1. chen.dalang:14:holder-continuity#
Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions
Le Chen and Robert C. Dalang
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.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear stochastic heat equation, rough initial data, sample path Hölder continuity, moments of increments.
[CD14a] Chen, Le & Dalang, Robert C. (2014) ‘Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions’, Stoch. Partial Differ. Equ. Anal. Comput. 2, 316–352. <https://doi.org/10.1007/s40072-014-0034-6>
@article{chen.dalang:14:holder-continuity, author = {Chen, Le and Dalang, Robert C.}, title = {H\"{o}lder-continuity for the nonlinear stochastic heat equation with rough initial conditions}, journal = {Stoch. Partial Differ. Equ. Anal. Comput.}, fjournal = {Stochastic Partial Differential Equations. Analysis and Computations}, volume = {2}, year = {2014}, number = {3}, pages = {316--352}, issn = {2194-0401}, mrclass = {60H15 (35B65 35K15 35R06 35R60 60G60)}, mrnumber = {3255231}, mrreviewer = {Anna Karczewska}, doi = {10.1007/s40072-014-0034-6}, url = {https://doi.org/10.1007/s40072-014-0034-6} }
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