In this paper, we study the motion of the free surface of a body of fluid over a variable bottom, in a long wave asymptotic regime. We assume that the bottom of the fluid region can be described by a stationary random process β(x, ω) whose variations take place on short length scales and which are decorrelated on the length scale of the long waves. This is a question of homogenization theory in the scaling regime for the Boussinesq and KdV equations.The analysis is performed from the point of view of perturbation theory for Hamiltonian PDEs with a small parameter, in the context of which we perform a careful analysis of the distributional convergence of stationary mixing random processes. We show in particular that the problem does not fully homogenize, and that the random effects are as important as dispersive and nonlinear phenomena in the scaling regime that is studied. Our principal result is the derivation of effective equations for surface water waves in the long wave small amplitude regime, and a consistency analysis of these equations, which are not necessarily Hamiltonian PDEs. In this analysis we compute the effects of random modulation of solutions, and give an explicit expression for the scattered component of the solution due to waves interacting with the random bottom. We show that the resulting influence of the random topography is expressed in terms of a canonical process, which is equivalent to a white noise through Donsker's invariance principle, with one free parameter being the variance of the random process β. This work is a reappraisal of the paper by Rosales & Papanicolaou [24] and its extension to general stationary mixing processes.2000 Mathematics Subject Classification. 76B15, 35Q53, 76M50, 60F17.

We study of the effect of weak noise on critical one dimensional maps; that is, maps with a renormalization theory.We establish a one dimensional central limit theorem for weak noises and obtain Berry-Esseen estimates for the rate of this convergence.We analyze in detail maps at the accumulation of period doubling and critical circle maps with golden mean rotation number. Using renormalization group methods, we derive scaling relations for several features of the effective noise after long times. We use these scaling relations to show that the central limit theorem for weak noise holds in both examples.We note that, for the results presented here, it is essential that the maps have parabolic behavior. They are false for hyperbolic orbits.where f is a map of a one dimensional space (R, T 1 or I = [−1, 1]) into itself, (ξ n ) is a sequence of real valued independent mean zero random variables of comparable sizes, and σ > 0 is a small parameter -called noise level-that controls the size of the noise.2000 Mathematics Subject Classification. Primary:37E20, 60F05 37C30 Secondary: 60B10 37H99 .

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