ON GIRSANOV AND GENERALIZED FEYNMAN–KAC TRANSFORMATIONS FOR SYMMETRIC MARKOV PROCESSES

2007 ◽  
Vol 10 (02) ◽  
pp. 141-163 ◽  
Author(s):  
CHUAN-ZHONG CHEN ◽  
ZHI-MING MA ◽  
WEI SUN
Keyword(s):  
Markov Process ◽  
Dirichlet Form ◽  
Sufficient Conditions ◽  
Dirichlet Space ◽  
Zero Energy ◽  
Continuous Version ◽  
Girsanov Transform ◽  
Energy Part ◽  
Decomposition Formula ◽  
Necessary And Sufficient

Let X be a Markov process, which is assumed to be associated with a symmetric Dirichlet form [Formula: see text]. For [Formula: see text], the extended Dirichlet space, we have the classical Fukushima's decomposition: [Formula: see text], where [Formula: see text] is a quasi-continuous version of u, [Formula: see text] the martingale part and [Formula: see text] the zero energy part. In this paper, we investigate two important transformations for X, the Girsanov transform induced by [Formula: see text] and the generalized Feynman–Kac transform induced by [Formula: see text]. For the Girsanov transform, we present necessary and sufficient conditions for which to induce a positive supermartingale and hence to determine another Markov process [Formula: see text]. Moreover, we characterize the symmetric Dirichlet form associated with the Girsanov transformed process [Formula: see text]. For the generalized Feynman–Kac transform, we give a necessary and sufficient condition for the generalized Feynman–Kac semigroup to be strongly continuous.

Basic Properties and Examples of Dirichlet Forms

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Markov Process ◽  
Potential Theory ◽  
Dirichlet Form ◽  
Dirichlet Space ◽  
Dirichlet Forms ◽  
Dirichlet Spaces ◽  
Basic Properties ◽  
Time Changes ◽  
Symmetric Operators ◽  
Analytic Potential

This chapter introduces the concepts of the transience, recurrence, and irreducibility of the semigroup for general Markovian symmetric operators and presents their characterizations by means of the associated Dirichlet form as well as the associated extended Dirichlet space. These notions are invariant under the time changes of the associated Markov process. The chapter then presents some basic examples of Dirichlet forms, with special attention paid to their basic properties as well as explicit expressions of the corresponding extended Dirichlet spaces. Hereafter the chapter discusses the analytic potential theory for regular Dirichlet forms, and presents some conditions for the demonstrated Dirichlet form (E,F) to be local.

Girsanov Transformations for Non-Symmetric Diffusions

2009 ◽  
Vol 61 (3) ◽  
pp. 534-547 ◽  
Author(s):  
Chuan-Zhong Chen ◽  
Wei Sun
Keyword(s):  
Diffusion Process ◽  
Dirichlet Form ◽  
Dirichlet Space ◽  
Explicit Representation ◽  
Dual Process ◽  
Sufficient Condition ◽  
Dual Processes ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Girsanov Transformations

Abstract.Let X be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form (ℰ, 𝓓 (ℰ)) on L2(E ;m). For u ∈ 𝓓(ℰ)e, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process Y of X . First, let be the dual process of X and Ŷ the Girsanov transformed process of . We give a necessary and sufficient condition for (Y , Ŷ to be in duality with respect to the measure e2um. We also construct a counterexample, which shows that this condition may not be satisfied and hence (Y , Ŷ ) may not be dual processes. Then we present a sufficient condition under which Y is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

scholarly journals A coupling technique for stochastic comparison of functions of Markov Processes

2000 ◽  
Vol 4 (1) ◽  
pp. 39-64 ◽  
Author(s):  
M. Doisy
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Sufficient Conditions ◽  
Reliability Theory ◽  
Stochastic Comparison ◽  
Transition Rates ◽  
Coupling Technique ◽  
Fixed Set ◽  
Multiple Component ◽  
Necessary And Sufficient

