Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions

Abstract

We study space-time regularity of the solution of the nonlinear stochastic heat equation in one spatial dimension driven by space-time white noise, with a rough initial condition. This initial condition is a locally finite measure $\mu$ with, possibly, exponentially growing tails. We show how this regularity depends, in a neighborhood of $t=0$, on the regularity of the initial condition. On compact sets in which $t>0$, the classical Hölder-continuity exponents $\frac{1}{4}-$ in time and $\frac{1}{2}-$ in space remain valid. However, on compact sets that include $t=0$, the Hölder continuity of the solution is $\left(\frac{\alpha }{2}\wedge \frac{1}{4}\right)-$ in time and $\left(\alpha \wedge \frac{1}{2}\right)-$ in space, provided $\mu$ is absolutely continuous with an $\alpha$-Hölder continuous density.

33 pages, 0 figures

References in corpus (1)

• Hitting Probabilities for Systems of Non-Linear Stochastic Heat Equations with Additive Noise

Cited by in corpus (1)

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