Large time behavior of the heat kernel

Abstract

In this paper we study the large time behavior of the (minimal) heat kernel $k_P^M(x,y,t)$ of a general time independent parabolic operator $L=u_t+P(x,{\partial }_x)$ which is defined on a noncompact manifold $M$. More precisely, we prove that $$\underset{t\to \infty }{\lim }{e}^{{\lambda }_{0}t}{k}_P^M(x,y,t)$$ always exists. Here ${\lambda }_0$ is the generalized principal eigenvalue of the operator $P$ in $M$.

15 pages