Small time asymptotics of Ornstein-Uhlenbeck densities in Hilbert spaces.

Jegaraj, Terence

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This text contains a slightly expanded version of my 6 hour mini-course at the PDE-meeting in Évian-les-Bains in June 2009. The first part gives some old and recent results on non-self-adjoint differential operators. The second part is devoted to recent results about Weyl distribution of eigenvalues of elliptic operators with small random perturbations. Part III, in collaboration with B. Helffer, gives explicit estimates in the Gearhardt-Prüss theorem for semi-groups.

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Acknowledgement of priority

Umberto Zannier (1996)

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We study the asymptotic behaviour of the semigroup of Markov operators generated by the equation $u_{t} + b u_{x} + c u = a \int_{0}^{a x} u (t, a x - y) μ (d y)$ . We prove that for a > 1 this semigroup is asymptotically stable. We show that for a ≤ 1 this semigroup, properly normalized, converges to a limit which depends only on a.

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