Large time behavior of the heat kernel

In this paper we study the large time behavior of the (minimal) heat kernel kPM(x,y,t) of a general time-independent parabolic operator Lu=ut+P(x,∂x)u which is defined on a noncompact manifold M. More precisely, we prove thatlimt→∞ eλ0tkPM(x,y,t)always exists. Here λ0 is the generalized principal eigenvalue of the operator P in M.