Large Deviations for Stochastic Generalized Porous Media Equations driven by L\'{e}vy Noise

TL;DR

This paper establishes a large deviation principle for stochastic porous media equations driven by Lévy noise, extending previous results by removing compactness assumptions and covering a broad class of operators and physical models.

Contribution

It introduces a new approach to prove LDPs for stochastic evolution equations with Lévy noise without relying on compactness conditions, applicable to various operators and physical problems.

Findings

First LDP result for Lévy-driven stochastic PDEs without compactness.

Applicable to Laplacian, fractional Laplacians, Schrödinger operators, and fractals.

Includes physical models like the Stefan problem.

Abstract

We establish a large deviation principle (LDP) for a class of stochastic porous media equations driven by L\'{e}vy-type noise on a $\sigma$-finite measure space $(E,\mathcal{B}(E),\mu)$, with the Laplacian replaced by a negative definite self-adjoint operator. One of the main contributions of this paper is that we do not assume the compactness of embeddings in the corresponding Gelfand triple, and to compensate for this generalization, a new procedure is provided. This is the first paper to deal with LDPs for stochastic evolution equations with L\'evy noise without compactness conditions. The coefficient $\Psi$ is assumed to satisfy nondecreasing Lipschitz nonlinearity, so an important physical problem covered by this case is the Stefan problem. Numerous examples of negative definite self-adjoint operators are applicable to our results, for example, for open $E\subset\Bbb{R}^d$, $L=$ Laplacian or fractional Laplacians, i.e., $L=-(-\Delta)^\alpha,\ \alpha\in(0,1]$, generalized Schr\"{o}dinger operators, i.e., $L=\Delta+2\frac{\nabla \rho}{\rho}\cdot\nabla$, Laplacians on fractals is also included.