Section: New Results

Stochastic PDE

Linearized wave turbulence convergence results for three-wave systems

In [30], E. Faou considers stochastic and deterministic three-wave semi-linear systems with bounded and almost continuous set of frequencies. Such systems can be obtained by considering nonlinear lattice dynamics or truncated partial differential equations on large periodic domains. We assume that the nonlinearity is small and that the noise is small or void and acting only in the angles of the Fourier modes (random phase forcing). We consider random initial data and assume that these systems possess natural invariant distributions corresponding to some Rayleigh-Jeans stationary solutions of the wave kinetic equation appearing in wave turbulence theory. We consider random initial modes drawn with probability laws that are perturbations of theses invariant distributions. In the stochastic case, we prove that in the asymptotic limit (small nonlinearity, continuous set of frequency and small noise), the renormalized fluctuations of the amplitudes of the Fourier modes converge in a weak sense towards the solution of the linearized wave kinetic equation around these Rayleigh-Jeans spectra. Moreover, we show that in absence of noise, the deterministic equation with the same random initial condition satisfies a generic Birkhoff reduction in a probabilistic sense, without kinetic description at least in some regime of parameters.

Large deviations for the dynamic ${\Phi }_d^{2n}$ model

In [5], we are dealing with the validity of a large deviation principle for a class of reaction-diffusion equations with polynomial non-linearity, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $\rho$ and $\delta \left(\rho \right)$, respectively, with $0<\rho ,\delta \left(\rho \right)<<1$. We prove that, under the assumption that $\rho$ and $\delta \left(\rho \right)$ satisfy a suitable scaling limit, a large deviation principle holds in the space of continuous trajectories with values both in the space of square-integrable functions and in Sobolev spaces of negative exponent. Our result is valid, without any restriction on the degree of the polynomial nor on the space dimension.

Kolmogorov equations and weak order analysis for SPDES with nonlinear diffusion coefficient

In [3], we provide new regularity results for the solutions of the Kolmogorov equation associated to a SPDE with nonlinear diffusion coefficients and a Burgers type nonlinearity. This generalizes previous results in the simpler cases of additive or affine noise. The basic tool is a discrete version of a two sided stochastic integral which allows a new formulation for the derivatives of these solutions. We show that this can be used to generalize the weak order analysis performed previously. The tools we develop are very general and can be used to study many other examples of applications.

Large deviations for the two-dimensional stochastic Navier-Stokes equation with vanishing noise correlation

In [6], we are dealing with the validity of a large deviation principle for the two-dimensional Navier-Stokes equation, with periodic boundary conditions, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $\epsilon$ and $\delta \left(\epsilon \right)$, respectively, with $0<\epsilon ,\delta \left(\epsilon \right)<<1$. Depending on the relationship between $\epsilon$ and $\delta \left(\epsilon \right)$ we will prove the validity of the large deviation principle in different functional spaces.

The Schrödinger equation with spatial white noise potential

In [13], we consider the linear and nonlinear Schrödinger equation with a spatial white noise as a potential in dimension 2. We prove existence and uniqueness of solutions thanks to a change of unknown originally used in a paper by Hairer and Labbé (2015) and conserved quantities.