Phase transition for fractional stochastic partial differential equations in a bounded domain

Abstract

In this dissertation, we study several stochastic partial differential equations (SPDEs) in the open and bounded domain $D$, subset of $\mathbb{R}^d$ for $d\geq 1$, driven by a multiplicative noise. We are interested in bounds and asymptotic properties of the random field solution.

We study the nonlinear stochastic fractional heat equation driven by three types of noise. Existence and uniqueness of the solution is proved using a Picard iteration scheme. Upper and lower bounds on all $p^{\text{th}}$ moments, for $p\geq 2$, of the solution are obtained when the noise is spatially homogeneous (or spatially colored) with the space correlation function given by the Riesz kernel and when the noise is space-time homogeneous, with the time correlation function given by the fractional Brownian motion (fBm) while the space correlation function is given by the Riesz kernel in space. We also show that under exterior boundary conditions, in the long run, the $p^{\text{th}}$-moment of the solution grows exponentially fast for large values of the noise level. However, for small values of the noise level, we observe eventually an exponential decay of the $p^{\text{th}}$-moment of this solution.