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April 2003 Brownian motion with singular drift
Richard F. Bass, Zhen-Qing Chen
Ann. Probab. 31(2): 791-817 (April 2003). DOI: 10.1214/aop/1048516536

Abstract

We consider the stochastic differential equation \[ dX_t=dW_t+dA_t, \] where $W_t$ is $d$-dimensional Brownian motion with $d\geq 2$ and the $i$th component of $A_t$ is a process of bounded variation that stands in the same relationship to a measure $\pi^i$ as $\int_0^t f(X_s)\, ds$ does to the measure $f(x)\, dx$. We prove weak existence and uniqueness for the above stochastic differential equation when the measures $\pi^i$ are members of the Kato class $\K_{d-1}$. As a typical example, we obtain a Brownian motion that has upward drift when in certain fractal-like sets and show that such a process is unique in law.

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Richard F. Bass. Zhen-Qing Chen. "Brownian motion with singular drift." Ann. Probab. 31 (2) 791 - 817, April 2003. https://doi.org/10.1214/aop/1048516536

Information

Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1029.60044
MathSciNet: MR1964949
Digital Object Identifier: 10.1214/aop/1048516536

Subjects:
Primary: 60H10
Secondary: 47D07 , 60J35 , 60J60

Keywords: diffusion , perturbation , resolvent , Revuz measure , Singular drift , Stochastic differential equations , Strong Feller property , weak convergeance , Weak solution

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2003
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