Transportation inequalities for stochastic differential equations of pure jumps
The law of stochastic diffusion equations with constant variance coefficient is known to satisfy a Poincaré variance inequality that shows the exponential decay of the associated semigroup under the invariant probability measure of the equation solution. This Poincaré inequality is a consequence of a transportation inequality which itself follows from the a logarithmic Sobolev inequality. An extension of these results has been given by the author and co-authors to the setting of a non-constant variance coefficient under the form of a transportation inequality that holds under a dissipativity condition. The goal of this paper is to extend such results to stochastic differential equations of pure jumps, for which the Poincaré inequality does not hold in general, by deriving \(W_1 H\) transportation inequalities under the \(L^1\) and \(L^\infty\)-Wasserstein distances. Concentration and deviation inequalities of Poisson type for the equation solution are obtained as a consequence. The proofs rely on a bound on the Laplace transform of the solution obtained by forward-backward stochastic calculus and the Clark representation formula of the Malliavin calculus for jump processes on the one hand, and on an equivalence result by \textit{N. Gozlan} and \textit{C. Léonard} [Probab. Theory Relat. Fields 139, No. 1--2, 235--283 (2007; Zbl 1126.60022)] between Laplace bounds and \(W_1 H\) transportation inequalities for the Wasserstein distance on the other hand.
scientific article; zbMATH DE number 5758857
scientific article