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Probability methods in potential theory. (English) Zbl 0677.60085

Potential theory, surveys and problems, Proc. Conf., Prague/Czech. 1987, Lect. Notes Math. 1344, 42-54 (1988).
[For the entire collection see Zbl 0642.00008.]
The paper gives a survey (without proofs) of probabilistic methods applied to the Dirichlet boundary value problem for the Schrödinger equation in a domain D in \({\mathbb{R}}^ d\). To state the main result let \((X_ t)_{t\geq 0}\) denote standard Brownian motion in \({\mathbb{R}}^ d\), and let \(\tau_ D\) denote the first exit time from D. Consider the Feynman-Kac multiplicative functional \[ e_ q(t):=\exp (\int^{t}_{0}q\circ X_ sds)\quad (t\geq 0), \] where q lies in a certain class of measurable functions containing \({\mathcal B}_ b({\mathbb{R}}^ d)\) (the bounded measurable functions). Let for \(\in {\mathcal B}_+(\partial D)\), \(u_ f\) denote the function \[ x\to P^ x[f\circ X_{\tau_ D}\cdot e_ q(\tau_ D);\quad \tau_ D<\infty]. \] Then \(u_ 1\) is called the gauge for (D,q), and (D,q) is called gaugeable if \(u_ 1\not\equiv \infty\) in D.
The main theorem states that provided (D,q) is gaugeable, \(u_ f\) is a continuous weak solution of the Dirichlet problem for the Schrödinger equation \((\Delta /2+q)\phi =0\) in D, which continuously approaches the boundary value f(x) at all regular boundary points \(x\in \partial D\) at which f is continuous.
The author gives several necessary and sufficient conditions for (D,q) to be gaugeable, presents some examples, and finally discusses the conditional gauge. For proofs the reader is referred to the literature (mainly papers by the author and coworkers). In addition there should be mentioned two references containing more general results, a paper by A. Boukricha, W. Hansen and H. Hueber [Expo. Math. 5, 97-135 (1987; Zbl 0659.35025)], and the Dissertation of K.-Th. Sturm, “Störung von Hunt-Prozessen durch signierte additive Funktionale”, Erlangen (1989).
Reviewer: J.Steffens

MSC:

60J45 Probabilistic potential theory
60J65 Brownian motion
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics