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

MSC:

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

Zbl 0575.00017