(α)∑X<n≤2Xe((h+α)nc)+O((logX)∑X<n≤2Xmin(1,1H||nc||)). |
Then by the expansion
min(1,1H||θ||)=∞∑h=−∞ahe(hθ), |
where
|ah|=min(log2HH, 1|h|, Hh2), |
we have
∑X<n≤2Xmin(1,1H||nc||)≤∞∑h=−∞|ah||∑X<n≤2Xe(hnc)|≪Xlog2HH+∑1≤h≤H1h((hXc)κXλ−κ+XhXc)+∑h≥HHh2((hXc)κXλ−κ+XhXc)≪Xκc+λ1+κlogX, |
where we estimated the sum over n by Lemma 2.2.
In a similar way, we have
∑|h|≤Hch(α)∑X<n≤2Xe((h+α)nc)=c0(α)∑X<n≤2Xe(αnc)+∑1≤h≤Hch(α)∑X<n≤2Xe((h+α)nc)≪Xκc+λ1+κlogX+XαXc. |
Then this lemma follows.
Lemma 2.10 ([42,Lemma 2.1]). Suppose that f(n) is a real-valued function in the interval [N,N1], where 2≤N<N1≤2N. If 0<c1λ1≤|f′(n)|≤c2λ1≤12, then we have
∑N<n≤N1e(f(n))≪λ−11. |
If |f(j)(n)|∼λ1N−j+1(j=1,2), then we have
∑N<n≤N1e(f(n))≪λ−11+N12λ121. |
If |f(j)(n)|∼λ1N−j+1(j=1,2,3,4,5,6), then we have
∑N<n≤N1e(f(n))≪λ−11+Nλλκ1, |
where (κ,λ) is any exponent pair.
Lemma 2.11 ([9,Lemma 6]). Suppose that 0<a<b≤2a and R is an open convex set in C containing the real segment [a,b]. Suppose further that f(z) is analytic on R. f(x) is real for real x∈R. f″(z)≤M for z∈R. There is a constant k>0 such that f″(x)≤−kM for all real x∈R. Let f′(b)=α and f′(a)=β, and define xυ for each integer υ in the range α<υ<β by f′(xυ)=υ. Then we have
∑a<n≤be(f(n))=e(−18)∑α<υ≤β|f″(xυ)|−12e(f(xυ)−υxυ)+O(M−12+log(2+M(b−a))). |
Lemma 3.1. Let P56≪X≪P, H=X1−(1+2κ)c+λ2+2κ and ch(α) denote complex numbers such that ch(α)≪(1+|h|)−1. Then uniformly for α∈(τ,1−τ), we have
SI=∑|h|∼Hch(α)∑M≤m≤2Ma(m)∑N≤n≤2Ne((h+α)(mn)c)≪X(1+2κ)c+λ2+2κ+2ε | (3.1) |
for any a(m)≪mε, where (κ,λ) is any exponent pair, X≪MN≪X and M≪Y with Y=min{X1,X2,X3,X4,X5,X6,X7,X8},
X1=X152(1+2κ)c+λ2+2κ−c2−112, X2=X5211(1+2κ)c+λ2+2κ−4c11−811,X3=X318(1+2κ)c+λ2+2κ−c8−238, X4=X2(1+2κ)c+2λ1+κ−3,X5=X4(1+2κ)c+4λ1+κ−467,X6=X167(1+2κ)c+λ1+κ−257,X7=X203(1+2κ)c+λ1+κ−343,X8=X73(1+2κ)c+λ1+κ−113. |
Proof. It is easy to deduce that
SI≪Mε∑h∼HKh, |
where Kh=∑m∼M|∑n∼Ne((α+h)(mn)c)|. According to Hölder's inequality, we have
K4h≪M3∑m∼M|∑n∼Ne((α+h)(mn)c)|4. | (3.2) |
Let zn=zn(m,α)=(α+h)(mn)c. Suppose that Q, J are two positive integers such that 1≤Q≤Nlog−1X, 1≤J≤Nlog−1X. For the inner sum in (3.2), applying Lemma 2.1 twice, we can get
K4h≪X4Q2+X4J+X3JQJ∑j=1Q∑q=1|Eq,j|, | (3.3) |
