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 μ 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 exponents41 − in time and21 − in space remain valid. However, on compact sets that include t = 0, the Hölder continuity of the solution is (α2 ∧41 ) − in time and (α ∧21 ) − in space, provided μ is absolutely continuous with an α-Hölder continuous density.