Optimal regularity of SPDEs with additive noise

Abstract

The sample-function regularity of the random-field solution to a stochastic partial differential equation (SPDE) depends naturally on the roughness of the external noise, as well as on the properties of the underlying integro-differential operator that is used to define the equation. In this paper, we consider parabolic and hyperbolic SPDEs on $0,\infty )\times {\mathbb{R}}^d$ of the form ${\partial }_{t}u=Lu+g(u)+\dot{F}\text{and}{\partial }_t^{2}u=Lu+c+\dot{F},$ with suitable initial data, forced with a space-time homogeneous Gaussian noise $\dot{F}$ that is white in its time variable and correlated in its space variable, and driven by the generator $L$ of a genuinely $d$-dimensional Lévy process $X$. We find optimal Hölder conditions for the respective random-field solutions to these SPDEs. Our conditions are stated in terms of indices that describe thresholds on the integrability of some functionals of the characteristic exponent of the process $X$ with respect to the spectral measure of the spatial covariance of $\dot{F}$. Those indices are suggested by references [45, 46] on the particular case that $L$ is the Laplace operator on ${\mathbb{R}}^d$.

27 pages, revised version

References in corpus (2)

• Gaussian fluctuations for the stochastic heat equation with colored noise
• Absolute continuity for SPDEs with irregular fundamental solution