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October, 1987 A Converse to a Theorem of P. Levy
P. J. Fitzsimmons
Ann. Probab. 15(4): 1515-1523 (October, 1987). DOI: 10.1214/aop/1176991990


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 $\inf I > -\infty$).


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P. J. Fitzsimmons. "A Converse to a Theorem of P. Levy." Ann. Probab. 15 (4) 1515 - 1523, October, 1987.


Published: October, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0652.60085
MathSciNet: MR905345
Digital Object Identifier: 10.1214/aop/1176991990

Primary: 60J65
Secondary: 60J25 , 60J35 , 60J60

Keywords: Brownian motion , generator , One-dimensional diffusion , Regenerative set , scale function , Speed measure

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 4 • October, 1987
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