ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 31, 60 Perišić, Vesna, Universität des Saarlandes, Saarbrücken, Germany On some energy inequalities We consider the Schrödinger operator H = −\frac{Δ}{2} + q in a bounded domain D in R^{d}, d ≥ 3, with q in Kato class K^{d}(D) and finite gauge g(x) = E_{x}[exp ∫_{0}^{τ} q(X_{s})ds], where (X_{t}) is a Brownian motion and τ the first exit time of the Brownian motion (X_{t}) from the domain D. Let K and G denote the Green operators in D for H and \frac{Δ}{2}, respectively. We prove that there is a positive constant α such that

\frac{1}{α}(Gf, f) ≤ (Kf, f) ≤ α(Gf, f),

where f is a real, measurable function for which (G|f|, |f|) is finite. As a direct consequence of this double inequality we have that the potential Gf is in the Sobolev space H_{0}^{1}(D) if and only if Kf ∈ H_{0}^{1}(D). References: K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger’s Equation, Springer Verlag, 1995. K. L. Chung and M. Rao, Feynman-Kac functional and the Schrödinger equation, Seminar on Stochastic Processes (1981), Birkhäuser, Boston, 1-