Large deviations for invariant measures of stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term
The authors prove a large deviations principle, as \(\varepsilon\) tends to zero, for the family of invariant measures of stochastic reaction-diffusion systems of the form \[ \begin{aligned} \frac{\partial u_i}{\partial t}(t,\xi) = &\mathcal{A}_i u_i(t,\xi)+f_i(\xi,u_1(t,\xi),\dots,u_r(t,\xi))+\\ & + \varepsilon\sum_{j=1}^r g_{ij}(\xi,u_1(t,\xi),\dots,u_r(t,\xi))\mathcal{Q}_j \frac{\partial w_j}{\partial t}(t,\xi),\end{aligned} \] \(t\geq 0\), \(\xi\in \bar{\mathcal{O}}\), with some initial and boundary conditions. \(\mathcal{O}\) is a bounded open set in \(\mathbb{R}^d\), \(d\geq 1\), the differential operators \(\mathcal{A}_i\) are uniformly elliptic, \(\mathcal{Q}_j\) are bounded operators on \(L^2(\mathcal{O})\) and the random perturbations \({\partial w_j}/{\partial t}\) are independent cylindrical Wiener processes. The drift \(f\) is locally Lipschitz and has polynomial growth and \(g\) is globally Lipschitz and may be degenerate. Due to the hypothesis on \(f\), \(g\) and the general space dimension \(d\geq 1\), the authors make use of new techniques and generalise related results of \textit{R. B. Sowers} [Ann. Probab. 20, No.~1, 504--537 (1992; Zbl 0749.60059) and Probab. Theory Relat. Fields 92, No.~3, 393--421 (1992; Zbl 0767.60025)].
stochastic partial differential equations
scientific article; zbMATH DE number 2141606