Note on basic features of large time behaviour of heat kernels
  Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework includes Laplacians on manifolds, metric graphs and discrete graphs. 
Matthias Keller, Daniel Lenz, Hendrik Vogt, Rados{\l}aw Wojciechowski
math.FA math-ph math.MP