Branching diffusion processes and systems of reaction-diffusion differential equations

R. G. Safaryan

Sbornik. Mathematics, Tome 62 (1989) no. 2, pp. 525-539

Résumé

Systems of reaction-diffusion differential equations of the form \begin{equation} \frac{\partial u_k}{\partial t}=L_ku_k+f_k(t,x,u),\quad x\in D\subseteq R^r,\ t>0,\ u=(u_1,\dots,u_n),\,1\leqslant k\leqslant n, \end{equation} are considered. Under certain special conditions on the nonlinear terms $f_k$ the solutions of the Cauchy problem and of mixed problems for systems of the type (1) have a representation in the form of an average value of a suitable functional of the sample paths of a corresponding branching process with diffusion. This representation is given, and it is used together with a direct probability investigation of the branching process with diffusion to obtain results on the behavior of solutions of certain problems with a small parameter for systems of the type (1). Bibliography: 12 titles.