XIII: 48, 557-569, LNM 721 (1979)
CARMONA, René
Processus de diffusion gouverné par la forme de Dirichlet de l'opérateur de Schrödinger (
Diffusion theory)
Standard conditions on the potential $V$ imply that the Schrödinger operator $-(1/2)ėlta+V$ (when suitably interpreted) is essentially self-adjoint on $L^2(
R^n,dx)$. Assume it has a ground state $\psi$. Then transferring everything on the Hilbert space $L^2(\mu)$ where $\mu$ has the density $\psi^2$ the operator becomes formally $Df=(-1/2)ėlta f + \nabla h.\nabla f$ where $h=-log\psi$. A problem which has aroused some excitement ( due in part to Nelson's ``stochastic mechanics'') was to construct true diffusions governed by this generator, whose meaning is not even clearly defined unless $\psi$ satisfies regularity conditions, unnatural in this problem. Here a reasonable positive answer is given
Comment: This problem, though difficult, is but the simplest case in Nelson's theory. In this seminar, see
1901,
1902,
2019. Seemingly definitive results on this subject are due to E.~Carlen,
Comm. Math. Phys.,
94, 1984. A recent reference is Aebi,
Schrödinger Diffusion Processes, Birkhäuser 1995
Keywords: Nelson's stochastic mechanics,
Schrödinger operatorsNature: Original
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