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On the large time behavior of heat kernels on Lie groups. (English) Zbl 1040.22004
Let \(G\) be a connected noncompact semisimple Lie group with finite center and let \(E_1,\dots ,E_n\) be left invariant vector fields on \(G\) that satisfy Hörmander’s condition; that is, they generate together with their successive Lie brackets \([E_{i_1},[ E_{i_2},[\dots, E_{i_{k-1}},E_{i_k}]\dots]]\) the tangent space \(T_xG\) at every point \(x\in G\). The corresponding sub-Laplacian is \(L=-(E_1^2+\dots ,+E_n^2)\). Denote by \(P_t(x,y)\) the associated heat kernel, that is the fundamental solution of the heat equation \((\frac{\partial}{\partial t}+L)u=0\). The authors study the large time behavior of the heat kernel \(P_t(x,y)\). The heat kernel \(P_t(x,y)\) is related to some other heat kernels defined by some subgroups \(A\) and \(N\) corresponding to some terms in the Iwasawa decomposition of the Lie algebra of \(G\). Next, some estimates for \(P_t(x,y)\) are obtained.

22E30 Analysis on real and complex Lie groups
43A90 Harmonic analysis and spherical functions
60B99 Probability theory on algebraic and topological structures
60J60 Diffusion processes
Full Text: DOI Euclid