H\"older-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 H\"older-continuity exponents $\frac{1}{4}-$ in time and $\frac{1}{2}-$ in space remain valid. However, on compact sets that include $t=0$, the H\"older 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 $\alpha$-H\"older continuous density.