Stable process with singular drift

Abstract

Suppose that d ≥ 2 and α ∈ (1, 2). Let µ = (µ 1 , . . . , µ d ) be such that each µ i is a signed measure on R d belonging to the Kato class K d,α−1 . In this paper, we consider the stochastic differential equationwhere S t is a symmetric α-stable process on R d and for each j = 1, . . . , d, the jth component A j t of A t is a continuous additive functional of finite variation with respect to X whose Revuz measure is µ j . The unique solution for the above stochastic differential equation is called an α-stable process with drift µ. We prove the existence and uniqueness, in the weak sense, of such an α-stable process with drift µ and establish sharp two-sided heat kernel estimates for such a process.

Brownian Motion with Singular Time-Dependent Drift

“…Sharp two-sided Dirichlet heat kernel estimates for such a drifted α-stable process were derived in [3]. As shown in [13], similar results hold when the drift is not a vector-valued function but a signed measure belonging to K d,α−1 . We refer also to [17,22] where the case of a more general α-stable process was discussed.…”

Quasi-stationarity and quasi-ergodicity of general Markov processes

“…Let Y = {Y t : t ≥ 0} is an α-stable with drift µ in R d , see [16]. It is known (see, again, [16]) that Y admits a continuous transition density q(t, x, y) with respect to the Lebesgue measure and that there exist c 1 , c 2 > 0 such that…”

Strong existence and uniqueness for stable stochastic differential equations with distributional drift

“…Weak existence and uniqueness results for (1.2) have been obtained in [15], [16], [14], and [34]. The question of weak uniqueness and existence for (1.1) was studied in Kulik [24] for b measurable and locally bounded and in Song [22], Chen, Wang [9] for b from a certain Kato class. Some of these results are also valid for the case when the SDEs have a non-trivial diffusion coefficient.…”