A singular large deviations phenomenon

Consider {X-t(epsilon): t greater than or equal to 0} (epsilon > 0), the solution starting from 0 of a stochastic differential equation, which is a small Brownian perturbation of the one-dimensional ordinary differential equation x(t)' = sgn(x(t))\x(t)\(gamma) (0 < <gamma> < 1). Denote by p(t)(epsilon)(x) the density of X-t(epsilon). We study the exponential decay of the density as epsilon --> 0. We prove that, for the points (t, x) lying between the extremal solutions of the ordinary differential equation, the rate of the convergence is different from the rate of convergence in large deviations theory (although respected for the points (t, x) which does not lie between the extremals). Proofs are based on probabilistic (large deviations theory) and analytic (viscosity solutions for Hamilton-Jacobi equations) tools. (C) 2001 Editions scientifiques et medicales Elsevier SAS.