Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values

Abstract

In this article we develop a framework for studying parabolic semilinear stochastic evolution equations (SEEs) with singularities in the initial condition and singularities at the initial time of the time-dependent coefficients of the considered SEE. We use this framework to establish existence, uniqueness, and regularity results for mild solutions of parabolic semilinear SEEs with singularities at the initial time. We also provide several counterexample SEEs that illustrate the optimality of our results.

References in corpus (3)

• Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
• Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients
• Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions

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• Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations
• Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
• Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients
• Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions
• Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
• Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces
• Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces
• On the differentiability of solutions of stochastic evolution equations with respect to their initial values