## Probabilistic approach to boundary value problems for Schrödinger’s equation.(English)Zbl 0577.60066

The paper contains a short survey of recent results on the probabilistic representation of the boundary problem solutions for the Schrödinger equation (1) $$(\Delta /2+q(x))\phi =0$$, $$x\in {\mathcal D}$$, where $${\mathcal D}$$ is a bounded domain in $${\mathbb{R}}^ d$$, $$d\geq 2$$, and (2) $$\phi =f|_{x\in \partial {\mathcal D}}$$ or (3) $$(\partial \phi /\partial n)|_{x\in \partial {\mathcal D}}=f$$. The gauge for the problem (1)-(2) is the function $u(x)=E^ x\{\exp \int_{0}^{\tau_{{\mathcal D}}}q(B_ t)dt\}$ where $$B_ t$$ is the Brownian motion in $${\mathbb{R}}^ d$$, $$\tau_{{\mathcal D}}$$ is the first exit time, $$E^ x$$ is the mathematical expectation starting at x.
The gauge theorem: If u is not identically $$+\infty$$ in $${\mathcal D}$$ then it is bounded there. In this case the unique solution of the Dirichlet problem (1)-(2) is given by $u_ f(x)=E^ x\{\exp [\int_{0}^{\tau_{{\mathcal D}}}q(B_ t)dt]f(B(\tau_{{\mathcal D}})\quad)\}.$ For a regular domain, bounded Hölder continuous q and continuous f the theorem was proved by the author and K. M. Rao [Stochastic processes, Semin. Evanston/Ill. 1981, Progr. Probab. Stat. 1, 1-29 (1981; Zbl 0492.60073)]. For the q(x) satisfying the condition $\lim_{t\downarrow 0}\sup_{x\in {\mathcal D}}E^ x\{\int^{t}_{0}1_ D| q(B_ t)| dt\}=\quad 0$ (the condition was introduced by A. Aizenman and B. Simon in Commun. Pure Appl. Math. 35, 209- 273 (1982; Zbl 0459.60069)], the theorem was extended by Zhongxin Zhao [Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 13-18 (1983; Zbl 0521.60074)] and for the Neumann boundary problem (1), (3) by the author and Hsu (to appear in Semin. Stoch. Proc.).
Reviewer: L.Pastur

### MSC:

 60H25 Random operators and equations (aspects of stochastic analysis) 60J65 Brownian motion 35J10 Schrödinger operator, Schrödinger equation 35J25 Boundary value problems for second-order elliptic equations

### Citations:

Zbl 0492.60073; Zbl 0459.60069; Zbl 0521.60074