A large deviation principle and wave front propagation for a reaction-diffusion equation

In this thesis we consider an asymptotic problem for the propagation of wave front for a reaction-diffusion equation depending on a small parameter ε > 0, as well as some generalizations. First we analyze the asymptotic behavior as ε ↓ 0 of the solution of a initial-boundary value problem in the region {(t, x, y): t > 0, x element-of IR, |y| ≤ b} formulated by means of a reaction-diffusion equation. This differential equation is characterized by a fast diffusion (coefficient of order 1/ε) in y-direction, a slow diffusion (coefficient of order ε) in x-direction, and a nonlinear term. In this analysis we use the same approach as in Freidlin (285a, 1991) where we study the generalized KPP (Kolomogorov-Petrovskii-Piskunov) equation. The main tools are the Feynman-Kac formula and a Large Deviation Principle for a class of random processes. The main result is the explicit description of the limit wave front as ε ↓ 0 for the solution of the problem under consideration. Secondly, some generalizations of the above mixed problem are considered. The motion in y-direction (fast motion) is described by a more geral Markov process in a compact subset D of IRtau. This process satisfies some suitable conditions formulated in terms of the semigroup of bounded operators associated with its transition probability function. The motion in x-direction (slow motion) can be a locally infinitely divisible process in IR, with frequent small jumps. Under certain assumptions, this class of processes obeys a Large Deviation Principle. Using the new fast and slow motions, a Cauchy problem analogous to the above mixed problem is studied. Again, in the analysis of the limit behavior of the solution of this problem, the Feynman-Kac formula and probabilities of Large Deviations for certain class of random processes are used