The First eigenvalue determine the behavior of the heat equation and then of the brownian motion, see reference therein http://link.springer.com/article/10.1007%2FBF02560531

The First eigenvalue determine the behavior of the heat equation and then of the brownian motion, see reference therein http://www.actamath.com/Jwk_sxxb_en/EN/article/downloadArticleFile.do?attachType=PDF&id=6458

Edit: The new link is Free? Else I can send you the pdf.

Of course there is the marvelous article of Kac, where the role of the eigenvalue in the diffusion process is make so clearly https://www.math.ucdavis.edu/~saito/courses/ACHA.READ.F03/kac-drum.pdf