Large time behavior of the heat kernel
  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 $$\lim_{t\to\infty} e^{\lambda_0 t}k_P^{M}(x,y,t)$$ always exists. Here $\lambda_0$ is the generalized principal eigenvalue of the operator $P$ in $M$. 
Yehuda Pinchover