Large Deviations for Stochastic Porous Media Equation on General Measure Spaces

Abstract

In this paper, we establish the large deviation principles for stochastic porous media equations driven by time-dependent multiplicative noise on $σ$-finite measure space $(E,\mathcal{B}(E),μ)$, 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 $L^2(μ)$-initial data or compact embeddings in the associated Gelfand triple. Applications include fractional powers of the Laplacian, i.e. $L=-(-Δ)^α,\ α\in(0,1]$, generalized $\rm Schr\ddot{o}dinger$ operators, i.e. $L=Δ+2\frac{\nabla ρ}ρ\cdot\nabla$, and Laplacians on fractals.