Large deviations for stochastic porous media equation on general measure spaces

Abstract

In this paper, we establish a large deviation principle for stochastic porous media equations driven by time-dependent multiplicative noise on a σ-finite measure space (E,B(E),μ), and with the Laplacian replaced by a negative definite self-adjoint operator. The coefficient Ψ is only assumed to satisfy the increasing Lipschitz nonlinearity assumption without the restriction rΨ(r)→∞ as r→∞ for L2(μ)-initial data. This paper also gets rid of the compact embedding assumption on the associated Gelfand triple. Examples of the negative definite self-adjoint operators include fractional powers of the Laplacian, i.e. L=−(−Δ)α,α∈(0,1], generalized Schro¨dinger operators, i.e. L=Δ+2∇ρρ⋅∇, and Laplacians on fractals.