On admissible singular drifts of symmetric α‐stable process

Abstract

We consider the problem of existence of a (unique) weak solution to the SDE describing symmetric 𝛼-stable process with a locally unbounded drift 𝑏 ∶ ℝ 𝑑 → ℝ 𝑑 , 𝑑 ≥ 3, 1 < 𝛼 < 2. In this paper, 𝑏 belongs to the class of weakly form-bounded vector fields, the class providing the 𝐿 2 theory of the non-local operator behind the SDE, i.e. (−Δ) 𝛼 2 + 𝑏 ⋅ ∇. It contains as proper sub-classes other classes of singular vector fields studied in the literature in connection with this operator, such as the Kato class, the weak 𝐿 𝑑∕(𝛼−1) class and the Campanato-Morrey class (in general, such 𝑏 makes invalid the standard heat kernel estimates in terms of the heat kernel of the fractional Laplacian). We show that the operator (−Δ) 𝛼 2 + 𝑏 ⋅ ∇ with weakly form-bounded 𝑏 admits a realization as (minus) Feller generator, and that the probability measures determined by the Feller semigroup (uniquely in appropriate sense) admit description as weak solutions to the corresponding SDE. The proof is based on detailed regularity theory of

Methods of the Perturbation Theory for Fundamental Solutions to the Generalization of the Fractional Laplaciane

“…There exist extensive literature dedicated to the partial differential fractional Laplacian operator, general questions can be found in [4,[7][8][9], a wide review is presented in [4], an interesting approach to nonlinear heat equations in modulation spaces and Navier-Stokes equations can be found in [10]; some aspects of weights inequalities are described in [2,11]; fractional Laplacian is considered in [2,12], the list of selected works consists of 29 works . In the recent works [1,2], authors proved sharp two-sided estimations on the heat kernel of the fractional Laplacian with the perturbation of drift having critical-order singularity, also authors show that the operator with the heat kernel of the fractional Laplacian can be expressed as a Feller generator so that the probability measures uniquely determined by the Feller semigroup admits description as weak solutions to the corresponding SDE.…”

Feller generators with measurable lower order terms

“…In particular, drift(-type) perturbations of Lévy processes (i.e. σ ≡ 1) have been studied quite intensively, [8,27,28,37,46,58] to mention just a few classical and recent works.…”