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On a probabilistic approach to a problem of semi-classical analysis. (Sur une approche probabiliste d’un problème d’analyse semi-classique.) (French. English summary) Zbl 1076.35019
Summary: We use probabilistic methods to study, under certain conditions, the asymptotic behavior (namely, the logarithmic equivalents), when \(\varepsilon\) goes to zero, of the solution \((\psi_\varepsilon(t, x))\) of the following Cauchy problem: \[ {\partial\over\partial t}\psi_\varepsilon(t, x)={\mathcal L}_\varepsilon\psi_\varepsilon(t, x)+{1\over\varepsilon^2} V(x)\psi_\varepsilon(t, x), \] \[ \psi_\varepsilon(0, x)= g(x)\exp\Biggl({-s(x)\over\varepsilon^2}\Biggr),\quad t\geq 0,\quad x\in\mathbb{R}^n, \] where \(V\) is a potential and \({\mathcal L}_\varepsilon\) is the infinitesimal generator of a perturbed diffusion process: \[ {\mathcal L}_\varepsilon= {\varepsilon^2\over 2} \sum^r_{i=1} A^2_i+ A_0+ {1\over 2} \sum^l_{j=1} \widetilde A^2_j \] with \(A_0, A_1,\dots, A_r\) are \(\widetilde A_1,\dots,\widetilde A_l\), \(r+ l+1\) vector fields on \(\mathbb{R}^n\) smooth enough.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
60J60 Diffusion processes
47D07 Markov semigroups and applications to diffusion processes
35K15 Initial value problems for second-order parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
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