Coefficient-matched random Lyapunov functionals and delay-robust small-gain criteria for stochastic delay systems

Delay and multiplicative noise complicate the analysis of long-time behavior in stochastic evolution equations. We study pathwise energy estimates when delayed states acquire random coefficients after an Ornstein–Uhlenbeck transformation. For positive delays and nonzero noise, the delay amplification factor is tempered but almost surely unbounded in time, so it has no uniform deterministic majorant. We construct a Lyapunov–Krasovskii functional whose history density uses the same random coefficient at a shifted time. Differentiation then reproduces the coefficient multiplying the delayed energy and closes the estimate. For a coupled fractional parabolic system with monotone polynomial damping and measure-valued memory, this gives $ L_D < \Lambda_{\mathrm{lin}} $, where $ L_D $ bounds the delay operator and $ \Lambda_{\mathrm{lin}} $ is the smallest componentwise first-mode linear damping. The boundary is exact in the deterministic scalar first-mode linear subclass for each fixed delay, although the decay rate can deteriorate as the delay grows. Gaussian concentration yields an explicit sufficient noise threshold for positive mean dissipation. Under the stated assumptions, the resulting cocycle has a measurable tempered pullback absorbing family and a unique pullback random attractor, with upper semicontinuity as delay and noise vanish. A retarded finite-dimensional gradient system with diagonal linear drift and a separable convex potential illustrates the construction through time equicontinuity.