TY - JOUR
T1 - Stable process with singular drift
AU - Kim, Panki
AU - Song, Renming
N1 - Funding Information:
Panki Kim’s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) ( 2013004822 ). Renming Song’s research was supported in part by a grant from the Simons Foundation ( 208236 ).
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2014/7
Y1 - 2014/7
N2 - Suppose that d≤2 and αε(1,2). Let μ=( μ1, ⋯, μd) be such that each μi is a signed measure on Rd belonging to the Kato class Kd,α-1. In this paper, we consider the stochastic differential equation dx t=dSt+dAt, where St is a symmetric α-stable process on Rd and for each j=1,⋯,d, the jth component Atj of At is a continuous additive functional of finite variation with respect to X whose Revuz measure is μj. 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.
AB - Suppose that d≤2 and αε(1,2). Let μ=( μ1, ⋯, μd) be such that each μi is a signed measure on Rd belonging to the Kato class Kd,α-1. In this paper, we consider the stochastic differential equation dx t=dSt+dAt, where St is a symmetric α-stable process on Rd and for each j=1,⋯,d, the jth component Atj of At is a continuous additive functional of finite variation with respect to X whose Revuz measure is μj. 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.
KW - Boundary Harnack inequality
KW - Exit time
KW - Gradient operator
KW - Green function
KW - Heat kernel
KW - Kato class
KW - Lévy system
KW - Symmetric α-stable process
KW - Transition density
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U2 - 10.1016/j.spa.2014.03.006
DO - 10.1016/j.spa.2014.03.006
M3 - Article
AN - SCOPUS:84897061402
SN - 0304-4149
VL - 124
SP - 2479
EP - 2516
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
IS - 7
ER -