Summary
LetE be a noncompact locally compact second countable Hausdorff space. We consider the question when, given a family of finite nonzero measures onE that behave like harmonic measures associated with all relatively compact open sets inE (i.e. that satisfy a certain consistency condition), one can construct a Markov process onE and a multiplicative functional with values in [0, ∞) such that the hitting distributions of the process “inflated” by the multiplicative functional yield the given harmonic measures. We achieve this construction under weak continuity and local transience conditions on these measures that are natural in the theory of Markov processes, and a mild growth restriction on them. In particular, if the spaceE equipped with the measures satisfies the conditions of a harmonic space, such a Markov process and associated multiplicative functional exist. The result extends in a new direction the work of many authors, in probability and in axiomatic potential theory, on constructing Markov processes from given hitting distributions (i.e. from harmonic measures that have total mass no more than 1).
References
Bliedtner, J., Hansen, W.: Potential theory. Berlin Heidelberg New York: Springer 1986
Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York: Academic Press 1968
Constantinescu, C., Cornea, A.: Potential theory in harmonic spaces. Berlin Heidelberg New York: Springer 1972
Getoor, R.K.: Markov processes: Ray processes and right processes. (Lect. Notes Math., vol. 440) Berlin Heidelberg New York: Springer 1975
Ito, K., Watanabe, S.: Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier15, 13–30 (1965)
Sharpe, M.J.: General theory of Markov processes. San Diego: Academic Press 1988
Shih, C.T.: Construction of right processes from hitting distributions. In: Cinlar, E. et al. (eds.) Seminar on stochastic processes, pp. 189–236. Boston Basel Stuttgart: Birkhäuser 1984
Shih, C.T.: On piecing together locally defined processes. In: Cinlar, E. et al. (eds.) Seminar on stochastic processes, pp. 321–333. Boston Basel Stuttgart: Birkhäuser 1991
Shih, C.T.: Construction of Markov processes from hitting distributions without quasi-leftcontinuity. In: Cinlar, E. et al. (eds.) Seminar on stochastic processes, pp. 187–231. Boston Basel Stuttgart: Birkhäuser 1992
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Shih, C.T. Construction of Markov processes and associated multiplicative functionals from given harmonic measures. Probab. Th. Rel. Fields 98, 185–207 (1994). https://doi.org/10.1007/BF01192513
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DOI: https://doi.org/10.1007/BF01192513
Mathematics Subject Classification (1991)
- 60J40
- 60J45
- 31D05