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We prove that the exit times of diffusion processes from a bounded open set Ω almost surely belong to the Besov space $B_{p,q}^{\alpha }\left(\Omega \right)$ provided that pα < 1 and 1 ≤ q < ∞.

In this article we prove the pathwise uniqueness for stochastic differential equations in ${ℝ}^d$ with time-dependent Sobolev drifts, and driven by symmetric $\alpha$-stable processes provided that $\alpha \in (1,2)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $\alpha \in (\frac{2d}{d+1},2)$. Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

We construct the heat kernel of the 1/2-order Laplacian perturbed by a first-order gradient term in Hölder spaces and a zero-order potential term in a generalized Kato class, and obtain sharp two-sided estimates as well as a gradient estimate of the heat kernel, where the proof of the lower bound is based on a probabilistic approach.