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.
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