Exponential ergodicity and finite-dimensional approximation for Markovian lifts of stochastic Volterra equations

Abstract: This paper investigates the long-time asymptotics and the existence of stationary solutions for a class of stochastic Volterra equations (SVEs). To address the non-Markovian nature of SVEs, we employ a Markovian lifting technique, formulating a Markovian lift as the solution to a stochastic evolution equation (SEE) on a Gelfand triplet. Our main objective is to establish the ergodicity of this Markovian lift via the generalized Harris' theorem, which in turn yields the asymptotic results for the original SVE. Despite the challenges posed by the highly degenerate, infinite-dimensional nature of the SEE, we achieve this by constructing a generalized coupling and a distance function that exploit the structural properties arising from the non-local operators in its coefficients. Furthermore, we prove that the invariant probability measure and, more generally, the stationary law on the path space of the SEE can be weakly approximated by those of finite-dimensional SDEs. This yields a novel approximation result for the stationary solution of the original SVE, while offering a rigorous mathematical framework that supports the validity of the Markovian embedding concept widely utilized in statistical physics.

• The paper proves a unique invariant measure, spectral gap, and exponential Wasserstein ergodicity for Markovian lifts under uniform ellipticity and Lyapunov conditions, without smallness assumptions on coefficient Lipschitz constants.
• The paper extends lifting methods to locally integrable drift kernels and locally square-integrable diffusion kernels, covering regimes such as generalized Langevin equations that exceed earlier frameworks.
• The paper uses the spectral gap to show stationary solutions of the infinite-dimensional model converge to finite-factor SDE approximations at rate O(ε_k^{1/2}), resolving an open problem for stationary Volterra dynamics.

Setting and motivation

This paper studies the long-time behavior of stochastic Volterra equations (SVEs) of the form

$$X_t=x(t)+\int_0^t K_b(t-s)b(X_s)\,ds+\int_0^t K_\sigma(t-s)\sigma(X_s)\,dW_s,$$

where $K_b,K_\sigma$ are matrix-valued kernels, $b$ and $\sigma$ are Lipschitz coefficients on $\mathbb{R}^n$, and the forcing term $x$ may be random but independent of $W$. SVEs of this type model rough volatility in mathematical finance and over-damped generalized Langevin equations (GLEs) in statistical physics. Two structural features obstruct a direct ergodic analysis: the solution is non-Markovian and generally not a semimartingale, so neither Harris-type theory nor Itô calculus applies directly.

The author's strategy is Markovian lifting. Under the assumption that the kernels admit Laplace-transform representations $K_b(t)=\int e^{-\theta t}M_b(\theta)\,\mu(d\theta)$ and similarly for $K_\sigma$, with respect to a Borel measure $\mu$ satisfying weighted integrability conditions, the SVE is lifted to a stochastic evolution equation (SEE) on a Gelfand triplet $K_b,K_\sigma$0 of weighted $K_b,K_\sigma$1 spaces:

$K_b,K_\sigma$2

where $K_b,K_\sigma$3 is a non-local operator. The original solution is recovered as $K_b,K_\sigma$4. A notable generalization relative to prior work by the same author is that the drift kernel need only be locally $K_b,K_\sigma$5 while the diffusion kernel is locally $K_b,K_\sigma$6—the minimal integrability for the Volterra integrals to be well-defined. This matters concretely: for the GLE derived under the fluctuation–dissipation theorem, the exponents satisfy $K_b,K_\sigma$7, so when $K_b,K_\sigma$8 the drift kernel fails to be square-integrable at the origin, a regime beyond all previously cited lifting frameworks.

Main ergodicity result

The main theorem combines two ingredients via the generalized Harris theorem of Hairer–Mattingly–Scheutzow.

Contractivity and small sets. Under Assumption (uniform ellipticity of $K_b,K_\sigma$9, positive definiteness of $b$0, $b$1, and Lipschitz coefficients), the paper constructs an admissible weight function $b$2 such that the distance $b$3 is contracting for $b$4 whenever $b$5, with contraction constant $b$6, and every $b$7-ball of radius $b$8 is $b$9-small for $\sigma$0 whenever $\sigma$1. The construction proceeds through a generalized coupling: a controlled SEE driven by a finite-dimensional control process, combined with a Girsanov change of measure whose Kullback–Leibler divergence is bounded by $\sigma$2. This is technically demanding because the SEE is highly degenerate—an infinite-dimensional state space driven by noise of dimension as low as one—so the standard "effectively elliptic" coupling arguments do not apply. The key device is a "change-of-norm" technique: the weight function $\sigma$3 is tuned so that the adjoint operator $\sigma$4 approximates the non-local operator $\sigma$5 on the relevant part of the spectrum.

Lyapunov function. An abstract coercivity condition on the coefficients, phrased in terms of a second admissible weight function $\sigma$6, yields $\sigma$7 as a Lyapunov function. A verifiable sufficient condition is given: sub-linear growth of $\sigma$8, linear growth plus coercivity of $\sigma$9 with slope $\mathbb{R}^n$0 small enough that $\mathbb{R}^n$1, and symmetric nonnegative-definite $\mathbb{R}^n$2. Notably, no restriction is placed on the Lipschitz constants of $\mathbb{R}^n$3 or $\mathbb{R}^n$4; dissipativity is achieved through the coercivity condition rather than smallness assumptions, in contrast to Bianchi–Bonaccorsi–Cañadas–Friesen and related works where smallness of Lipschitz/growth constants is required and no spectral gap is obtained.

