Large Deviations for Stochastic Porous Media Equation on General Measure Spaces

TL;DR

This paper establishes large deviation principles for stochastic porous media equations with multiplicative noise on general measure spaces, extending classical results to more general operators and nonlinearities.

Contribution

It introduces large deviation principles for stochastic porous media equations driven by time-dependent noise on measure spaces with minimal restrictions on nonlinearities.

Findings

Large deviation principles are proven for a broad class of stochastic porous media equations.

Applications include fractional Laplacians, generalized Schrödinger operators, and fractal Laplacians.

The results extend classical theory to more general operators and nonlinearities.

Abstract

In this paper, we establish the large deviation principles for stochastic porous media equations driven by time-dependent multiplicative noise on $\sigma$-finite measure space $(E,\mathcal{B}(E),\mu)$, and the Laplacian replaced by a negative definite self-adjoint operator. The coefficient is only assumed to satisfy the increasing Lipschitz nonlinearity assumption without the restrictions to its monotone behavior at infinity for $L^2(\mu)$-initial data or compact embeddings in the associated Gelfand triple. Applications include fractional powers of the Laplacian, i.e. $L=-(-\Delta)^\alpha,\ \alpha\in(0,1]$, generalized $\rm Schr\ddot{o}dinger$ operators, i.e. $L=\Delta+2\frac{\nabla \rho}{\rho}\cdot\nabla$, and Laplacians on fractals.