Stability of solutions to stochastic partial differential equations

Abstract

We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift given as the subdifferential of a convex function and prove continuous dependence of the solutions with regard to random Mosco convergence of the convex potentials. In particular, we identify the concept of stochastic variational inequalities (SVI) as a well-suited framework to study such stability properties. The generality of the developed framework is then laid out by deducing Trotter type and homogenization results for stochastic fast diffusion and stochastic singular p-Laplace equations. In addition, we provide an SVI treatment for stochastic nonlocal p-Laplace equations and prove their convergence to the respective local models.

39 pages, to appear in: J. Differential Equations

References in corpus (2)

• Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations
• Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations

Cited by in corpus (10)

• Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations
• Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise
• Convergent numerical approximation of the stochastic total variation flow
• Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions
• Non-local SPDE limits of spatially-correlated-noise driven spin systems derived to sample a canonical distribution
• Well-posedness of SVI solutions to singular-degenerate stochastic porous media equations arising in self-organised criticality
• The Energy-Dissipation Principle for stochastic parabolic equations
• Stability and moment estimates for the stochastic singular $\Phi$-Laplace equation
• Singular limits for stochastic equations
• Improved regularity for the stochastic fast diffusion equation