Sample path properties of parabolic SPDEs with non constant coefficients
Robert C. Dalang, Marta Sanz-Solé
TL;DR
This work analyzes a stochastic parabolic PDE with non-constant coefficients driven by a Gaussian noise that is white in time and spatially colored. It develops a Dalang-type random-field framework, proves existence and uniqueness of the mild solution under precise assumptions on the Green's function and noise, and establishes optimal Hölder continuity in time and space via the factorization method. The results yield explicit Hölder exponents that depend on the spatial covariance through the spectral measure $\mu$ and accommodate kernels such as Riesz kernels, connecting Green's function bounds, stochastic integration, and regularity. Overall, the paper extends SPDE regularity theory to non-constant coefficient parabolic operators and provides a rigorous link between noise structure and sample-path properties with potential applications to spatially inhomogeneous systems.
Abstract
We consider an SPDE driven by a parabolic second order partial differential operator with a nonlinear random external forcing defined by a Gaussian noise that is white in time and has a spatially homogeneous covariance. We prove existence and uniqueness of a random field solution to this SPDE. Our main result concerns the space-time sample path regularity of its solution.