Central limit theorems for nonlinear stochastic wave equations in dimension three
The author considers the nonlinear stochastic wave equation on \([0,T]\times\mathbb{R}^3\) given by\N\[\N\left(\frac{\partial^2}{\partial t^2}-\Delta\right)u(t,x)=\sigma(u(t,x))\dot{W}(t,x)\,,\N\]\Nwith \(u(0,x)=1\) and \(\frac{\partial u}{\partial t}(0,x)=0\), where \(\Delta\) is the Laplacian on \(\mathbb{R}^3\), the function \(\sigma:\mathbb{R}\to\mathbb{R}\) is Lipschitz continuous and continuously differentiable with \(\sigma(1)\not=0\), and \(\dot{W}(t,x)\) is centred Gaussian noise with covariance given by\N\[\N\mathbb{E}\left[\dot{W}(t,x)\dot{W}(s,y)\right]=\delta_0(t-s)\gamma(x-y)\,,\N\]\Nwhere \(\delta_0(\cdot)\) is the Dirac delta and \(\gamma\) is a spatial correlation function. The first main results of the present paper are central limit theorems (in Wasserstein distance) for the spatial averages\N\[\N\int_{B_R}\left(u(t,x)-1\right)\text{d}x\N\]\Nas \(R\to\infty\), where \(B_R\) is a ball in \(\mathbb{R}^3\) centred at the origin and with radius \(R\). Central limit theorems are established for the cases where either (i) \(\gamma\in L^1(\mathbb{R}^3)\) such that \(\gamma(x)>0\) for all \(x\), or (ii) \(\gamma(x)=|x|^{-\beta}\) for some \(\beta\in(0,2)\). These are established using the Malliavin-Stein method. The second main results of the paper are corresponding functional central limit theorems in each case.
scientific article; zbMATH DE number 7880265
scientific article