Ma, Li (2011) Generalized FeynmanKac Transformation and Fukashima's Decomposition for Nearly Symmetric Markov Processes. PhD thesis, Concordia University.

PDF
 Accepted Version
901kB 
Abstract
In this thesis, we study some problems about nearly symmetric Markov processes,which are associated with nonsymmetric Dirichlet forms or semiDirichlet forms. For a Markov process (Xt, Px) associated with a nonsymmetric Dirichlet form (E,D(E)) on L2(E;m), we study the strong continuity of the generalized Feynman Kac semigroup (Pu
t )t≥0, which is defined by Pu t f(x) := Ex[eNu t f(Xt)], f ≥ 0 and t ≥ 0.Here u ∈ D(E), Nu t is the continuous additive functional of zero energy in the Fukushima’s decomposition. We give two sufficient conditions for (Pu
t )t≥0 to be strongly continuous.The first sufficient condition is that there exists a constant α0 ≥ 0 such that for any f ∈ D(E)b, Qu(f, f) ≥ −α0(f, f)m, where (Qu,D(E)b) is defined by Qu(f, g) := E(f, g) + E(u, fg), f,g∈ D(E)b := D(E) ∩ L∞(E;m). The second sufficient condition is that there exists a constant α0 ≥ 0 such that �Pu
t �2 ≤ eα0t, ∀t > 0. For a Markov process associated with a semiDirichlet form, we establish Fukushima’s
decomposition and give a transformation formula for local martingale additive functionals.
Divisions:  Concordia University > Faculty of Arts and Science > Mathematics and Statistics 

Item Type:  Thesis (PhD) 
Authors:  Ma, Li 
Institution:  Concordia University 
Degree Name:  Ph. D. 
Program:  Mathematics 
Date:  29 June 2011 
Thesis Supervisor(s):  Sun, Wei 
Keywords:  Dirichlet forms, strong continuous, generalized FeynmanKac semigroup, semiDirichlet forms, Fukushima's decomposition, local martingale additive functionals, zero energy, transformation formula. 
ID Code:  7696 
Deposited By:  LI MA 
Deposited On:  22 Nov 2011 13:44 
Last Modified:  04 Jan 2012 20:06 
References:  [AFRS1995] S. Albeverio, R.Z. Fan, M. R¨ockner and W. Stannat: A remark on coercive forms and associated semigroups, Partial Differential Operators and Mathematical
Physics, Operator Theory Advances and Applications. 78, (1995), 18. [AM1991] S. Albeverio and Z.M. Ma: Perturbation of Dirichlet formslower semiboundedness, closablility, and form cores. J. Funct. Anal. 99, (1991), 332356. [AM1992] S. Albeverio and Z.M. Ma: Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms. Osaka J. Math. 29, (1992), 247265. [C2007] C.Z. Chen: A note on perturbation of nonsymmetric Dirichlet forms by signed smooth measures. Acta Math. Scientia 27B, (2007), 219224. [CHM2010] C.Z. Chen, X.F. Han and L. Ma : Some new results about asymptotic properties of additive functionals of Brownian Motion. Acta Mathematica Scientia. 30 (6) (A), (2010), 14851494. [CMS2007] C.Z. Chen, Z.M. Ma and W. Sun: On Girsanov and generalized FeynmanKac transfromations for symmetric Markov processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10, (2007), 141163. [CS2006] C.Z. Chen and W. Sun: Strong continuity of generalized FeynmanKac semigroups: necessary and sufficient conditions. J. Funct. Anal. 237, (2006), 446 465. [CS2009] C.Z. Chen and W. Sun: Girsanov transformations for nonsymmetric diffusions. Canad. J. Math. 61, (2009), 534547. [Z2005] Z.Q. Chen: On FeynmanKac perturbation of symmetric Markov processes. Proceedings of Functional Analysis IX, Dubrovnik, Croatia. (2005), 3943. [CFKZ2008a] Z.Q. Chen, P.J. Fitzsimmons, K. Kuwae and T.S. Zhang: Stochastic calculus for symmetric Markov processes. Ann. Probab. 36, (2008), 931970. [CFKZ2008b] Z.Q. Chen, P.J. Fitzsimmons, K. Kuwae and T.S. Zhang: Perturbation of symmetric Markov Processes. Probab. Theory Relat. Fields. 