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

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## Abstract

In this thesis, we study some problems about nearly symmetric Markov processes,which are associated with non-symmetric Dirichlet forms or semi-Dirichlet forms. For a Markov process (Xt, Px) associated with a non-symmetric 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 semi-Dirichlet 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 |
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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 Feynman-Kac semigroup, semi-Dirichlet forms, Fukushima's decomposition, local martingale additive functionals, zero energy, transformation formula. |

ID Code: | 7696 |

Deposited By: | LI MA |

Deposited On: | 22 Nov 2011 08:44 |

Last Modified: | 04 Jan 2012 15:06 |

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