Wavefront propagation for reaction-diffusion systems and backward SDEs

Abstract

We first show a large deviation principle for degenerate diffusion-transmutation processes and study the Riemannian metric associated with the action functional under a Hörmander-type assumption. Then we study the behavior of the solution $u^{\varepsilon}$ of a system of strongly coupled scaled KPP equations. Using backward stochastic differential equations and the theory of Hamilton-Jacobi equations, we show that, when the parabolic operator satisfies a Hörmander-type hypothesis or when the nonlinearity depends on the gradient, the wavefront location is given by the same formula as that in Freidlin and Lee or Barles, Evans and Souganidis. We obtain the exact logarithmic rates of convergence to the unstable equilibrium state in the general case and to the stable equilibrium state when the equations are uniformly positively coupled.