L\'evy-driven Volterra equations in space and time

TL;DR

This paper studies nonlinear stochastic Volterra equations driven by Le9vy bases, establishing existence, uniqueness, and stability criteria in weighted spaces, with applications to the stochastic heat equation.

Contribution

It provides new existence, uniqueness, and stability results for Le9vy-driven Volterra equations with infinite memory, extending classical conditions.

Findings

Existence and uniqueness depend on kernel and Le9vy characteristics.

Solutions are stable under small kernel and characteristic conditions.

Application to stochastic heat equation demonstrates practical relevance.

Abstract

We investigate nonlinear stochastic Volterra equations in space and time that are driven by L\'evy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the L\'evy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infinity. Here, in contrast to the case where time is positive, the usual integrability conditions on the kernel are no longer sufficient for the existence and uniqueness of solutions, but we have to impose additional size conditions on the kernel and the L\'evy characteristics. Furthermore, once the existence of a solution is guaranteed, we analyse its asymptotic stability, that is, whether its moments remain bounded when time goes to infinity. Stability is proved whenever kernel and characteristics are small enough, or the nonlinearity of the equation exhibits a fractional growth of order strictly smaller than one. The results are applied to the stochastic heat equation for illustration.