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A formula on scattering length of dual Markov processes
Author:
Ping He
Journal:
Proc. Amer. Math. Soc. 139 (2011), 1871-1877
MSC (2010):
Primary 60J40; Secondary 60J45
Posted:
November 1, 2010
MathSciNet review:
2763774
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Abstract: A formula on the scattering length for 3-dimensional Brownian motion was conjectured by M. Kac and proved by others later. It was recently proved under the framework of symmetric Markov processes by Takeda. In this paper, we shall prove that this formula holds for Markov processes under weak duality by the machinery developed mainly by Fitzsimmons and Getoor.
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- 2.
- R.M. Blumenthal, R.K. Getoor, Additive functionals of Markov processes in duality. Trans. Amer. Math. Soc. 112 (1964), 131-163. MR 0160269 (28:3483)
- 3.
- Z. Chen, M. Fukushima, J. Ying, Entrance law, exit system and Lévy system of time change processes, Illinois J. Math. 50, 2 (2006), 269-312. MR 2247830 (2007k:60242)
- 4.
- P.J. Fitzsimmons and R.K. Getoor, Revuz measures and time changes, Math. Zeit. 199 (1988), 233-256. MR 958650 (89h:60124)
- 5.
- M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, 1994. MR 1303354 (96f:60126)
- 6.
- R.K. Getoor, Excessive measures, Birkhäuser, 1990. MR 1093669 (92i:60135)
- 7.
- R.K. Getoor, Duality Theory for Markov Processes, Part I, preprint, 2010.
- 8.
- R.K. Getoor, M.J. Sharpe, Naturality, standardness, and weak duality for Markov processes, Z. W. Verw. Geb. 67 (1984), 1-62. MR 756804 (86f:60093)
- 9.
- P. He, J. Ying, Revuz measure under time change, Sci. China. Ser. A 51 (2008), 321-328. MR 2395426 (2009e:60178)
- 10.
- M.W. Jin, J. Ying, Additive functionals and perturbation of semigroup. Chinese Ann. Math. Ser. B 22 (2001), no. 4, 503-512. MR 1870075 (2003f:60137)
- 11.
- M. Kac, Probabilistic methods in some problems of scattering theory, Rocky Mountain J. Math. 4 (1974), 511-537. MR 0510113 (58:23170)
- 12.
- M. Kac and J.-M. Luttinger, Scattering length and capacity, Ann. Inst. Fourier (Grenoble) 25 (1975), 317-321. MR 0402079 (53:5902)
- 13.
- P.A. Meyer, Note sur l'interpretation des mesures d'équilibre. Sém. de Prob. VII. Lecture Notes in Math. 321, pp. 210-216, Springer, Berlin, 1973. MR 0373030 (51:9232)
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- 15.
- D.W. Stroock, The Kac approach to potential theory. I, J. Math. Mech. 16 (1967), 829-852. MR 0208690 (34:8499)
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- Y. Takahashi, An integral representation on the path space for scattering length, Osaka J. Math. 7 (1990), 373-379. MR 1066632 (91j:35083)
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 by positive potentials, J. Math. Anal. Appl. 53 (1976), 291-312. MR 0477504 (57:17028)
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- J. Ying, Bivariate Revuz measure and the Feynman-Kac formula, Ann. Inst. Henri Poincaré 32, 2 (1996), 251-287. MR 1386221 (97j:60139)
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Additional Information
Ping He
Affiliation:
Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, People’s Republic of China
Email:
pinghe@mail.shufe.edu.cn
DOI:
http://dx.doi.org/10.1090/S0002-9939-2010-10618-4
PII:
S 0002-9939(2010)10618-4
Received by editor(s):
March 2, 2010
Received by editor(s) in revised form:
May 26, 2010
Posted:
November 1, 2010
Additional Notes:
This research supported in part by the National Natural Science Foundation of China (Grant No. 10771131)
Communicated by:
Richard C. Bradley
Article copyright:
© Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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