140, (2008), 239275. [CFKZ2009] Z.Q. Chen, P.J. Fitzsimmons, K. Kuwae and T.S. Zhang: On general perturbations of symmetric Markov processes. J. Math. Pures et Appliqu´ees. 92, (2009), 363374. [CMR1994] Z.Q. Chen, Z.M. Ma and M. R¨ockner: Quasihomeomorphisms of Dirichlet forms. Nagoya Math. J. 136, (1994), 115. [CS2003] Z.Q. Chen, R.M. Song: Conditional gauge theorem for nonlocal FeynmanKac transforms. Probab. Theory Relat. Fields. 125, (2003), 4572. [CZ2002] Z.Q. Chen and T.S. Zhang: Girsanov and FeynmanKac type transformations for symmetric Markov processes. Ann. Inst. H. Poincar´e Probab. Statist. 38,(2002), 475505. [F2001] P.J. Fitzsimmons: On the quasiregularity of semiDirichlet forms. Potential Anal. 15, (2001), 158185. [FK2004] P.J. Fitzsimmons and K. Kuwae: Nonsymmetric perturbations of symmetric Dirichlet forms. J. Funct. Anal. 208, (2004), 140162. [F1979] M. Fukushima: A decomposition of additive functionals of finite energy. Nagoya Math. J. 74, (1979), 137168. [FOT1994] M. Fukushima, Y. Oshima and M. Takeda: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyrer. Berlin, 1994. [FU2010] M. Fukushima and T. Uemura: Jumptype Hunt processes generated by lower bounded semiDirichlet Forms. Preprint. [GRSS1994] J. Glover, M. Rao, H. ˇSiki´c and R. Song: Quadratic forms corresponding to the generalized Schr¨odinger semigroups. J. Funct. Anal. 125, (1994), 358378. [HWY1992] S.W. He, J.G.Wang and J.A. Yan: Semimartingale Theory and Stochastic Calculus. Science Press. Beijing, 1992. [HMS2006] Z.C. Hu, Z.M. Ma and W. Sun: Extensions of L´evyKhintchine formula and BeurlingDeny formula in semiDirichlet forms setting. J. Funct. Anal. 239, (2006), 179213. [HMS2010] Z.C. Hu, Z.M. Ma and W. Sun: On representations of nonsymmetric Dirichlet forms. Potential Anal. 32, (2010), 101131. [HS2010] Z.C. Hu and W. Sun: Balayage of SemiDirichlet Forms. Preprint. [K1987] J.H. Kim: Stochastic calculus related to nonsymmetric Dirichlet forms, Osaka J. Math. 24, (1987), 331371. [K2008] K. Kuwae: Maximum principles for subharmonic functions via local semiDirichlet forms. Can. J. Math. 60, (2008), 822874. [K2010] K. Kuwae: stochastic calculus over symmetric Markov processes without time reversal. Ann. Prob. 38(4), (2010), 15321569. [MS2010a] L. Ma and W. Sun: On the generalized FeynmanKac transformation for nearly symmetric Markov processes. Journal of Theoretical Probability, DOI: 10.1007/s1095901003183. 17 [MMS2011] L. Ma, Z.M. Ma and W. Sun: Fukushima’s decomposition for diffusions associated with semiDirichlet forms. submitted. [MOR1995] Z.M. Ma, L. Overbeck and M. R¨ockner: Markov processes associated with semiDirichlet forms. Osaka J. Math. 32, (1995), 97119. [MR1992] Z.M. Ma and M. R¨ockner: Introduction to the Theory of (NonSymmetric) Dirichlet Forms, SpringerVerlag, Berlin, 1992. [MR1995] Z.M. Ma and M. R¨ockner: Markov processes associated with positivity preserving coercive forms. Can. J. Math. 47, (1995), 817840. [MS2010b] Z.M. Ma and W. Sun: Some topics on Dirichlet forms. Preprint, 2010. [ORS1995] L. Overbeck, M. R¨ockner and B. Schmuland: An analytic approach to FlemingViot processes with interactive selection. Ann. Probab. 23, (1995), 136. [O1988] Y. Oshima: Lecture on Dirichlet Spaces. Univ. ErlangenN¨urnberg, 1988. [P2005] P.E. Protter: Stochastic Integration and Differential Equations, Springer,Berlin Heidelberg New York, 2005. [RS1995] M. R¨ockner and B. Schmuland: Quasiregular Dirichlet forms: examples and counterexamples. Canadian Journal of Mathematics. 47(1), (1995), 165200. [V1991] Z. Vondracek.: An estimate for the L2−norm of a quasi continuous function with respect to a smooth measure. Arch. Math. 67, (1991), 408414. [T2001] T.S. Zhang: Generalized FeynmanKac semigroups, associated quadratic forms and asymptotic properties. Potential Anal. 14, (2001), 387408. 
Repository Staff Only: item control page