On the support of solution of a stochastic differential equation without drift (Q1323254)

Let \(M\) be a \(\sigma\)-compact \(C^ \infty\)-manifold and \(V_ 0, V_ 1, \dots, V_ r\) be \(C^ \infty\)-vector fields on \(M\). The author considers the stochastic differential equation of Stratonovich type \[ dx_ t=\sum_{k=1}^ r V_ k (x_ t)\circ dw^ k (t)+ V_ 0 (x_ t)dt, \qquad x_ 0=x, \quad 0\leq t<\tau, \] where \((w^ 1,\dots, w^ r)\) is a Wiener process, \(x\in M\) is fixed, and \(\tau\) is the explosion time. Let \(E\) be the orbit of \(\{V_ 0, \dots, V_ r\}\) with its differential structure as defined by \textit{H. J. Sussmann} [Trans. Am. Math. Soc. 180, 171-188 (1973; Zbl 0274.58002)]. Let \(\text{vol}_ E (\cdot)\) denote the Riemannian volume on \(E\). The main result of the paper consists in proving that if \(V_ 0= 0\), then for any compact subset \(K\) of \(E\) there exists \(c_ K>0\) such that \(P(\tau>1, x_ 1\in B)\geq c_ K \text{vol}_ E(B)\), for any Borel \(B\subset K\). This theorem implies that if \(p\) denotes the density of the absolutely continuous part of the distribution of \(x_ 1\) with respect to the volume on \(E\), then \(\text{ess inf}_{y\in K} p(y)>0\) for any compact \(K\subset E\). On the other hand, the distribution of \(x_ 1\) need not be absolutely continuous with respect to the volume on \(E\).