Abstract. We describe an expansion of the solution of the wave equation in the De SitterSchwarzschild metric in terms of resonances. The main term in the expansion is due to a zero resonance. The error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an ε derivative loss in the angular directions.

Abstract. We study a semiclassical random walk with respect to a probability measure with a finite number n0 of wells. We show that the associated operator has exactly n0 exponentially close to 1 eigenvalues (in the semiclassical sense), and that the other are O(h) away from 1. We also give an asymptotic of these small eigenvalues. The key ingredient in our approach is a general factorization result of pseudodifferential operators, which allows us to use recent results on the Witten Laplacian.

We study the microlocal kernel of h-pseudodifferential operators Op h (p) − z, where z belongs to some neighborhood of size O(h) of a critical value of its principal symbol p 0 (x, ξ ). We suppose that this critical value corresponds to a hyperbolic fixed point of the Hamiltonian flow H p 0 . First we describe propagation of singularities at such a hyperbolic fixed point, both in the analytic and in the C ∞ category. In both cases, we show that the null solution is the only element of this microlocal kernel which vanishes on the stable incoming manifold, but for energies z in some discrete set. For energies z out of this set, we build the element of the microlocal kernel with given data on the incoming manifold. We describe completely the operator which associate the value of this null solution on the outgoing manifold to the initial data on the incoming one. In particular it appears to be a semiclassical Fourier integral operator associated to some natural canonical relation.