Proceedings of the American Mathematical Society

Gaugeability for Feynman-Kac functionals with applications to symmetric $\alpha$ -stable processes

Author(s): Masayoshi Takeda
Journal: Proc. Amer. Math. Soc. 134 (2006), 2729-2738.
MSC (2000): Primary 60J45, 60J40, 35J10
Posted: March 22, 2006
Retrieve article in: PDF

Abstract: For symmetric $\alpha$ -stable processes, an analytic criterion for a measure being gaugeable was obtained by Z.-Q. Chen (2002), M. Takeda (2002) and M. Takeda and T. Uemura (2004). Applying it, we consider the ultracontractivity of Feynman-Kac semigroups and expectations of the number of branches hitting closed sets in branching symmetric $\alpha$ -stable processes.

1.: S. Albeverio, P. Blanchard and Z.M. Ma, Feynman-Kac semigroups in terms of signed smooth measures, Random Partial Differential Equations, eds: U. Hornung et al., Birkhäuser, 1991 MR 1185735 (93i:60140)
2.: A. Bendikov, Asymptotic formulas for symmetric stable semigroups, Expo. Math., 12 (1994), 381-384. MR 1297844 (95j:60029)
3.: Z.-Q. Chen, Gaugeability and Conditional Gaugeability, Trans. Amer. Math. Soc., 354 (2002), 4639-4679. MR 1926893 (2003i:60127)
4.: Z.-Q. Chen and R.M. Song, General gauge and conditional gauge theorems, Ann. Probab., 30 (2002), 1313-1339. MR 1920109 (2003f:60135)
5.: Z.-Q. Chen and S.T. Zhang, Girsanov and Feynman-Kac type transformations for symmetric Markov processes, Ann. Inst. Henri Poincare, 38 (2002), 475-505. MR 1914937 (2004e:60128)
6.: K.L. Chung and Z. Zhao, From Brownian Motion to Schrödinger's Equation, Springer (1995). MR 1329992 (96f:60140)
7.: E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, U.K. (1989). MR 0990239 (90e:35123)
8.: D.A. Dawson, Measure-valued Markov processes, Lectures Notes in Math., 1541, Springer, (1993), 1-260. MR 1242575 (94m:60101)
9.: M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, 1994. MR 1303354 (96f:60126)
10.: A. Grigor'yan and M. Kelbert, Recurrence and transience of branching diffusion processes on Riemannian manifolds, Ann. Probab., 31 (2003), 244-284. MR 1959793 (2003k:60211)
11.: D. Revuz and M. Yor, Continuous martingales and Brownian motion, Third edition, Grundlehren der Mathematischen Wissenschaften, 293, Springer, 1999. MR 1725357 (2000h:60050)
12.: Y. Shiozawa, Principal eigenvalues for time changed processes of one-dimensional $\alpha$ -stable processes, Probability and Mathematical Statistics, 24 (2004), 111-122.
13.: B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7, 447-536, (1982). MR 0670130 (86b:81001a)
14.: P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Analysis, 5 (1996) 109-138. MR 1378151 (97e:47065)
15.: M. Takeda, Exponential decay of lifetime and a Theorem of Kac on total occupation times, Potential Analysis, 11 (1999), 235-247. MR 1717103 (2000i:60084)
16.: M. Takeda, Subcriticality and conditional gaugeability of generalized Schrödinger operators, J. Funct. Anal., 191 (2002), 343-376. MR 1911190 (2003e:60176)
17.: M. Takeda and K. Tsuchida, Differentiability of spectral functions for symmetric $\alpha$ -stable process, to appear in Trans. Amer. Math. Soc.
18.: M. Takeda and T. Uemura, Subcriticality and gaugeability for symmetric $\alpha$ -stable processes, Forum Math. 16 (2004), 505-517. MR 2044025 (2005d:60124)
19.: Z. Zhao, Subcriticality and gaugeability of the Schrödinger operator, Trans. Amer. Math. Soc., 334 (1992), 75-96. MR 1068934 (93a:81041)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 60J45, 60J40, 35J10

Masayoshi Takeda
Affiliation: Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan
Email: takeda@math.tohoku.ac.jp

DOI: 10.1090/S0002-9939-06-08281-5
PII: S 0002-9939(06)08281-5
Keywords: Symmetric stable process, gaugeability, subcriticality, ultracontractivity, branching process
Received by editor(s): September 17, 2004
Received by editor(s) in revised form: March 30, 2005
Posted: March 22, 2006
Additional Notes: The author was supported in part by Grant-in-Aid for Scientific Research (No.15530229 (C)(2)), Japan Society for the Promotion of Science.
Communicated by: Richard C. Bradley
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS