Asymptotic behavior of the stochastic heat equation over large intervals

Abstract: We consider a nonlinear stochastic heat equation on $[0,T]\times [-L,L]$, driven by a space-time white noise $W$, with a given initial condition $u_0: \mathbb{R} \to \mathbb{R}$ and three different types of (vanishing) boundary conditions: Dirichlet, Mixed and Neumann. We prove that as $L\to\infty$, the random field solution at any space-time position converges in the $L^p(\Omega)$-norm ($p\ge 1$) to the solution of the stochastic heat equation on $\mathbb{R}$ (with the same initial condition $u_0$), and we determine the (near optimal) rate of convergence. The proof relies on estimates of differences between the corresponding Green's functions on $[-L, L]$ and the heat kernel on $\mathbb{R}$, and on a space-time version of a Gronwall-type lemma.