The short memory limit for long time statistics in a stochastic Coleman-Gurtin model of heat conduction

Abstract: We consider a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. Our main study is the long time statistics of the system in the singular regime as the memory kernel collapses to a Dirac function. Specifically, we show that provided that sufficiently many directions in the phase space are stochastically forced, there is a unique invariant probability measure to which the system converges, with respect to a suitable Wasserstein-type topology, and at an exponential rate which is independent of the decay rate of the memory kernel. We then prove the convergence of this unique statistically steady state to the unique invariant probability measure of the classical stochastic reaction-diffusion equation in the zero-memory limit. Consequently, we establish the global-in-time validity of the short memory approximation.

• The paper establishes the existence and uniqueness of statistically steady states by embedding the memory system into a Markovian framework under non-degenerate noise.
• The paper quantifies convergence by showing that finite-time solution statistics and invariant measures approach those of the classical model at rates of O(ε^(1/3)) and O(ε^(1/12)) respectively.
• The paper leverages advanced coupling methods, Lyapunov functions, and optimal transport metrics to rigorously justify the transition from non-Markovian to Markovian dynamics.

The Short Memory Limit for Long-Time Statistics in a Stochastic Coleman-Gurtin Model

Introduction and Model Formulation

This work develops a rigorous stochastic analysis of a class of semilinear differential Volterra equations, modeling heat conduction in viscoelastic media with memory, under the influence of additive Gaussian noise. The focus is on the long-time asymptotic statistical properties when the memory kernel becomes highly localized (i.e., in the "short memory" limit, as the kernel converges to a Dirac delta function). In this regime, the system transitions from non-Markovian dynamics (intrinsic to equations with memory terms) to Markovian behavior characteristic of the classical stochastic reaction-diffusion equation.

The main equation of interest is

$$du(t) = \kappa \Delta u(t)dt + (1-\kappa)\int_0^{\infty} K(s)\Delta u(t-s)dsdt + \varphi(u(t))dt + Qdw(t),$$

where $K$ is the memory kernel, $\varphi$ is a polynomial-like nonlinearity, $Qdw(t)$ is an infinite-dimensional temporal white noise, and $\kappa \in (0,1)$. The model is set over a bounded domain $O$ with appropriate (e.g., Dirichlet) boundary conditions. As $K$ collapses to the Dirac delta, the model formally reduces to the well-studied stochastic reaction-diffusion equation.

To overcome the non-Markovianity induced by the memory convolution, the authors utilize an abstract phase space embedding based on a "history variable," producing an autonomous Markovian system in a suitably extended space. This enables leveraging ergodic and coupling techniques for Markov processes.

Main Results

The paper achieves several substantial results on the stochastic dynamics in the short memory regime:

1. Existence and Uniqueness of Statistically Steady States:

It is proven that for sufficiently "non-degenerate" noise (ensuring enough directions in the phase space are forced directly), the extended system admits a unique invariant probability measure. This measure attracts solutions (with arbitrary initial data) at an exponential rate, uniform in the short memory parameter $\varepsilon$ (the parameter controlling the collapse of the memory kernel) (Theorem 1.1 and Theorem 3.6). The analysis applies to a broad class of nonlinearities, including, for instance, the Allen–Cahn cubic potential.

2. Quantitative Convergence to the Memoryless Model:

The main quantitative results establish that as $\varepsilon \to 0$, i.e., as the memory kernel approaches the Dirac delta, the invariant measure of the system with memory (and even finite-time solution statistics) converges to those of the classical stochastic reaction-diffusion equation. Explicit uniform rates are provided in suitable Wasserstein-type metrics.

• For solutions starting from compatible random initial data, the $L^2$-distance between the solution with memory and the solution without memory vanishes as $O(\varepsilon^{1/3})$ on finite time intervals (Theorem 3.9).
• For the invariant measures, the distance (in a Wasserstein metric) between the marginal of the extended-system invariant measure and the invariant measure of the memoryless system vanishes at a rate $O(\varepsilon^{1/12})$ (Theorem 3.11).
• For more general observables $f$ (satisfying suitable Lipschitz properties), an explicit rate of convergence on the difference of expected values is also established, uniformly in time (Theorem 3.12).

The argument relies on combining uniform-in-$\varepsilon$ geometric ergodicity for the transition semigroup, Lyapunov function methods, careful coupling constructions (particularly generalized asymptotic couplings with Girsanov transforms), and moment bounds uniform in the memory parameter.

3. Structural and Dimensional Constraints:

The technical machinery covers a wide class of nonlinearities but places explicit constraints on the polynomial degree, dependent on the spatial dimension. In $d = 1,2$, the degree is unrestricted, in $d = 3$ at most fifth-order, and for $d \geq 4$ linear growth is required.

Analytical Methods

The paper utilizes an extended phase space—embedding the memory system into a Markovian format by augmenting with the integrated past $\eta(t, s)$. This enables:

• Application of Markov semigroup theory and the weak Harris theorem;
• Construction of contractive metrics (pseudo-distances tailored to the energy/phase space) for analyzing ergodicity and convergence rates;
• Adaptation of the coupling method, including generalized asymptotic coupling, along with Lyapunov structures (functionals controlling growth at infinity).

Control over the effects of nonlinearity and memory is achieved through careful, dimension-dependent a priori moment bounds and estimates, with all constants tracked uniformly with respect to $\varepsilon$. The analysis relies crucially on exponential decay assumptions on the memory kernel; sub-exponential or algebraic memory behavior would require qualitatively different methods.

The use of optimal transport (Wasserstein) metrics allows explicit rates for convergence of distributions, even in infinite dimensions, and is essential for the statistical comparisons.

Implications and Perspectives

Theoretical Implications:

This work provides the first quantitative, global-in-time justification for the short memory approximation in nonlinear stochastic PDEs with memory effects. It rigorously connects the ergodic/statistical properties of viscoelastic (memory) heat conduction models with those of standard reaction-diffusion equations as the memory effect becomes negligible. This establishes that for a broad class of stochastically forced systems, the "Markovianization" of the dynamics and associated statistical effects occur uniformly on all time scales as the kernel collapses.

Practical Implications:

These results rigorously support the analytical tractability and physical appropriateness of using memoryless reaction-diffusion models to approximate systems with extremely short memory—important in both numerical modeling of heat conduction in viscoelastic media and inference for systems where the precise structure of the memory kernel is unknown but known to be strongly localized.

Broader Directions:

Future research could extend these results to:

• Multiplicative noise regimes or physically more complex potentials;
• Memory kernels with slower decay (sub-exponential or algebraic), where recent ergodic results are less developed;
• Further refinements on the rates of convergence, possibly exploiting more intricate optimal transport structures or improved regularity of invariant measures.

Additionally, open questions remain regarding explicit formulas or characterizations for invariant measures in more complex memory-driven systems—a direction of interest for both rigorous mathematics and nonequilibrium statistical physics.

Conclusion

This paper provides a comprehensive, mathematically rigorous treatment of the short memory limit in nonlinear SPDEs representing viscoelastic heat conduction, demonstrating uniform ergodic convergence properties and precise quantitative rates for the approach to memoryless models. The results rest on advanced probabilistic and analytic tools suitable for nonlinear, infinite-dimensional, and non-Markovian systems, significantly advancing the theory of stochastic PDEs with memory.