Abstract
SupposeX is a Borel right process andm is a σ-finite excessive measure forX. Given a positive measure μ not chargingm-semipolars we associate an exact multiplicative functionalM(μ). No finiteness assumptions are made on μ. Given two such measures μ and ν,M(μ)=M(ν) if and only if μ and ν agree on all finely open measurable sets. The equation (q−L)u+uμ=f whereL is the generator of (a subprocess of)X may be solved for appropriatef by means of the Feynman-Kac formula based onM(μ). Both uniqueness and existence are considered.
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References
Azéma, J.:Une remarque sur les temps de retour. Trois applications, Sém. de Prob. VI. Lecture Notes in Math.258, Springer-Verlag, Berlin, Heidelberg, New York, 1972, pp. 35–50.
Azéma, J.: ‘Théorie générale des processus et retournement du temps’,Ann. Sci. de l'Ecole Norm. Sup. 6 (1973), 459–519.
Baxter, J., Dal Maso, G., and Mosco, U.: ‘Stopping times and Γ-convergence’,Trans. Amer. Math. Soc. 303 (1987), 1–38.
Blumenthal, R. M. and Getoor, R. K.:Markov Processes and Potential Theory, Academic Press, New York, 1968.
Chung, K. L.:Lectures from Markov Processes to Brownian Motion, Springer-Verlag, Berlin, Heidelberg, New York, 1982.
Dellacherie, C. and Meyer, P.:Probabilités et Potential, Ch. XII–XVI, Hermann, Paris, 1987.
Fitzsimmons, P. J.: ‘Homogeneous random measures and a weak order for excessive measures of a Markov process’,Trans. Amer. Math. Soc. 303 (1987), 431–478.
Fitzsimmons, P. J.: ‘On the equivalence of three potential principles for right Markov processes’,Probab. Th. Rel. Fields 84 (1990), 251–265.
Fitzsimmons, P. J. and Getoor, R. K.: ‘Revuz measures and time changes’,Math. Z. 199 (1988), 233–256.
Fitzsimmons, P. J. and Getoor, R. K.: ‘A fine domination principle for excessive measures’,Math. Z. 207 (1991), 137–151.
Getoor, R. K.:Markov Processes: Ray Processes and Right Processes, Lecture Notes in Math.,440, Springer-Verlag, Berlin, Heidelberg, New York, 1975.
Getoor, R. K.:Excessive Measures, Birkhäuser, Boston, 1990.
Getoor, R. K. and Rao, Murali: ‘Another look at Doob's theorem’,Ann. Math. Stat. 41 (1970), 503–506.
Getoor, R. K. and Sharpe, M. J.: ‘Balayage and multiplicative functionals’,Z. Wahrsheinlichkeitstheorie verw. Geb. 28 (1974), 139–164.
Getoor, R. K. and Sharpe, M. J.: ‘Naturality, standardness and weak duality’,Z. Wahrsheinlichkeitstheorie verw. Geb. 67 (1984), 1–62.
Ma, Z. and Röckner, M.:Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer-Verlag, Berlin, Heidelberg, New York, 1992.
Ma, Z., Overbeck, L., and Röckner, M.: ‘Markov processes associated with semi-Dirichlet forms’, to appear.
Meyer, P. A.:Probability and Potentials, Blaisedell, Waltham, Toronto, London, 1966.
Meyer, P. A.:Processus de Markov, Lecture Notes in Math.26, Springer-Verlag, Berlin, Heidelberg, New York, 1967.
Rebuz, D.: ‘Mesures associées aux functionelles additives de Markov I’,Trans. Amer. Math. Soc. 148 (1970), 501–531.
Sharpe, M. J.:General Theory of Markov Processes, Academic Press, Boston, San Diego, New York, 1988.
Sturm, K. T.: ‘Measures charging no polar sets and additive functionals of Brownian motion’,Forum Math. 4 (1992), 257–297.
Dellacherie, C.:Autour des ensembles semipolaires, Seminar on Stochastic Processes 1987, 65–82, Birkhäuser, Boston, 1988.
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Supported in part by NSF Grant DMS 92-24990.
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Getoor, R.K. Measures not charging semipolars and equations of schrödinger type. Potential Anal 4, 79–100 (1995). https://doi.org/10.1007/BF01048968
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DOI: https://doi.org/10.1007/BF01048968