Asymptotic behavior of the invariant measure for a diffusion related to an $NA$ group
Volume 104 / 2006
Colloquium Mathematicum 104 (2006), 285-309
MSC: 22E25, 22E30, 31B25, 43A80, 60J60.
DOI: 10.4064/cm104-2-6
Abstract
On a Lie group $NA$ that is a split extension of a nilpotent Lie group $N$ by a one-parameter group of automorphisms $A$, the heat semigroup $\mu _t$ generated by a second order subelliptic left-invariant operator $\sum _{j=0}^mY_j +Y$ is considered. Under natural conditions there is a $\check \mu _t$-invariant measure $m$ on $N$, i.e. $\check \mu _t*m=m$. Precise asymptotics of $m$ at infinity is given for a large class of operators with $Y_0,\dots,Y_m$ generating the Lie algebra of $S$.
