Quantitative heat kernel estimates for diffusions with distributional drift

Abstract

We consider the stochastic differential equation on R d given bywhere B is a Brownian motion and b is considered to be a distribution of regularity > − 1 2 . We show that the martingale solution of the SDE has a transition kernel Γ t and prove upper and lower heat kernel bounds for Γ t with explicit dependence on t and the norm of b.

Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q -- L $ρ$ Drift Coefficient and Additive Noise

“…Let us mention that similar Gaussian estimates were obtained for even rougher drifts in Besov spaces with negative regularity index by Perkowski and Van Zuijlen in [41] using Littlewood-Paley decompositions for the drift. For time-homogeneous drifts, heat kernel estimates of the same type were obtained by Zhang and Zhao [49], see Theorem 5.1 therein, under the condition…”

Analysis of the Anderson operator

“…We justify Proposition 25 by proving an r-uniform similar estimate for the heat kernel p r (t, x, y) of H r . The continuity of p − p ∆ as a function of ξ in item (i) of Theorem 17 allows us to pass to the limit in the corresponding inequalities for each fixed positive t. For a fixed positive r we use the idea of conjugating the operator to a simpler operator for which one can use well-known heat kernel bounds with good control on its parameters as functions of r. The reader will find in Section 1.1 of [34] more references on works about diffusions with distributional drifts.…”

2D random magnetic Laplacian with white noise magnetic field

“…For example, the eigenvalues are random variables and a direct application of the method used in [17] would give tail estimates. One could also consider the martingale problem associated to rough differential equations (RDEs) in the case of a time-independant distributional drift, see [18] and the references therein.…”