Limiting measure and stationarity of solutions to stochastic evolution equations with Volterra noise

TL;DR

This paper investigates the long-term behavior of solutions to stochastic evolution equations driven by Volterra noise, establishing conditions for the existence of limiting measures and stationarity, with applications to heat equations driven by Rosenblatt processes.

Contribution

It provides new sufficient conditions for limiting measure existence and stationarity of solutions to stochastic Volterra equations, including an example where these conditions are necessary.

Findings

Conditions for limiting measure existence are established.

Strict stationarity of solutions is characterized.

Application to heat equations driven by Rosenblatt process is demonstrated.

Abstract

Large-time behaviour of solutions to stochastic evolution equations driven by two-sided regular Volterra processes is studied. The solution is understood in the mild sense and takes values in a separable Hilbert space. Sufficient conditions for the existence of limiting measure and strict stationarity of the solution process are found and an example for which these conditions are also necessary is provided. The results are further applied to the heat equation driven by the two-sided Rosenblatt process.

Equations

$$\left\{\begin{array}[]{cl}\,\mathrm{d}{X}_{t}&=AX_{t}\,\mathrm{d}{t}+\varPhi\,\mathrm{d}{B}_{t},\quad t\geq 0,\\ X_{0}&=x,\end{array}\right.$$

$$X_t^x:=S(t)x+{\int }_0^{t}S(t-r)\Phi \mathrm{d}{B}_r,t\ge 0.$$

Full text

The author was supported by the Charles University, project GAUK No. 322715, and by the Czech Science Foundation, project GAČR No. 15-08819S

1. Introduction

Consider the stochastic evolution equation

[TABLE]

where $A$ generates a $C_{0}$-semigroup of bounded linear operators $S=(S(t),t\geq 0)$ acting on a separable Hilbert space and its mild solution which is defined by the variation of constants formula

[TABLE]

The noise process is a two-sided Hilbert space valued $\alpha$-regular Volterra process $B$ (see LABEL:def:cylindrical_process). It is shown (see LABEL:prop:limiting_measure) that if the process $B$ has stationary and reflexive increments (see LABEL:def:stationarity_reflexivity) and the equation satisfies certain stability conditions (see formula (LABEL:eq:condition_on_inv_measure)), there is a limiting measure $\mu_{\infty}$ such that the law of $X^{0}_{t}$ and converges to $\mu_{\infty}$ as $t\rightarrow\infty$. Furthermore, we provide an example for which the stability condition is also a necessary one (see LABEL:ex:non_exp_stable). Additionally, if the semigroup is strongly stable, we have (see LABEL:prop:limiting_measure_2) that the law of $X_{t}^{x}$ tends to $\mu_{\infty}$ as $t\rightarrow\infty$ for each initial condition $x\in L^{2}(\Omega;V)$. Also, it is shown (see LABEL:prop:strict_stationarity) that there exists an initial condition $x_{\infty}$, such that the solution $X^{x_{\infty}}$ is a strictly stationary process.

Volterra processes have been considered in the pioneering work [AlosMazNua01] where the authors considered Gaussian Volterra processes (see also [BauNua03, ErrEss09, Hida60]). Regular Volterra processes which might not be Gaussian and stochastic evolution equations driven by them were studied in the literature as well. In particular, existence and regularity results were given in [BonTud11, CouMas16, CouMasOnd17, CouMasSnup17] and the present paper can be viewed as a continuation of the work. For specific cases of the driving noise, stationarity and large-time behavior of the solutions have been already treated in the literature (see e.g. [DunMasDun02, MasNua03, MasPos07, MasPos08] and others for equations driven by the fractional Brownian motion (fBm)).

It is not a priori clear how the two-sided Volterra processes should be defined. We propose such a definition (LABEL:def:Volterra_process) after analysis of two main examples of two-sided $\alpha$-regular Volterra processes - the fractional Brownian motion with Hurst parameter $H\in(1/2,1)$ (see e.g. [AlosNua03, DecrUstu99, MandelbrotVanNess] for its definition and properties) and the Rosenblatt process (see e.g. [Taqqu11, Tud08] for its definition and properties).

The paper is organized as follows.

In LABEL:sec:prelim, we define two-sided $\alpha$-regular Volterra processes and give two examples - the (two-sided) fBm of Hurst parameter $H\in(1/2,1)$ and the (two-sided) Rosenblatt process. Then we modify the already existing stochastic integral with respect to one-sided $\alpha$-regular Volterra processes to the case when the integrator is two-sided and give basic properties of the integral.