Intermittency and stochastic pseudo-differential equation with spatially inhomogeneous white noise
Consider the nonlinear stochastic partial differential equation in (1+1)-dimension \[ \bigl(\partial/\partial t+q(x,D_x)\bigr)u(t,x)=g(u(t,x))\bigl(\partial^2/\partial t\,\partial x\bigr)w_\rho(t,x) \] with initial condition \(u(0,x)=u_0(x)\), where \(q(x,D_x)\) is a pseudo-differential operator generating a stable-like process, and \(\partial^2w_\rho/\partial t\,\partial x\) is a space-time white noise with intensity measure \(dt\,\rho(dx)\). The authors first prove under some conditions the existence, uniqueness, and Hölder continuity of a mild solution. Then, and this is the main result, they study the intermittency of the solution. More precisely, by studying the moments of \(u(t,x)\), they obtain the existence of a smallest integer \(p\) such that \[ \inf_x\Bigl(\liminf_{t\to\infty}\frac{1}{t}\log\mathbb{E}\bigl[|u(t,x)|^p\bigr]\Bigr)0. \]
scientific article; zbMATH DE number 7031082
scientific article