Large deviation principle for catalytic processes associated with nonlinear catalytic noise equations (Q2735161)

In this conference article the author considers the following nonlinear reaction diffusion equation with catalytic noise [\textit{D. A. Dawson} and \textit{K. Fleischmann}, J. Theor. Probab. 10, No.~1, 213-276 (1997; Zbl 0877.60030)]: NEWLINE\[NEWLINE\frac{\partial}{\partial s}u(s,y)+\frac{\kappa}{2}\Delta u(s,y)+\psi(s,y)=X_t(\omega)u^2(s,y).\tag{1}NEWLINE\]NEWLINE Here the coefficient function \( X_t\) of the nonlinear term is a super-Brownian motion [\textit{D. A. Dawson}, in: École d'Été de probabilités de Saint-Flour XXI-1991. Lect. Notes Math. 1541, 1-260 (1993; Zbl 0799.60080)]. The main purpose is the existence of the catalytic superprocess associated with equation (1), an exponential moment formula, a probabilistic representation of the solution and a large deviation principle for the catalytic process. After some preliminaries the author provides in Section 3 the existence via implicit function theorem and the standard iteration scheme and the uniqueness of the solutions of equation (1). With a certain class of measure-valued continuous paths the author constructs in the next section the Brownian collision local time. Section 5 consists of the reconstruction of the Laplace-functional formalism with the \(\log\)-Laplace equation in order to derive a probabilistic representation of the solutions for (1). The main result of the article, the large deviation principle for the catalytic process, is found in Section 6. The key ingredient for the proof of LDP is the weak large deviation property established in Section 7 such that via exponential tightness [\textit{J.-D. Deuschel} and \textit{D. W. Stroock}, AMS (2001; to appear), Lemma 2.1.5] the LDP follows. In the final section the author discusses some scaling properties of the catalytic process.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00044].