Hypercontractivity for space-time white noise driven SPDEs with reflection
A semilinear heat equation \[ \partial_tu=\frac{1}{2}\Delta u+b(x,u)+\dot W+\eta(dt,dx),\qquad u(0,x)=h(x) \] on \((0,1)\) with the Dirichlet boundary condition and the obstacle \(u\ge 0\) is considered. Here \(\dot W\) is a Gaussian space-time white noise, \(\eta\) is an unknown random adapted non-negative measure, \(b\) is a sum of a measurable function \(b_1(x,u)\), being a uniformly Lipschitz continuous function in the second variable, with \(b_1(x,0)=0\) and a continuous function \(b_2\) satisfying \[(u-v)(b_2(x,u)-b_2(x,v))\le -K|u-v|^2,\qquad u,v\in\mathbb R\] for some \(K>0\), and \(b_2(x,0)=0\). If \((u^h,\eta^h)\) is the unique solution with the initial condition \(h\in K_0\) then \(u^h\) defines a Markov semi-group on \(K_0=\{h\in L^2(0,1):\,h\ge 0\}\) and the author proves that there exist a unique invariant probability measure \(\mu\) and \(\tau>0\) such that \[ \|P_\tau\|_{\mathscr L(L^2(\mu),L^4(\mu))}=1, \] \(P_t\) is compact on \(L^2(\mu)\) for large \(t>0\), there exist positive constants \(C\), \(\kappa\), \(\lambda\) and \(T\) such that \[ \int_{K_0}(P_t\Phi)\log(P_t\Phi)\,d\mu\le Ce^{-\lambda t}\int_{K_0}\Phi\log(\Phi)\,d\mu \] holds for every \(t\ge 0\) and every non-negative bounded \(\Phi\) such that \(\|\Phi\|_{L^1(\mu)}=1\), \[ \left\|P_t\Phi-\int_{K_0}\Phi\,d\mu\right\|_{L^2(\mu)}\le e^{-\kappa t}\left\|\Phi-\int_{K_0}\Phi\,d\mu\right\|_{L^2(\mu)} \] holds for every \(t\ge 0\) and every \(\Phi\in L^2(\mu)\), \[ \|P^*_t\delta_h-P^*_t\delta_{\tilde h}\|_{TV}\le C\|h-\tilde h\|_{K_0}e^{-\kappa t} \] holds for every \(t\ge T\) and every \(h,\tilde h\in K_0\). Here \(\|\cdot\|_{TV}\) stands for the total-variation norm. As side results, the author proves a Harnack inequality for \((P_t)\) and the Gaussian concentration of \(\mu\), i.e. \(\exp\{\delta\|\cdot\|^2_{K_0}\}\in L^1(\mu)\) for some \(\delta>0\).
scientific article; zbMATH DE number 7041834
scientific article