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Greenian bounds for Markov processes. (English) Zbl 1081.60543

Summary: In Section 1, a temporal bound is estimated by a spatial bound, for a Markov process whose transition density satisfies a simple condition. This includes the Brownian motion, for which comparison with a more special method is made. In Section 2, the result is related to the Green operator and examples are given. In Section 3, the result is applied to an old problem of eigenvalues of the Laplacian. In Section 4, recent extensions from the Laplacian to the Schrödinger circle-of-ideas are briefly described. In this case, time is measured by an exponential functional of the process, commonly known under the name Feynman-Kac.
This article was written as script for two talks, one general and one technical, given in Taiwan in January 1991. In the spirit of the occasion, the style of exposition is deliberately paced and discursive.

MSC:

60J45 Probabilistic potential theory
35J10 Schrödinger operator, Schrödinger equation
60J35 Transition functions, generators and resolvents
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References:

[1] Lectures From Markov Processes to Brownian Motion, Grundlehren der Mathematischen Wissenschaften 249, Springer-Verlag, 1982. · Zbl 0503.60073
[2] Doubly-Feller Process with Multiplicative Functional, Seminar on Stochastic Processes 1985, pp. 63-78, Birkhäuser.
[3] Gauge Theorem for Unbounded Domains, Seminar on Stochastic Processes 1988, 87-98, Birkhäuser.
[4] Probability Methods in Potential Theory, Potential Theory Surveys and Problems, Proceedings, Prague 1987. Lecture Notes in Mathematics 1344, 42-54, Springer-Verlag, 1987.
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