Doss, Halim Démonstration probabiliste de certains développements asymptotiques quasi classiques. (Probabilistic proof of some quasi-classical asymptotic expansions). (French) Zbl 0564.35101 Bull. Sci. Math., II. Sér. 109, 179-208 (1985). We study, by probabilistic methods, the behavior, when \(\epsilon\) tends to zero (asymptotic expansions of W.K.B. type) of the solution of the following Cauchy problem: \[ \partial \psi (t,x)/\partial t={\mathcal L}_{\epsilon} \psi (t,x);\quad \psi (0,x)=f(x)\exp (-s(x)/\epsilon^ 2), \] where: \({\mathcal L}_{\epsilon}=(\epsilon^ 2/2)\sum^{r}_{1}A^ 2_ i+A_ 0+(1/\epsilon^ 2)V\), V is a potential and \(A_ 0,A_ 1,...,A_ r\) are \(r+1\) regular vector fields. In the real case, the obtained estimates (without strict ellipticity assumptions) allow one to consider, in the same fashion, some situations where the differential operator \({\mathcal L}_{\epsilon}\) has complex analytic coefficients. Cited in 1 ReviewCited in 6 Documents MSC: 35R60 PDEs with randomness, stochastic partial differential equations 35C20 Asymptotic expansions of solutions to PDEs 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) Keywords:asymptotic expansions; W.K.B. type; Cauchy problem PDFBibTeX XMLCite \textit{H. Doss}, Bull. Sci. Math., II. Sér. 109, 179--208 (1985; Zbl 0564.35101)