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XIII: 48, 557-569, LNM 721 (1979)
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 operators
Nature: Original
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XVI-S: 57, 165-207, LNM 921 (1982)
MEYER, Paul-André
Géométrie différentielle stochastique (bis) (Stochastic differential geometry)
A sequel to 1505. The main theme is that an ordinary differential equation has a non unique extension as a stochastic differential equation: besides the Stratonovich one, given by the ``transfer principle'', there are other possibilities: choosing among them requires some additional, connection-like, structure. The most striking application is the Dohrn-Guerra correction to the parallel transport along a semimartingale
Comment: For complements, see Émery 1658, Hakim-Dowek-Lépingle 2023, Émery's monography Stochastic Calculus in Manifolds (Springer, 1989) and article 2428, and Arnaudon-Thalmaier 3214
Keywords: Semimartingales in manifolds, Stochastic differential equations, Local characteristics, Nelson's stochastic mechanics, Transfer principle
Nature: Original
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