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

Davies, Ian M.

Electron. J. Probab. 13, 1283-1306 (2008).

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