Notes on the inhomogeneous Schrödinger equation. (English) Zbl 0585.60063

Stochastic processes, Semin. Evanston/Ill. 1984, Prog. Probab. Stat. 9, 55-62 (1986).
[For the entire collection see Zbl 0575.00017.]
Let \((X_ t: t\geq 0)\) be an \({\mathbb{R}}^ d\)-Brownian motion; \(q\in L^{\infty}({\mathbb{R}}^ d)\); D be a domain in \({\mathbb{R}}^ d\) with finite volume; \(\tau\) be the first exit from D. The semigroup \((L^ q_ t: t\geq 0)\), given by \[ L^ q_ tf(x)={\mathbb{E}}^ x(t<\tau: f(X_ t)\exp [\int^{t}_{0}q(X_ s)ds]), \] is considered and the properties of the potential \(V^ q(f)=\int^{\infty}_{0}L^ q_ tfdt\) are studied. Under some smoothness conditions for q and \(\phi\) it is proved that \((\Delta /2+q)V^ q\phi =-\phi\).
Reviewer: O.Enchev


60H25 Random operators and equations (aspects of stochastic analysis)
60J45 Probabilistic potential theory
60J65 Brownian motion


Zbl 0575.00017