Nonattainability of a set by a diffusion process

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.