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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.

##### MSC:
 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
##### Keywords:
Lie groups; heat kernels; sub-Laplacian; homogenization theory
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