Combining these ingredients gives the central result: the Markov semigroup $\mathbb{R}^n$5 on $\mathbb{R}^n$6 admits a unique invariant probability measure $\mathbb{R}^n$7, satisfies a spectral gap estimate

$\mathbb{R}^n$8

and exponential ergodicity $\mathbb{R}^n$9 in the $x$0-Wasserstein distance. The invariant measure charges $x$1 with finite second moment. As a corollary, the stationary law on path space exists and shifted laws converge weakly to it; moreover, a strictly stationary solution of the original SVE is constructed via Kolmogorov extension, so all finite-dimensional distributions are time-shift invariant—a stronger notion than the "fake stationarity" available for linear drift.

Two further results deserve emphasis. First, the proof yields an asymptotic log-Harnack inequality extending earlier scalar-kernel results to matrix-valued kernels with distinct $x$2, implying uniqueness of the invariant measure, asymptotic strong Feller property, asymptotic irreducibility, gradient estimates, and asymptotic heat kernel bounds. Second, the stationary distributions of the SVE are shown to be independent of the particular lifting basis chosen to generate the kernels, resolving a potential ambiguity inherent in the non-uniqueness of liftings.

Finite-dimensional approximation of stationary solutions

The spectral gap estimate is then used to approximate the infinite-dimensional stationary objects by finite-dimensional ones. Given an approximating component—finite partitions of $x$3 with piecewise-constant approximations of $x$4 and representative decay rates $x$5, with approximation error $x$6—one obtains finite-dimensional SDEs on $x$7 whose solutions solve approximating SVEs with sum-of-exponentials kernels. These SDEs admit invariant measures $x$8 for large $x$9, uniformly bounded in $W$0-moment.

The main approximation theorem states that, uniformly over all invariant measures $W$1,

$W$2

with quantitative rate $W$3 (not claimed optimal). The proof exploits the spectral gap in an essential way: writing the Wasserstein distance between invariant measures at a fixed large time $W$4 and using contraction absorbs the unknown distance into itself, leaving only the finite-time stability error between the SEE and its discretization, which is controlled by $W$5 via an Itô estimate uniform in $W$6. This mechanism is precisely why the spectral gap result matters beyond qualitative ergodicity—it provides the stability needed for the approximation argument.

The convergence transfers to the original SVE: the laws of the $W$7-equivalence classes converge in $W$8-Wasserstein distance, and the finite-dimensional distributions converge uniformly over times in compact intervals. The paper notes that this resolves an open problem: previous Markovian approximation results for SVEs addressed finite-horizon solutions only, not stationary solutions. It also supplies rigorous justification for the Mori–Zwanzig Markovian embedding heuristic in statistical physics, under which infinite-dimensional auxiliary dynamics arising from general kernels were treated as formal limits; here the convergence of the associated stationary solutions is proved, albeit within a lifting formulation that differs formally from the classical Mori–Zwanzig construction.

Limitations and open questions

Several restrictions bound the applicability of the main results. The ergodicity and approximation theorems require $W$9, i.e., exponentially decaying kernels; pure power-law kernels—the case directly relevant to the fractional GLE—are excluded, and only tempered fractional kernels fall within scope. Uniform ellipticity of $K_b(t)=\int e^{-\theta t}M_b(\theta)\,\mu(d\theta)$0 is essential to the coupling construction, and the paper does not address hypoelliptic or degenerate diffusion regimes. The embedding $K_b(t)=\int e^{-\theta t}M_b(\theta)\,\mu(d\theta)$1 is characterized as compact if and only if $K_b(t)=\int e^{-\theta t}M_b(\theta)\,\mu(d\theta)$2 is finite for every $K_b(t)=\int e^{-\theta t}M_b(\theta)\,\mu(d\theta)$3; since typical cases of interest (e.g., tempered fractional kernels) have non-compact embeddings, classical ultimate-boundedness arguments fail, which is why the Lyapunov construction must be combined with the contractivity/small-set machinery rather than invoked alone. The convergence rate $K_b(t)=\int e^{-\theta t}M_b(\theta)\,\mu(d\theta)$4 is acknowledged as possibly suboptimal, and weak convergence of $K_b(t)=\int e^{-\theta t}M_b(\theta)\,\mu(d\theta)$5 in the topology of $K_b(t)=\int e^{-\theta t}M_b(\theta)\,\mu(d\theta)$6 is not obtained because evaluation maps are discontinuous in $K_b(t)=\int e^{-\theta t}M_b(\theta)\,\mu(d\theta)$7. Finally, the author notes that limit theorems (law of large numbers, CLT, averaging, diffusion approximation) enabled by the exponential ergodicity are deferred to future work.

Conclusion

The paper establishes exponential ergodicity, with a spectral gap, for Markovian lifts of SVEs under natural Lyapunov and uniform ellipticity conditions, without smallness constraints on the coefficients and allowing drift kernels that are merely locally integrable. The spectral gap is then put to work: it yields weak approximation of the invariant measure and stationary path law by those of finite-dimensional Markovian SDEs, uniformly over invariant measures and with an explicit rate. Beyond stochastic analysis, the results give a rigorous footing to Markovian embedding procedures used informally in statistical physics, and they settle the previously open problem of approximating stationary solutions of SVEs by multi-factor models.