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 Access
Open Access
Received
1 October 2014
1 October 2014
Revised
10 May 2015
10 May 2015
Accepted
26 May 2015
26 May 2015
Published
18 June 2015
18 June 2015
Abstract
For a (non-symmetric) strong Markov process X, consider the Feynman–Kac semigroup
\[{T_{t}^{A}}f(x):={\mathbb{E}}^{x}\big[{e}^{A_{t}}f(X_{t})\big],\hspace{1em}x\in {\mathbb{R}}^{n},\hspace{2.5pt}t>0,\]
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 ${T_{t}^{A}}$ possesses the density ${p_{t}^{A}}(x,y)$ with respect to the Lebesgue measure and construct upper and lower bounds for ${p_{t}^{A}}(x,y)$. Some examples are provided.References
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
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
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
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
Bogdan, K., Szczypkowski, K.: Gaussian estimates for Schroedinger perturbations. Available at http://arxiv.org/pdf/1301.4627.pdf
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
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
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
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
Chung, K.L., Rao, M.: General gauge theorem for multiplicative functionals. Trans. Am. Math. Soc. 306, 819–836 (1988). MR0933320. doi:10.2307/2000825
Chung, K.L., Zhao, Z.: From Brownian Motion to Schrödinger’s Equation. Springer, Berlin (1995). MR1329992. doi:10.1007/978-3-642-57856-4
Dynkin, E.B.: Markov Processes, vols. 1–2. Springer, Berlin (1965). MR3111220. doi:10.1007/978-1-4614-6240-8_1
Fukushima, M.: Dirichlet Forms and Markov Processes. North-Holland Publishing Company, Amsterdam (1980). MR0569058
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter & Co., Berlin (1994). MR1303354. doi:10.1515/9783110889741
Jacob, N.: Pseudo Differential Operators and Markov Processes, I: Fourier Analysis and Semigroups. Imperial College Press, London (2001). MR1873235. doi:10.1142/9781860949746
Knopova, V.: Compound kernel estimates for the transition probability density of a Lévy process in ${\mathbb{R}}^{n}$. Theory Probab. Math. Stat. 89, 57–70 (2014). MR3235175. doi:10.1090/s0094-9000-2015-00935-2
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
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
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
Sznitman, A.-Z.: Brownian Motion, Obstacles and Random Media. Springer, Berlin (1998). MR1717054. doi:10.1007/978-3-662-11281-6
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
