The Annals of Probability

A Converse to a Theorem of P. Levy

P. J. Fitzsimmons

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By a well-known theorem of P. Levy, if $(X_t)$ is a standard Brownian motion on $\mathbb{R}$ with $X_0 = 0$ and if $H_t = \min_{u \leq t}X_u$, then $(Y_t) = (X_t - H_t)$ is Brownian motion with 0 as a reflecting lower boundary. More generally, if $X$ is allowed to have nonzero drift or a reflecting lower boundary at $A < 0$, then the process $Y = X - H$ is still a diffusion process. We prove the converse result: If $X$ is a diffusion on an interval $I \subset \mathbb{R}$ which contains 0 as an interior point, and if $(Y_t) = (X_t - H_t)$ is a time homogeneous strong Markov process (when $X_0 = 0$), then $X$ must be a Brownian motion on $I$ (with drift $\mu$, variance parameter $\sigma^2 > 0$, killing rate $c \geq 0$ and reflection at $\inf I$ in case $\inf I > -\infty$).

Article information

Ann. Probab., Volume 15, Number 4 (1987), 1515-1523.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60: Diffusion processes [See also 58J65]

Brownian motion one-dimensional diffusion generator scale function speed measure regenerative set


Fitzsimmons, P. J. A Converse to a Theorem of P. Levy. Ann. Probab. 15 (1987), no. 4, 1515--1523. doi:10.1214/aop/1176991990.

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