Random perturbations of reaction-diffusion equations: the quasideterministic approximation

Abstract:

Random fields ${u^\varepsilon }(t, x) = (u_1^\varepsilon (t, x), \ldots ,u_n^\varepsilon (t, x))$, defined as the solutions of a system of the PDE due. \[ \frac {{\partial u_k^\varepsilon }} {{\partial t}} = {L_k}u_k^\varepsilon + {f_k}(x; u_1^\varepsilon , \ldots ,u_n^\varepsilon ) + \varepsilon {\zeta _k}(t, x)\] are considered. Here ${L_k}$ are second-order linear elliptic operators, ${\zeta _k}$ are Gaussian white-noise fields, independent for different $k$, and $\varepsilon$ is a small parameter. The most attention is given to the problem of determining the behavior of the invariant measure ${\mu ^\varepsilon }$ of the Markov process $u_t^\varepsilon = (u_1^\varepsilon (t, \cdot ), \ldots ,u_n^\varepsilon (t, \cdot ))$ in the space of continuous functions as $\varepsilon \to 0$, and also of describing transitions of $u_t^\varepsilon$ between stable stationary solutions of nonperturbed systems of PDE. The behavior of ${\mu ^\varepsilon }$ and the transitions are defined by large deviations for the field ${u^\varepsilon }(t, x)$.