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

By Le Chen, Robert C. Dalang. 2013

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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 $$\mu $$μ with, possibly, exponentially growing tails. We show how this regularity depends, in a neighborhood of $$t=0$$t=0, on the regularity of the initial condition. On compact sets in which $$t>0$$t>0, the classical Hölder-continuity exponents $$\frac{1}{4}-$$14- in time and $$\frac{1}{2}-$$12- in space remain valid. However, on compact sets that include $$t=0$$t=0, the Hölder continuity of the solution is $$\left( \frac{\alpha }{2}\wedge \frac{1}{4}\right) -$$α2∧14- in time and $$\left( \alpha \wedge \frac{1}{2}\right) -$$α∧12- in space, provided $$\mu $$μ is absolutely continuous with an $$\alpha $$α-Hölder continuous density.

Subfields: Computational Theory and Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cb2880c451a43f9e2141e6a6bc845a7a618ba5fc

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In Computational Theory and Mathematics

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Citations in the corpus

Global solutions to stochastic reaction–diffusion equations with super-linear drift and multiplicative noise (2019, background): “A related statement is given in [6], Theorem 3.”