## Tohoku Mathematical Journal 2003 June SECOND SERIES VOL. 55, NO. 2

 Tohoku Math. J. 55 (2003), 255-269

Title LONG TIME BEHAVIOR OF THE TRANSITION PROBABILITY OF A RANDOM WALK WITH DRIFT ON AN ABELIAN COVERING GRAPH

Author Tomoyuki Shirai

(Received May 16, 2001, revised June 26, 2002)
Abstract. For a certain class of reversible random walks possibly with drift on an abelian covering graph of a finite graph, using the technique of twisted transition operator, we obtain the asymptotic behavior of the $n$-step transition probability $p_n(x,y)$ as $n \to \infty$ and give an expression of the constant which appears in the asymptotics.

2000 Mathematics Subject Classification. Primary 60J10; Secondary 60G50.