Pejman Mahboubi | Searcharxiv

arXiv2014

Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion

Consider the stochastic partial differential equation $\partial_t u = Lu+σ(u)ξ$, where $ξ$ denotes space-time white noise and $L:=-(-Δ)^{α/2}$ denotes the fractional Laplace operator of index $α/2\in(\nicefrac12\,,1]$. We study the detailed behavior of the approximate spatial gradient $u_t(x)-u_t(x-\varepsilon)$ at fixed times $t>0$, as $\varepsilon\downarrow0$. We discuss a few applications of this work to the study of the sample functions of the solution to the KPZ equation as well.

math.PR↗

arXiv2011

Regularity of the density for a stochastic heat equation

We study the smoothness of the density of the solution to the nonlinear heat equation u_t=Lu(t,x)+σ(u(t,x))W on a torus with a periodic boundary condition, where L is the generator of a Levy process on the torus, and W is white noise. We use Malliavin calculus techniques to show that the law of the solution has a density with respect to the Lebesgue measure for all t >0 and x in R.

math.PR↗