For a second-order (hypoelliptic) operator $\mathscr{A} = A_0 + \frac{1}{2} \sum^m_{i=1} A_i$ on a $d$-dimensional manifold $M^d$, let $x_t$ be the diffusion governed by $\mathscr{A}$ and $\varphi (t)$ its associated deterministic control system. We investigate the relations between transience, recurrence and (finite) invariant measures for $x_t$ using the control theoretic decomposition of $M^d$ with respect to $\varphi (t)$. On the invariant control sets for $\varphi (t)$ we obtain the same classification for $x_t$ as is well known for the nondegenerate case, while outside these sets the diffusion $x_t$ is transient.
Publié le : 1987-04-14
Classification:
Degenerate diffusions,
hypoellipticity,
recurrence,
invariant measures,
geometric control theory,
60J60
@article{1176992166,
author = {Kliemann, Wolfgang},
title = {Recurrence and Invariant Measures for Degenerate Diffusions},
journal = {Ann. Probab.},
volume = {15},
number = {4},
year = {1987},
pages = { 690-707},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992166}
}
Kliemann, Wolfgang. Recurrence and Invariant Measures for Degenerate Diffusions. Ann. Probab., Tome 15 (1987) no. 4, pp. 690-707. http://gdmltest.u-ga.fr/item/1176992166/