Abstract
A process $(X_t)$ is equivalent to an Ornstein-Uhlenbeck process if and only if $e^{-\lambda t}f(X_t)$ is a martingale for every $f \geq 0$ on $\mathbb{R}^d$ such that $\Delta f(x) - \langle x, \nabla f(x)\rangle = \lambda f(x)$.
Citation
J. C. Taylor. "The Minimal Eigenfunctions Characterize the Ornstein-Uhlenbeck Process." Ann. Probab. 17 (3) 1055 - 1062, July, 1989. https://doi.org/10.1214/aop/1176991256
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