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

• DOI: 10.1007/s40072-014-0034-6
• Publication Date: 2014-08-13T22:23:29Z
• Authors: Le Chen; Robert C. Dalang
• Keywords: Hölder condition; Wedge (geometry); Heat equation; Absolute continuity

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 $��$ 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��lder-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��lder continuity of the solution is $\left(\frac��{2}\wedge \frac{1}{4}\right)-$ in time and $\left(��\wedge \frac{1}{2}\right)-$ in space, provided $��$ is absolutely continuous with an $��$-H��lder continuous density.<br/>33 pages, 0 figures<br/>