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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Nonattainability of a set by a diffusion process


Author: Avner Friedman
Journal: Trans. Amer. Math. Soc. 197 (1974), 245-271
MSC: Primary 60H10; Secondary 60J60
DOI: https://doi.org/10.1090/S0002-9947-1974-0346903-7
MathSciNet review: 0346903
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Abstract: Consider a system of n stochastic differential equations $ d\xi = b(\xi )dt + \sigma (\xi )dw$. Let M be a k-dimensional submanifold in $ {R^n},k \leq n - 1$. For $ x \in M$, denote by $ d(x)$ the rank of $ \sigma {\sigma ^ \ast }$ restricted to the linear space of all normals to M at x. It is proved that if $ d(x) \geq 2$ for all $ x \in M$, then $ \xi (t)$ does not hit M at finite time, given $ \xi (0) \notin M$, i.e., M is nonattainable. The cases $ d(x) \geq 1,d(x) \geq 0$ are also studied.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0346903-7
Keywords: Diffusion process, stochastic differential equations, Brownian motion, Dirichlet problem
Article copyright: © Copyright 1974 American Mathematical Society