Large Deviation Principle for Mild Solutions of Stochastic Evolution Equations with Multiplicative L\'{e}vy Noise

TL;DR

This paper establishes a large deviation principle for the mild solutions of stochastic evolution equations with multiplicative Lévy noise in the small noise limit, using the weak convergent method and variational representations.

Contribution

It introduces a novel application of the weak convergent method to derive large deviations for a broad class of stochastic evolution equations with Lévy noise.

Findings

Large deviation principle proven for stochastic evolution equations with Lévy noise.

Method applicable to semilinear parabolic, hyperbolic, and delay differential equations.

Examples demonstrate practical applications of the theoretical results.

Abstract

We demonstrate the large deviation principle in the small noise limit for the mild solution of stochastic evolution equations with monotone nonlinearity. A recently developed method, weak convergent method, has been employed in studying the large deviations. we have used essentially the main result of Budhiraja et al., [4] which discloses the variational representation of exponential integrals w.r.t. the L\'{e}vy noise. An It\^{o}-type inequality is a main tool in our proofs. Our framework covers a wide range of semilinear parabolic, hyperbolic and delay differential equations. We give some examples to illustrate the applications of the results.