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Démonstration probabiliste de certains développements asymptotiques quasi classiques. (Probabilistic proof of some quasi-classical asymptotic expansions). (French) Zbl 0564.35101

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.

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)
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