Semiclassical Analysis and a New Result for Poisson-Lévy Excursion Measures

Abstract

The Poisson-Lévy excursion measure for the diffusion process with small noise satisfying the Ito equation $$ dX^{\varepsilon} = b(X^{\varepsilon}(t))dt + \sqrt\varepsilon\,dB(t) $$ is studied and the asymptotic behaviour in $\varepsilon$ is investigated. The leading order term is obtained exactly and it is shown that at an equilibrium point there are only two possible forms for this term - Levy or Hawkes-Truman. We also compute the next to leading order.