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A formula on scattering length of dual Markov processes. (English) Zbl 1221.60108
The scattering length formula is extended to the case of right Markov processes in weak duality. This formula was conjectured by M. Kac and J. M. Luttinger [Ann. Inst. Fourier 25, No. 3–4, 317–321 (1975; Zbl 0303.28011)] and extended recently to the case of symmetric Markov process by M. Takeda [Proc. Am. Math. Soc. 138, No. 4, 1491–1494 (2010; Zbl 1193.60095)].

MSC:
60J45 Probabilistic potential theory
60J40 Right processes
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[1] R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. · Zbl 0169.49204
[2] R. M. Blumenthal and R. K. Getoor, Additive functionals of Markov processes in duality, Trans. Amer. Math. Soc. 112 (1964), 131 – 163. · Zbl 0133.40904
[3] Zhen-Qing Chen, Masatoshi Fukushima, and Jiangang Ying, Entrance law, exit system and Lévy system of time changed processes, Illinois J. Math. 50 (2006), no. 1-4, 269 – 312. · Zbl 1098.60076
[4] P. J. Fitzsimmons and R. K. Getoor, Revuz measures and time changes, Math. Z. 199 (1988), no. 2, 233 – 256. · Zbl 0631.60070
[5] Masatoshi Fukushima, Yōichi Ōshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. · Zbl 0838.31001
[6] R. K. Getoor, Excessive measures, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. · Zbl 0982.31500
[7] R.K. Getoor, Duality Theory for Markov Processes, Part I, preprint, 2010.
[8] R. K. Getoor and M. J. Sharpe, Naturality, standardness, and weak duality for Markov processes, Z. Wahrsch. Verw. Gebiete 67 (1984), no. 1, 1 – 62. · Zbl 0553.60070
[9] Ping He and JianGang Ying, Revuz measures under time change, Sci. China Ser. A 51 (2008), no. 3, 321 – 328. · Zbl 1145.60042
[10] Mengwei Jin and Jiangang Ying, Additive functionals and perturbation of semigroup, Chinese Ann. Math. Ser. B 22 (2001), no. 4, 503 – 512. · Zbl 0993.60076
[11] M. Kac, Probabilistic methods in some problems of scattering theory, Rocky Mountain J. Math. 4 (1974), 511 – 537. Notes by Jack Macki and Reuben Hersh; Papers arising from a Conference on Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972). · Zbl 0314.47006
[12] M. Kac and J.-M. Luttinger, Scattering length and capacity, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 3-4, xvi, 317 – 321 (English, with French summary). Collection of articles dedicated to Marcel Brelot on the occasion of his 70th birthday. · Zbl 0303.28011
[13] P. A. Meyer, Note sur l’interprétation des mesures d’équilibre, Séminaire de Probabilités, VII (Univ. Strasbourg, année universitaire 1971 – 1972), Springer, Berlin, 1973, pp. 210 – 216. Lecture Notes in Math., Vol. 321 (French).
[14] Michael Sharpe, General theory of Markov processes, Pure and Applied Mathematics, vol. 133, Academic Press, Inc., Boston, MA, 1988. · Zbl 0649.60079
[15] Daniel W. Stroock, The Kac approach to potential theory. I, J. Math. Mech. 16 (1967), 829 – 852. · Zbl 0148.36202
[16] Yōichirō Takahashi, An integral representation on the path space for scattering length, Osaka J. Math. 27 (1990), no. 2, 373 – 379. · Zbl 0712.60088
[17] Hideo Tamura, Semi-classical limit of scattering length, Lett. Math. Phys. 24 (1992), no. 3, 205 – 209. · Zbl 0813.35074
[18] Michael E. Taylor, Scattering length and perturbations of -\? by positive potentials, J. Math. Anal. Appl. 53 (1976), no. 2, 291 – 312. · Zbl 0336.31005
[19] Masayoshi Takeda, A formula on scattering length of positive smooth measures, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1491 – 1494. · Zbl 1193.60095
[20] Jiangang Ying, Bivariate Revuz measures and the Feynman-Kac formula, Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), no. 2, 251 – 287. · Zbl 0861.60082
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