A central limit theorem for the stochastic heat equation (Q2229683)

The authors consider the stochastic heat equation \(\partial u/\partial t=0.5\Delta u+\sigma (u)\dot{W}\). Under mild conditions this equation has a solution \(u(t,x)\) so that \(E\left\vert u(t,x)\right\vert^{2}<\infty\). In this paper, the authors are interested in the asymptotic behavior as \(R\rightarrow \infty\) of the integral \(X(R,t)=\int_{-R}^{R}u(t,x)dx\). Let \(\mu (R,t)=EX(R,t)\) and \(\sigma^{2}(R,t)=\mathrm{Var}(X(R,t))\), and let \(F(R,t)=(X(R,t)-\mu (R,t))/\sigma (R,t)\). In the main result of the paper, the authors prove that \(d_{TV}(F(R,t),Z)\leq C(t)R^{-1/2}\), where \(Z\sim N(0,1)\), \(d_{TV}\) is the total variation distance and \(C(t)\) depends only on \(t\). The authors use Malliavin calculus and Stein's method.