Processing math: 100%
  1. Home
  2. Issues
  3. Volume 2, Issue 2 (2015)
  4. On the Feynman–Kac semigroup for some Ma ...

Modern Stochastics: Theory and Applications


On the Feynman–Kac semigroup for some Markov processes
Volume 2, Issue 2 (2015), pp. 107–129
Pub. online: 18 June 2015      Type: Research Article      Open accessOpen Access
Received
1 October 2014
Revised
10 May 2015
Accepted
26 May 2015
Published
18 June 2015

Abstract

For a (non-symmetric) strong Markov process X, consider the Feynman–Kac semigroup
TAtf(x):=Ex[eAtf(Xt)],x∈Rn,t>0,
where A is a continuous additive functional of X associated with some signed measure. Under the assumption that X admits a transition probability density that possesses upper and lower bounds of certain type, we show that the kernel corresponding to TAt possesses the density pAt(x,y) with respect to the Lebesgue measure and construct upper and lower bounds for pAt(x,y). Some examples are provided.

References

[1] 
Albeverio, S., Ma, Z.-M.: Perturbation of Dirichlet forms—lower semiboundedness, closability, and form cores. J. Funct. Anal. 99, 332–356 (1991). MR1121617. doi:10.1016/0022-1236(91)90044-6
[2] 
Albeverio, S., Ma, Z.-M.: Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms. Osaka J. Math. 29, 247–265 (1992). MR1173989
[3] 
Blanchard, P., Ma, Z.-M.: Semigroup of Schrödinger operators with potentials given by Radon measures. In: Stochastic Processes, Physics and Geometry. L’Ecuyer, Pierre and Owen, Art B. World Sci. Publishing, Teaneck, NJ (1990). MR1124210
[4] 
Bogdan, K., Jakubowski, T.: Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Commun. Math. Phys. 271(1), 179–198 (2007). MR2283957. doi:10.1007/s00220-006-0178-y
[5] 
Bogdan, K., Szczypkowski, K.: Gaussian estimates for Schroedinger perturbations. Available at http://arxiv.org/pdf/1301.4627.pdf
[6] 
Bogdan, K., Sztonyk, P.: Estimates of the potential kernel and Harnack’s inequality for the anisotropic fractional Laplacian. Stud. Math. 181(2), 101–123 (2007). MR2320691. doi:10.4064/sm181-2-1
[7] 
Bogdan, K., Sztonyk, P., Knopova, V.: Transition densities of parastable Markov processes. In preparation
[8] 
Chaumont, L., Bravo, G.U.: Markovian bridges: Weak continuity and pathwise constructions. Ann. Probab. 39(2), 609–647 (2011). MR2789508. doi:10.1214/10-AOP562
[9] 
Chen, Z.-Q., Song, R.: Conditional gauge theorem for non-local Feynman–Kac transforms. Probab. Theory Relat. Fields 125, 45–72 (2003). MR1952456. doi:10.1007/ s004400200219
[10] 
Chen, Z.-Q., Kim, P., Song, R.: Stability of Dirichlet heat kernel estimates for non-local operators under Feynman–Kac perturbation. Trans. Am. Math. Soc. 367(7), 5237–5270 (2015). MR3335416. doi:10.1090/S0002-9947-2014-06190-4
[11] 
Chung, K.L., Rao, M.: General gauge theorem for multiplicative functionals. Trans. Am. Math. Soc. 306, 819–836 (1988). MR0933320. doi:10.2307/2000825
[12] 
Chung, K.L., Zhao, Z.: From Brownian Motion to Schrödinger’s Equation. Springer, Berlin (1995). MR1329992. doi:10.1007/978-3-642-57856-4
[13] 
Dynkin, E.B.: Markov Processes, vols. 1–2. Springer, Berlin (1965). MR3111220. doi:10.1007/978-1-4614-6240-8_1
[14] 
Fukushima, M.: Dirichlet Forms and Markov Processes. North-Holland Publishing Company, Amsterdam (1980). MR0569058
[15] 
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter & Co., Berlin (1994). MR1303354. doi:10.1515/9783110889741
[16] 
Jacob, N.: Pseudo Differential Operators and Markov Processes, I: Fourier Analysis and Semigroups. Imperial College Press, London (2001). MR1873235. doi:10.1142/9781860949746
[17] 
Knopova, V.: Compound kernel estimates for the transition probability density of a Lévy process in Rn. Theory Probab. Math. Stat. 89, 57–70 (2014). MR3235175. doi:10.1090/s0094-9000-2015-00935-2
[18] 
Knopova, V., Kulik, A.: Intrinsic compound kernel estimates for the transition probability density of a Lévy type processes and their applications. Available at http://arxiv.org/abs/1308.0310
[19] 
Knopova, V., Kulik, A.: Parametrix construction for certain Lévy-type processes. Random Oper. Stoch. Equ., in press.
[20] 
Knopova, V., Kulik, A.: Intrinsic small time estimates for distribution densities of Lévy processes. Random Oper. Stoch. Equ. 21(4), 321–344 (2013). MR3139314. doi:10.1515/rose-2013-0015
[21] 
Ma, Z.M., Roekner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992)
[22] 
Song, R.: Two-sided estimates on the density of the Feynman–Kac semigroups of stable-like processes. Electron. J. Probab. 7, 146–161 (2006). MR2217813. doi:10.1214/ EJP.v11-308
[23] 
Sznitman, A.-Z.: Brownian Motion, Obstacles and Random Media. Springer, Berlin (1998). MR1717054. doi:10.1007/978-3-662-11281-6
[24] 
Van Casteren, J.A., Demuth, M.: Stochastic Spectral Theory for Selfadjoint Feller Operators: A Functional Integration Approach. Birkhäuser, Berlin (2000). MR1772266. doi:10.1007/978-3-0348-8460-0

Full article Cited by PDF XML
Full article Cited by PDF XML

Copyright
© 2015 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
Transition probability density continuous additive functional Kato class Feynman–Kac semigroup

MSC2010
60J35 (primary) 60J45 60J55 60J57 60J75

Metrics
since March 2018
781

Article info
views

631

Full article
views

356

PDF
downloads

171

XML
downloads


Share


RSS