where
Eq,j=∑m∼M∑N<n≤2N−q−je(zn−zn+q+zn+q+j−zn+j). | (3.4) |
Let Δ(nc;q,j)=(n+q+j)c−(n+q)c−(n+j)c+nc, G(m,n)=(α+h)mcΔ(nc;q,j). Then zn−zn+q−zn+j+zn+q+j=G(m,n). Thus we have
Eq,j=∑m∑ne(G(m,n)). | (3.5) |
For any t≠1,0, we have
Δ(nt;q,j)=t(t−1)qjnt−2+O(Nt−3qj(q+j)), | (3.6) |
then
∂G∂n=c(c−1)(c−2)(α+h)qjmcnc−3(1+O(q+jN)) |
and
∂2G∂n2=c(c−1)(c−2)(c−3)(α+h)qjmcnc−4(1+O(q+jN)). | (3.7) |
If c(c−1)(c−2)(α+h)qjMcNc−3≤1100, by Lemma 2.5 we have
∑m∑ne(G(m,n))≪MN3((α+h)qjMcNc)−1. |
From now we always suppose that c(c−1)(c−2)(α+h)qjMcNc−3≥1100. By Lemma 2.7 we have
∑N<n≤2N−j−qe(G(m,n))=e(18)∑α<υ<β|∂2G∂n2(m,nυ)|−12e(G(m,nυ)−υnυ)+R(m,q,j), |
where
∂G∂n(m,nυ)=υ,β=∂G∂n(m,N),α=∂G∂n(m,2N−q−j),R=N4[(h+α)qjXc]−1,υ∼(h+α)qjMcNc−3,R(m,q,j)=O(logX+RN−1+min(√R,max(1‖α‖,1‖β‖))). | (3.8) |
By Lemma 2.8, the contribution of R(m,q,j) to E(q,j) is
≪MlogX+MRN−1+∑m∼Mmin(√R,1‖α‖)+∑m∼Mmin(√R,1‖β‖)≪MlogX+X3−c[(h+α)qjM2]−1+[(h+α)qj]12MXc2−1logX. | (3.9) |
Then we only need to deal with the following exponential sum
∑m∼M∑α<υ<β|∂2G∂n2(m,nυ)|−12e(G(m,nυ)−υnυ)=∑υ∑m∈Iυ|∂2G∂n2(m,nυ)|−12e(G(m,nυ)−υnυ), |
where Iυ is a subinterval of [M,2M]. For a fixed υ, we define Δλ′=Δ(nλ′υ;q,j), where λ′ is an arbitrary real number. We take the derivative of m in (3.8) and get
n′υ=−cΔc−1(c−1)mΔc−2. | (3.10) |
It follows from (3.7) that
ddm(∂2G∂n2(m,nυ))=c2(h+α)mc−1Δc−2((c−1)Δ2c−2−(c−2)Δc−1Δc−3). |
Recalling (3.6), we can get
ddm(∂2G∂n2(m,nυ))=c2(c−1)(c−2)(c−3)(h+α)qjmc−1nc−4υ(1+O(q+jN)). |
Thus for m, |∂2G∂n2(m,nυ)|−12 is monotonic. Let g(m)=G(m,nυ(m))−υnυ(m). Then we have
g′(m)=c(α+h)mc−1Δc,g″(m)=c(α+h)c−1(c−1)2ΔcΔc−2−c2Δ2c−1m2−cΔc−2=c(α+h)(c−1)g1(m)−g2(m)g0(m),g‴(m)=c(α+h)(c−1)(g′1−g′2)g0−g′0(g1−g2)g20, |
where
g′1=((c−1)2cΔc−1Δc−2+(c−1)2(c−2)ΔcΔc−3)n′υ(m),g′2=2c2(c−1)Δc−1Δc−2n′υ(m),g′0=(2−c)m1−c(c−1)Δc−2((c−1)Δ2c−2+cΔc−1Δc−3). |
From the above formulas we can obtain
g‴(m)∼(h+α)qjM−1Xc−2. |
Using Lemma 2.5 and partial summation we can get
∑m∼M∑υ|∂2G∂n2(m,nυ)|−12e(G(m,nυ)−υnυ)≪(M((h+α)qjM−1Xc−2)16+((h+α)qjM−1Xc−2)−13)×(h+α)qjMcNc−3((h+α)qjMcNc−4)−12≪((h+α)qj)23M116X2c3−43+((h+α)qj)16M43Xc6−13. | (3.11) |
By (3.5), (3.9) and (3.11), we have
Eq,jlog−1X≪M+((h+α)qjM2)−1X3−c+((h+α)qj)12MXc2−1+((h+α)qj)23M116X2c3−43+((h+α)qj)16M43Xc6−13. | (3.12) |
Inserting (3.12) into (3.3), we obtain
K4hlog−1X≪X4Q2+X4J+MX3+((h+α)QJM2)−1X6−c+((h+α)QJ)12MXc2+2+((h+α)QJ)23M116X2c3+53+((h+α)QJ)16M43Xc6+83. |
Then choosing optimal J∈[0,Nlog−1X] and Q∈[0,Nlog−1X] and using Lemma 6 twice we can get
Khlog−3X≪B(h), |
where
B(h)=X56+(α+h)114M17Xc14+57+(α+h)112M1148Xc12+1724+(α+h)130M415Xc30+1115+X34M14+(α+h)−14X1−c4+X2328M18+X2532M732+X1720M340+X1114M314. |
Recalling the definitions of H and Y, we have
SIlog−3X≪MεHB(H)≪X(1+2κ)c+λ2+2κ+2ε, |
and Lemma 3.1 is proved.
Lemma 3.2. Let P56≪X≪P, H=X1−(1+2κ)c+λ2+2κ, F=(h+α)Xcand ch(α) denote complex numbers such that ch(α)≪(1+|h|)−1. Then uniformly for α∈(τ,1−τ), we have
SII=∑|h|∼Hch(α)∑M≤m≤2Ma(m)∑N≤n≤2Nb(n)e((h+α)(mn)c)≪X(1+2κ)c+λ2+2κ+2ε, | (3.13) |
for any a(m)≪mε, b(n)≪nε, where (κ,λ) is any exponent pair, X≪MN≪X and
max{X3−(1+2κ)c+λ1+κF−1,X4−2(1+2κ)c+2λ1+κ,X26−533(1+2κ)c+λ1+κF539}≪M≪X(1+2κ)c+λ1+κ−1. |
Proof. Taking Q=[X2−(1+2κ)c+λ1+κlog−1X], then we have Q=o(N). By Cauchy's inequality and Lemma 2.1, we have
|SII|2≪∑m∼M|a(m)|2∑m∼M|∑n∼Nb(n)e(f(mn))|2≪M2N2log2A+2BXQ+MNlog2AXQQ∑q=1Eq, | (3.14) |
where Eq=∑n∼N|b(n+q)b(n)||∑m∼Me(G(mn))| and G(m,n)=G(m,n,q)=(h+α)mcΔ(n,q;c), Δ(n,q;c)=(n+q)c−nc.
If |∂G∂m|≤103Mq−2, by Lemma 2.10 we have
Eq≪∑n∼N|b(n+q)b(n)|(MNFq+(FqMN)12M12)≪∑n∼N(|b(n+q)|2+|b(n)|2)(MNFq+Mq)≪MNqlog2BX | (3.15) |
noting that M≫XF.
Now we suppose |∂G/∂m|>103Mq−2. By Lemma 11 we get
∑m∼Me(G(m,n))≪MN1/2(Fq)1/2|∑r1(n)≤r≤r2(n)φ(n,r)e(s(r,n))|+logX+MN1/2(Fq)−1/2, |
where s(r,n)=G(M(r,n),n)−rm(r,n), φ(r,n)=(Fq)12MN12|∂2G(m(r,n),n)∂m2|−12 and
r1(n)=∂G∂m(M,n), r2(n)=∂G∂m(2M,n). |
Thus we have
Eq≪MN1/2(Fq)1/2∑n∼N|b(n+q)b(n)||∑r1(n)<r≤r2(n)φ(n,r)e(s(r,n))|+Nlog2B+1X+MN3/2(Fq)−1/2log2BX. | (3.16) |
So it suffices to bound the sum
Σ1=∑n∼N|b(n+q)b(n)||∑r1(n)<r≤r2(n)φ(n,r)e(s(r,n))|. |
Let T=[Fq3/M2N] and R=Fq/MN. By Cauchy's inequality and Lemma 2.1 again we get
Σ12≪∑n∼N|b(n+q)b(n)|2∑n∼N|∑r1(n)<r≤r2(n)φ(n,r)e(s(r,n))|2≪N2R2log4BXT+NRlog4BXTΣ2, | (3.17) |
where
Σ2=T∑t=1|∑n∼N∑r1(n)<r≤r2(n)−tφ(n,r+t)φ(n,r)e(s(r+t)−s(r,n))| |
and where we used the estimate
∑n∼N|b(n+q)b(n)|2≪∑n∼N(|b(n+q)|4+|b(n)|4)≪Nlog4BX. |
It is easy to check that 10<T=o(R).
Recall that s(r,n)=G(m(r,n),n)−rm(r,n), where m(r,n) denotes the solution of
∂G∂m(m,n)=r. |
It is easy to deduce that
∂s∂r(r,n)=∂G∂m∂m∂r−m(r,n)−r∂m∂r=−m(r,n). |
So we can obtain
H(n):=Hr,t,q(n)=s(r+t,n)−s(r,n)=∫r+tr∂s∂u(u,n)du=−∫r+trm(u,n)du, |
which implies that |H(j)|∼tMN−j, (j=0,1,2,3,4,5,6). Denote by I(r,t) the interval N<n≤2N, r1(n)<n≤r2(n)−t. Then we have
Σ2≪T∑t=1∑r∼R|∑n∈I(r,t)φ(n,r+t)φ(n,r)e(s(r+t,n)−s(r,n))|. |
Thus using partial summation, we get
Σ2≪T∑t=1∑r∼R(tMN)κNλ≪RMκNλ−κT1+κ≪NR | (3.18) |
with the exponent pair (κ,λ), if we note that M≫X26−533(1+2κ)c+λ1+κF539. From (3.15)–(3.18) we get that for any 1≤q≤Q,
Eq≪MNlog2B+1Xq+Nlog2B+1X+MN32(Fq)−12log2BX. | (3.19) |
Now this lemma follows from inserting (3.19) into (3.14).
Lemma 3.3. For τ≤α≤1−τ, we have
S(α)≪P(1+2κ)c+λ2+2κ+4ε, |
where (κ,λ) is any exponent pair.
Proof. Throughout the proof of this lemma, we write H=X1−(1+2κ)c+λ2+2κ for convenience. By a dissection argument we only need to prove that
∑X<n≤2XΛ(n)e(α[nc])≪X(1+2κ)c+λ2+2κ+3ε | (3.20) |
holds for P56≤X≤P and τ≤α≤1−τ. According to Lemma 2.3, we have
∑X<n≤2XΛ(n)e(α[nc])=∑|h|≤Hch(α)∑X<n≤2XΛ(n)e((h+α)nc)+O((logX)∑X<n≤2Xmin(1,1H‖nc‖)). | (3.21) |
By the expansion
min(1,1H‖nc‖)=∞∑h=−∞ahe(hnc), |
where
|ah|≤min(log2HH,1|h|,Hh2), |
we get
∑X<n≤2Xmin(1,1H‖nc‖)≤∞∑h=−∞ah|∑X<n≤2Xe(hnc)|≪Xlog2HH+∑1≤h≤H1h((hXc)12+XhXc)+∑h≥HHh2((hXc)12+XhXc)≪X(1+2κ)c+λ2+2κlogX, | (3.22) |
where we estimated the sum over n by Lemma 2.2 with the exponent pair (12,12).
Let R=max{X3−(1+2κ)c+λ1+κF−1,X4−2(1+2κ)c+2λ1+κ,X26−533(1+2κ)c+λ1+κF539}. Recall the definition of Y in Lemma 3.1. Let U=R, V=X(1+2κ)c+λ1+κ−1, Z=[XY−1]+12. By Lemma 4 with F(n)=e((h+α)nc), then we reduce the estimate of
∑|h|≤Hch(α)∑X<n≤2XΛ(n)e((h+α)nc) |
to the estimates of sums of type I
S′I=∑|h|≤Hch(α)∑M<m≤2Ma(m)∑N<n≤2Ne((h+α)(mn)c), N>Z, |
and sums of type II
S′II=∑|h|≤Hch(α)∑M<m≤2Ma(m)∑N<n≤2Nb(n)e((h+α)(mn)c), U<M<V, |
where a(m)≪mε, b(n)≪nε, X≪MN≪X. By Lemma 3.1, we get
S′I≪X(1+2κ)c+λ2+2κ+2ε. | (3.23) |
By Lemma 3.2, we get
S′II≪X(1+2κ)c+λ2+2κ+3ε. | (3.24) |
From (3.23) and (3.24) we can obtain
∑|h|≤Hch(α)∑X<n≤2XΛ(n)e((h+α)nc)≪X(1+2κ)c+λ2+2κ+3ε. | (3.25) |
Now (3.20) follows from (3.21), (3.22) and (3.25). Thus we complete the proof of this Lemma.
It is easy to see that
R(N)=∫1−τ−τS3(α)e(−αN)dα=∫τ−τS3(α)e(−αN)dα+∫1−ττS3(α)e(−αN)dα=R1(N)+R2(N). | (4.1) |
Following the argument of Laporta and Tolev [18,pages 928–929], we can get that
R1(N)=Γ3(1+1c)Γ(3c)N3c−1+O(N3c−1exp(−log13−δN)) | (4.2) |
for 1<c<32 and any 0<δ<13, where the implied constant in the O−symbol depends only on c.
Let
S(α,X)=∑X<p≤2Xe(α[pc])logp, T(α,X)=∑X<n≤2Xe(α[nc]). |
We can get
R2(N)=∫1−ττS3(α)e(−αN)dα≪(logX)maxP56≤X≤0.5P|∫1−ττS2(α)S(α,X)e(−αN)dα|+P116log2P, | (4.3) |
where we used
∫1−ττ|S2(α)|dα≪∫10|S2(α)|dα≪Plog2P. | (4.4) |
Now, we start to estimate the absolute value on the right hand in (4.3). By Cauchy's inequality we have
|∫1−ττS2(α)S(α,X)e(−αN)dα|=|∑X<p≤2X(logp)∫1−ττS2(α)e(α[pc]−αN)dα|≤∑X<p≤2X(logp)|∫1−ττS2(α)e(α[pc]−αN)dα|≪(logX)∑X<n≤2X|∫1−ττS2(α)e(α[nc]−αN)dα|≪X12(logX)(∑X<n≤2X|∫1−ττS2(α)e(α[nc]−αN)dα|2)12=X12(logX)(∫1−ττ¯S2(β)e(−βN)dβ∫1−ττS2(α)T(α−β,X)e(−αN)dα)12≪X12(logX)(∫1−ττ|S(β)|2dβ∫1−ττ|S(α)|2|T(α−β,X)|dα)12. | (4.5) |
Then we have
∫1−ττ|S(α)|2|T(α−β,X)|dα≪∫τ<α<1−τ|α−β|≤X−c|S(α)|2|T(α−β,X)|dα+∫τ<α<1−τ|α−β|>X−c|S(α)|2|T(α−β,X)|dα. | (4.6) |
By Lemma 3.3, we have
∫τ<α<1−τ|α−β|≤X−c|S(α)|2|T(α−β,X)|dα≪Xmaxα∈(τ,1−τ)|S(α)|2∫|α−β|≤X−c1dα≪X1−cP(1+2κ)c+λ1+κ+8ε, | (4.7) |
where we used the trivial bound T(α,X)≪X. By Lemma 2.9, Lemma 3.3 and (4.4), we get
∫τ<α<1−τ|α−β|>X−c|S(α)|2|T(α−β,X)|dα≪∫τ<α<1−τ|α−β|>X−c|S(α)|2(Xκc+λ1+κlogX+X|α−β|Xc)dα≪Xκc+λ1+κ(logX)∫1−ττ|S(α)|2dα+maxα∈(τ,1−τ)|S(α)|2∫|α−β|>X−cX|α−β|Xcdα≪Xκc+λ1+κPlog3P+X1−cP(1+2κ)c+λ1+κ+9ε. | (4.8) |
Thus, combining (4.6)–(4.8) we obtain
∫1−ττ|S(α)|2|T(α−β,X)|dα≪Xκc+λ1+κPlog3P+X1−cP(1+2κ)c+λ1+κ+9ε. | (4.9) |
By (4.3), (4.5) and (4.9), we can obtain
R2(N)≪P3−c−ε. | (4.10) |
Now putting (4.1), (4.2) and (4.10) into together, we have
R(N)=Γ3(1+1c)Γ(3c)N3c−1+O(N3c−1exp(−log13−δN)) |
follows for any 0<δ<13, where the implied constant in the O−symbol depends only on c. Thus we complete the proof of Theorem 1.1.
The authors would like to thank the referees for their many useful comments. This work is supported by National Natural Science Foundation of China (Grant Nos. 11801328 and 11771256).
The authors declare no conflict of interest.
[1] |
M. M. Meerschaert, H. P. Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Probab., 41 (2004), 623–638. https://doi.org/10.1239/jap/1091543414 doi: 10.1239/jap/1091543414
![]() |
[2] | B. Baeumer, M. M. Meerchaert, Stochastic solutions for fractional Cauchy problems, Fract. Calc. Appl. Anal., 4 (2001), 481–500. Available from: https://stt.msu.edu/users/mcubed/FracCauchyJM.pdf |
[3] |
M. Allen, L. Caffarelli, A. Vasseur, A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221 (2016), 603–630. https://doi.org/10.1007/s00205-016-0969-z doi: 10.1007/s00205-016-0969-z
![]() |
[4] |
B. Baeumer, T. Luks, M. M. Meerschaert, Space-time fractional Dirichlet problems, Math. Nachr., 291 (2018), 2516–2535. https://doi.org/10.1002/mana.201700111 doi: 10.1002/mana.201700111
![]() |
[5] | M. Foondun, E. Nane, Asymptotic properties of some space-time fractional stochastic equations, Math. Z., 287 (2017) 493–519. https://doi.org/10.1007/s00209-016-1834-3 |
[6] |
M. Grothaus, F. Jahnert, Mittag-Leffler analysis, II: application to the fractional heat equation, J. Funct. Anal., 270 (2016), 2732–2768. https://doi.org/10.1016/j.jfa.2016.01.018 doi: 10.1016/j.jfa.2016.01.018
![]() |
[7] |
Z. Q. Chen, K. H. Kim, P. Kim, Fractional time stochastic partial differential equations, Stochastic Process. Appl., 125 (2015), 1470–1499. https://doi.org/10.1016/j.spa.2014.11.005 doi: 10.1016/j.spa.2014.11.005
![]() |
[8] |
K. Li, J. Peng, J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263 (2012), 476–510. https://doi.org/10.1016/j.jfa.2012.04.011 doi: 10.1016/j.jfa.2012.04.011
![]() |
[9] |
J. Mijena, E. Nane, Space time fractional stochastic partial differential equations, Stochastic Process. Appl., 125 (2015), 3301–3326. https://doi.org/10.1016/j.spa.2015.04.008 doi: 10.1016/j.spa.2015.04.008
![]() |
[10] | Z. C. Fang, J. Zhao, H. Li, Y. Liu, A fast time two-mesh finite volume element algorithm for the nonlinear time-fractional coupled diffusion model. Numer. Algorithms, 93 (2023), 863–898. https://doi.org/10.1007/s11075-022-01444-2 |
[11] |
M. Kovács, S. Larsson, F. Saedpanah, Mittag-Leffler Euler integrator for a stochastic fractional order equation with additive noise, SIAM J. Numer. Anal., 58 (2020), 66–85. https://doi.org/10.1137/18M1177895 doi: 10.1137/18M1177895
![]() |
[12] |
D. L. Wang, M. Stynes, Optimal long-time decay rate of numerical solutions for nonlinear time-fractional evolutionary equations, SIAM J. Numer. Anal., 61 (2023), 2011–2034. https://doi.org/10.1137/22M1494361 doi: 10.1137/22M1494361
![]() |
[13] |
B. Y. Zhou, X. L. Chen, D. F. Li, Nonuniform Alikhanov linearized Galerkin finite element methods for nonlinear time-fractional parabolic equations, J. Sci. Comput., 85 (2020), 39. https://doi.org/10.1007/s10915-020-01350-6 doi: 10.1007/s10915-020-01350-6
![]() |
[14] | M. M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, vol. 43, Walter de Gruyter, Berlin/Boston, 2012. https://doi.org/10.1515/9783110560244 |
[15] | K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993. Available from: https://www.gbv.de/dms/ilmenau/toc/122837029.PDF |
[16] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, New York and London, 1993. Available from: https://book.douban.com/subject/4185601/ |
