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Volume of sub-Riemannian balls and explosion of the heat kernel in the sense of Stein. (Volume de boules sous-riemanniennes et explosion du noyau de la chaleur au sens de Stein.) (French) Zbl 0764.60053
Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 426-447 (1989).
[For the entire collection see Zbl 0722.00030.]
Consider a hypoelliptic \(d\)-dimensional diffusion process \(X(t,x)\) and let \(p(t,x,y)\) be the density of its transition probability. One wants to study the behaviour of \(p(t,x,y)\) as the time \(t\) tends to 0 and \(y\) tends to \(x\). More precisely, \(X(t,x)\) can be associated to a function defined on the Cameron-Martin space, and one considers the images by this function of small balls in the Cameron-Martin space; then one looks for conditions on \(y\) ensuring that \(p(t,x,y)\) tends to infinity at a rate linked with the volumes of the above-mentioned images. This work generalizes the case where the drift coefficient is in the vector space generated by the diffusion coefficients and their Lie brackets of order 2. The study is based on a deterministic Malliavin matrix introduced by Bismut.

MSC:
60H07 Stochastic calculus of variations and the Malliavin calculus
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