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Exponential decay of the heat kernel over the diagonal. II. (Décroissance exponentielle du noyau de la chaleur sur la diagonale. II.) (French) Zbl 0734.60027
[For part I see the preceding review, Zbl 0734.60026.]
We give some conditions for the heat kernel to have an asymptotic expansion in small time such that all coefficients vanish, although the phenomenon seems difficult to understand by large deviations theory. The fact that the leading term is not zero is strongly related to Bismut’s condition. These examples are related to the Varadhan estimates of the density of a dynamical system submitted to small random perturbations. To understand that type of asymptotic, one must modify the definition of the distance by adding the Bismut condition (unnoticed, but hidden, in classical cases).
Reviewer: G.Ben Arous

60F10 Large deviations
60J60 Diffusion processes
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