Uniqueness of stable processes with drift

Zhen Qing Chen; Longmin Wang

doi:10.1090/proc/12909

Uniqueness of stable processes with drift

Zhen Qing Chen
, Longmin Wang

University of Washington
Nankai University

Research output: Contribution to journal › Article › peer-review

31 Citations (Scopus)

Abstract

Suppose that d ≥ 1 and α ∈ (1, 2). Let L^b = −(−Δ)^α/2 + b · ∇, where b is an ℝ^d-valued measurable function on Rd belonging to a certain Kato class of the rotationally symmetric α-stable process Y on ℝ^d. We show that the martingale problem for (L^b, C^∞_c (ℝ^d)) has a unique solution for every starting point x ∈ ℝ^d. Furthermore, we show that the stochastic differential equation dXt = dY_t + b(X_t)dt with X₀ = x has a unique weak solution for every x ∈ ℝ^d.

Original language	English
Pages (from-to)	2661-2675
Number of pages	15
Journal	Proceedings of the American Mathematical Society
Volume	144
Issue number	6
DOIs	https://doi.org/10.1090/proc/12909
Publication status	Published - Jun 2016
Externally published	Yes

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abstract = "Suppose that d ≥ 1 and α ∈ (1, 2). Let Lb = −(−Δ)α/2 + b · ∇, where b is an ℝd-valued measurable function on Rd belonging to a certain Kato class of the rotationally symmetric α-stable process Y on ℝd. We show that the martingale problem for (Lb, C∞c (ℝd)) has a unique solution for every starting point x ∈ ℝd. Furthermore, we show that the stochastic differential equation dXt = dYt + b(Xt)dt with X0 = x has a unique weak solution for every x ∈ ℝd.",

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AB - Suppose that d ≥ 1 and α ∈ (1, 2). Let Lb = −(−Δ)α/2 + b · ∇, where b is an ℝd-valued measurable function on Rd belonging to a certain Kato class of the rotationally symmetric α-stable process Y on ℝd. We show that the martingale problem for (Lb, C∞c (ℝd)) has a unique solution for every starting point x ∈ ℝd. Furthermore, we show that the stochastic differential equation dXt = dYt + b(Xt)dt with X0 = x has a unique weak solution for every x ∈ ℝd.

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Uniqueness of stable processes with drift

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