The Annals of Probability

Brownian motion with singular drift

Richard F. Bass and Zhen-Qing Chen

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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.

Article information

Ann. Probab., Volume 31, Number 2 (2003), 791-817.

First available in Project Euclid: 24 March 2003

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60J60: Diffusion processes [See also 58J65]

Stochastic differential equations weak solution diffusion Revuz measure perturbation weak convergeance resolvent strong Feller property singular drift


Bass, Richard F.; Chen, Zhen-Qing. Brownian motion with singular drift. Ann. Probab. 31 (2003), no. 2, 791--817. doi:10.1214/aop/1048516536.

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