Borel summation of the small time expansion of some SDE’s driven by Gaussian white noise

Abstract

We consider stochastic differential equations with a drift term of gradient type and driven by Gaussian white noise on ${\mathbb{R}}^{\mathit{d}}$ . Particular attention is given to the kernel ${\mathit{p}}_{\mathit{t}}$ , $\mathit{t}\mathrm{>}0$ of the transition semigroup associated with the solution process.

Under some rather strong regularity and growth assumptions on the coefficients, we adapt previous work by Thierry Hargé on Schrödinger operators and prove that the small time asymptotic expansion of ${\mathit{p}}_{\mathit{t}}$ , $\mathit{t}\mathrm{>}0$ is Borel summable.

We also briefly indicate some extensions and applications.