Probability Theory and Related Fields

, Volume 90, Issue 3, pp 377-402

First online:

Décroissance exponentielle du noyau de la chaleur sur la diagonale (II)

  • G. Ben ArousAffiliated withDépartement de mathématiques, Université Paris Sud
  • , R. LéandreAffiliated withDépartement de mathématiques, Université Louis Pasteur

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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).