No abstract

We provide here some sharp Schauder estimates for degenerate PDEs of Kolmogorov type when the coefficients lie in some suitable anisotropic Hölder spaces and the first order term is non-linear and unbounded. We proceed through a perturbative approach based on forward parametrix expansions.Due to the low regularizing properties of the degenerate variables, for the procedure to work, we heavily exploit duality results between appropriate Besov spaces.Our method can be seen as constructive and provides, even in the non-degenerate case, an alternative approach to Schauder estimates.

For an ergodic Brownian diffusion with invariant measure ν, we consider a sequence of empirical distributions (νn) n≥1 associated with an approximation scheme with decreasing time step (γn) n≥1 along an adapted regular enough class of test functions f such that f −ν(f ) is a coboundary of the infinitesimal generator A. Denote by σ the diffusion coefficient and ϕ the solution of the Poisson equation Aϕ = f − ν(f ). When the square norm |σ * ∇ϕ| 2 lies in the same coboundary class as f , we establish sharp non-asymptotic concentration bounds for suitable normalizations of νn(f ) − ν(f ). Our bounds are optimal in the sense that they match the asymptotic limit obtained by Lamberton and Pagès in [LP02], for a certain large deviation regime. In particular, this allows us to derive sharp non-asymptotic confidence intervals. Eventually, we are able to handle, up to an additional constraint on the time steps, Lipschitz sources f in an appropriate non-degenerate setting.

In this article, we approximate the invariant distribution  $\nu$ of an ergodic Jump Diffusion driven by the sum of a Brownian  motion and a Compound Poisson process with sub-Gaussian jumps.  We first construct  an Euler discretization scheme with decreasing time steps. This scheme is similar to those introduced by Lamberton and Pagès in \cite{lamb:page:02} for a Brownian diffusion and extended by Panloup in \cite{panl:08:AAP} to a diffusion with  Lévy jumps. We obtain a non-asymptotic \textit{quasi} Gaussian (asymptotically Gaussian) concentration bound for the difference between the invariant distribution and the empirical distribution computed with the scheme of decreasing time step along  appropriate test functions $f$ such that $f-\nu(f)$ is is a coboundary of the infinitesimal generator.