SMALL STOCHASTIC PERTURBATION OF A ONE-DIMENSIONAL WAVE EQUATION by R´ emi L ´ EANDRE and Francesco RUSSO R.L. Universit´ e de Strasbourg D´ epartement de Math´ ematiques 7, rue Ren´ e Descartes F- 67084 Strasbourg F.R. Universit´ e de Provence UFR-MIM 3, Place Victor Hugo F-13331 Marseille Cedex 3 Universitaet Bielefeld Forschungszentrum BIBOS D-4800 Bielefeld 1 Part of the research of the second author has been done during his visit at the “ ´ Ecole Nationale Sup´ erieure des T´ el´ ecommunications” in Paris. During his stay, he was financially supported by a grant of the “Fonds National Suisse de la Recherche Scientifique”. Abstract : Carmona and Nualart have considered an ordinary wave equation which is perturbed by a non-linear forcing term. Under certain technical conditions, they have proved that the law of the solution at a fixed point has a smooth density. We study how the logarithm of this density behaves asymptotically if the stochastic perturbation is weak and we evaluate an expansion of the density around the unperturbed deterministic wave equation. 1 Introduction We consider the stochastic partial differential equation (1.1) X (t,x)= εa(X (t,x))ξ (t,x)+ b(X (t,x)) where is the d’Alembert operator ∂ 2 ∂t 2 − ∂ 2 ∂x 2 and ξ is a space-time white noise. The equation is an ordinary wave equation which is slightly perturbed by a random non-linear forcing ; ε is a small parameter, the “time variable” varies in [0, ∞[, and the “space variable” x varies in IR. The basic study of (1.1) was initiated by Walsh in [W] and carried out by Carmona and Nualart [CN 1; CN 2]. As in [CN 1], we suppose that a and b are regular. The physical model of (1.1) is a vibrating string. Among the first scientists who studied this problem there are Cabana and Orsingher [C 1; C 2; 0] who considered the linear case a =1,b = 0. Concerning the linear case, two recent preprints have to be mentioned : [DW] and [MM]. The first studies the L´ evy Markov properties of solutions when the forcing term is quite general, the second one considers stochasticity in the initial conditions. (See [H], [MaMi] for other approaches). In this paper, we examine the asymptotic behaviour for ε → 0 of the solution in X = X ε of (1.1) under deterministic initial conditions. 1