Summary
Let (X x t,P ) be anm-symmetric Markov process with a strictly positive transition density. Consider the additive functionalA t : = ∫ t0 f (X s ) wheref:E→[0, ∞] is a universally measurable function on the state spaceE. Among others, we prove thatP x(A t <∞)=1, for somex∈E and somet>0, already impliesP x(A t <∞)=1, for quasi everyx∈E and allt>0. The latter is also equivalent toP x(A t <∞)>0, for quasi everyx∈E and allt>0, and to the analytic condition\(\smallint _{F_n } fdm< \infty \), for a sequence of finely open Borel setsF n such thatE∪F n is polar. In the special cases of Brownian motion and Bessel process, these results were obtained earlier by H.J. Engelbert, W. Schmidt, X.-X. Xue and the authors.
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Höhnle, R., Sturm, KT. Some zero-one laws for additive functionals of Markov processes. Probab. Th. Rel. Fields 100, 407–416 (1994). https://doi.org/10.1007/BF01268986
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DOI: https://doi.org/10.1007/BF01268986
Mathematics Subject Classification
- 60 J 55
- 60 J 57
- 60 J 40