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 $\sigma$-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 $\mathrm{Schr}\ddot{\mathrm{o}}\mathrm{dinger}$ operators, i.e. $L=\Delta +2\frac{\nabla \rho }{\rho }\cdot \nabla$, and Laplacians on fractals.

A revised version, 27 pages, the proof of Claim 4.1 is revised by using the added assumption (H2)(iii). The statements of Theorem 3.2, Theorem 4.1 and Lemma 4.1 are revised

References in corpus (2)

• Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise
• Large deviation principles for first-order scalar conservation laws with stochastic forcing