[17] | V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Volume I: Background and Theory, Springer & Higher Education Press, Berlin & Beijing, 2013. https://doi.org/10.1007/978-3-642-33911-0 |
[18] | V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Volume II: Applications, Springer & Higher Education Press, Berlin & Beijing, 2013. https://doi.org/10.1007/978-3-642-33911-0 |
[19] |
H. Y. Xu, Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials, Commun. Anal. Mech., 15 (2023), 132–161. https://doi.org/10.3934/cam.2023008 doi: 10.3934/cam.2023008
![]() |
[20] | Z. Q. Chen, M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, Princeton University Press, 2012. https://doi.org/10.1515/9781400840564 |
[21] | M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, 2nd ed., De Gruyter, 2011. https://doi.org/10.1515/9783110218091 |
[22] |
Z. Q. Chen, Time fractional equations and probabilistic representation, Chaos Solitons Fractals, 102 (2017), 168–174. https://doi.org/10.1016/j.chaos.2017.04.029 doi: 10.1016/j.chaos.2017.04.029
![]() |
[23] |
Z. Q. Chen, P. Kim, T. Kumagai, J. Wang, Heat kernel estimates for time fractional equations, Forum Math., 30 (2018), 1163–1192. https://doi.org/10.1515/forum-2017-0192 doi: 10.1515/forum-2017-0192
![]() |
[24] |
Z. Q. Chen, W. H. Deng, P. B. Xu, Feynman-Kac transform for anomalous processes, SIAM J. Math. Anal., 53 (2021), 6017–6047. https://doi.org/10.1137/21M1401528 doi: 10.1137/21M1401528
![]() |
[25] |
R. Sun, W. H. Deng, Unified stochastic representation, well-posedness analysis, and regularity analysis for the equations modeling anomalous diffusions, Discrete Contin. Dyn. Syst. Ser. B, 29 (2024), 991–1018. https://doi.org/10.3934/dcdsb.2023121 doi: 10.3934/dcdsb.2023121
![]() |
[26] | K. I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999. Available from: http://assets.cambridge.org/97805215/53025/frontmatter/9780521553025_frontmatter.pdf |
[27] | J. Bertoin, Lévy Processes, Cambridge University Press, 1996. Available from: https://cambridge.readlink.com/front-book.html?id = 590255 |
1. | Qian Sima, Shan Wu, Rahim Khan, The Acceptability of Traditional Culture under the Background of Deep Learning, 2022, 2022, 1687-5273, 1, 10.1155/2022/4010099 | |
2. | Min Hong, Jiajia He, Kexian Zhang, Zhidou Guo, Does digital transformation of enterprises help reduce the cost of equity capital, 2023, 20, 1551-0018, 6498, 10.3934/mbe.2023280 | |
3. | Kaimeng Zhang, Xihe Liu, Jingjing Wang, Exploring the relationship between corporate ESG information disclosure and audit fees: evidence from non-financial A-share listed companies in China, 2023, 11, 2296-665X, 10.3389/fenvs.2023.1196728 | |
4. | Tingfang Zhang, Model Design and Discourse Construction of Intercultural Communication of Chinese Cultural Community in Globalized Business Environment, 2024, 9, 2444-8656, 10.2478/amns-2024-3259 | |
5. | Guangbin Liu, Wei Nie, The role of Marxist ecological view on environmental protection in China, 2023, 0958-305X, 10.1177/0958305X231177738 | |
6. | Yang Xu, Conghao Zhu, Runze Yang, Qiying Ran, Xiaodong Yang, Applications of linear regression models in exploring the relationship between media attention, economic policy uncertainty and corporate green innovation, 2023, 8, 2473-6988, 18734, 10.3934/math.2023954 | |
7. | Shan Huang, Khor Teik Huat, Yue Liu, Study on the influence of Chinese traditional culture on corporate environmental responsibility, 2023, 20, 1551-0018, 14281, 10.3934/mbe.2023639 | |
8. | Tinghui Li, Xin Shu, Gaoke Liao, Does corporate greenwashing affect investors' decisions?, 2024, 67, 15446123, 105877, 10.1016/j.frl.2024.105877 |