## Semiclassical analysis and a new result for Poisson-Lévy excursion measures.(English)Zbl 1197.60051

Summary: The Poisson-Lévy excursion measure for the diffusion process with small noise satisfying the Itô 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 – Lévy or Hawkes-Truman. We also compute the next to leading order term and demonstrate the remarkable fact that it is identically zero.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 60J75 Jump processes (MSC2010)

### Keywords:

excursion measures; asymptotic expansions
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