Uniform large deviations for a class of semilinear stochastic partial differential equations driven by a Brownian sheet
The author considers a family of parabolic semilinear SPDEs indexed by noise intensity \({\varepsilon\in(0,1]}\) \[ \frac{\partial u^\varepsilon}{\partial t}(t,x)=\frac{\partial^2u^\varepsilon}{\partial^2x}(t,x)+\sqrt{\varepsilon}\sigma(t,x,u^\varepsilon)\frac{\partial^2 W}{\partial t\partial x}(t,x)+\frac{\partial}{\partial x}g(t,x,u^\varepsilon)+f(t,x,u^\varepsilon) \] with conditions \[ u^\varepsilon(t,0)=u^\varepsilon(t,1)=0,\ \ t\in[0,T]; \ \ u^\varepsilon(0,x)=\eta(x)\in L^2([0,1]). \] Here \(f,g,\sigma\) are Borel functions defined on \({\mathbb{R}_+\times[0,1]\times\mathbb{R}}\) and satisfying some regularity conditions, and \(W\) is the Brownian sheet (see [\textit{A. Budhiraja} et al., Ann. Probab. 36, No. 4, 1390--1420 (2008; Zbl 1155.60024)]). The existence and uniqueness of a solution \(u^\varepsilon\) was proven by \textit{I. Gyöngy} [Stochastic Processes Appl. 73, No. 2, 271--299 (1998; Zbl 0942.60058)]. The author proves a uniform large deviation principle for the law of the solutions \({u^\varepsilon}.\) The uniformity is with respect to initial conditions that are bounded and do not necessarily belong to a compact set. The proof is based on an equivalent formulation to the Freidlin-Wentzell uniform large deviation principle (so called the equicontinuous uniform Laplace principle) due to \textit{M. Salins} [Probab. Surv. 16, 99--142 (2019; Zbl 1481.60068)].
stochastic partial differential equations
scientific article; zbMATH DE number 7634972
scientific article