The aim of this work is to obtain explicit conditions (i.e., conditions on the transition rates) for the stochastic comparison of Markov Processes. A general coupling technique is used to obtain necessary and sufficient conditions for the construction of a coupling Markov Process which stays in a fixed set K for all times and with given marginal processes. The strong stochastic comparison—or, more generally, the stochastic comparison through states functions—appears as a particular case. An example in the Reliability Theory is developed and proves the efficiency of the method. Systems with multiple component types and redundant units are stochastically compared directly or through particular functions.

scholarly journals Condensor Principle and the Unit Contraction

1967 ◽  
Vol 30 ◽  
pp. 9-28 ◽  
Author(s):  
Masayuki Itô
Keyword(s):  
Positive Measure ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Dirichlet Space ◽  
Functional Space ◽  
Dirichlet Spaces ◽  
Locally Compact ◽  
Necessary And Sufficient ◽  
Regular Functional ◽  
Locally Compact Hausdorff Space

Deny introduced in [4] the notion of functional spaces by generalizing Dirichlet spaces. In this paper, we shall give the following necessary and sufficient conditions for a functional space to be a real Dirichlet space.Let be a regular functional space with respect to a locally compact Hausdorff space X and a positive measure ξ in X. The following four conditions are equivalent.

Necessary and sufficient conditions for the two-point phase probability function of two-phase random media

2008 ◽  
Vol 464 (2095) ◽  
pp. 1761-1779 ◽  
Author(s):  
John A Quintanilla
Keyword(s):  
Random Media ◽  
Probability Function ◽  
Recent Literature ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Condition ◽  
Two Phase ◽  
Decomposition Formula ◽  
Positive Matrices ◽  
Necessary And Sufficient

Constructing realizations of random media with a specified two-point phase probability function S 2 has attracted considerable attention in the recent literature. However, little is known about conditions under which a prescribed S 2 is realizable. The only known necessary and sufficient condition, due to McMillan, involves a class of square matrices, called corner-positive matrices, about which almost nothing is known except their definition. As a result, McMillan's theorem has gone mostly unused in the literature for over 50 years. In this paper, we present a general decomposition formula for corner-positive matrices, which allows McMillan's theorem to be written in a significantly more tractable and testable form. We also connect McMillan's theorem with many known but heretofore unrelated necessary conditions on S 2 , extending many of these conditions.

Lifting modules with finite internal exchange property and direct sums of hollow modules

2020 ◽  
pp. 2150189
Author(s):  
Yosuke Kuratomi
Keyword(s):  
Direct Summand ◽  
Sufficient Conditions ◽  
Exchange Property ◽  
Direct Sums ◽  
Perfect Ring ◽  
Direct Sum ◽  
Decomposition Formula ◽  
Direct Sum Decomposition ◽  
Extending Module ◽  
Necessary And Sufficient

A module [Formula: see text] is said to be lifting if, for any submodule [Formula: see text] of [Formula: see text], there exists a decomposition [Formula: see text] such that [Formula: see text] and [Formula: see text] is a small submodule of [Formula: see text]. A lifting module is defined as a dual concept of the extending module. A module [Formula: see text] is said to have the finite internal exchange property if, for any direct summand [Formula: see text] of [Formula: see text] and any finite direct sum decomposition [Formula: see text], there exists a direct summand [Formula: see text] of [Formula: see text] [Formula: see text] such that [Formula: see text]. This paper is concerned with the following two fundamental unsolved problems of lifting modules: “Classify those rings all of whose lifting modules have the finite internal exchange property” and “When is a direct sum of indecomposable lifting modules lifting?”. In this paper, we prove that any [Formula: see text]-square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property.

Symmetric Markovian Semigroups and Dirichlet Forms

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Markov Process ◽  
Dirichlet Form ◽  
Measure Space ◽  
Dirichlet Space ◽  
Finite Measure ◽  
Dirichlet Forms ◽  
Linear Operators ◽  
Dirichlet Spaces ◽  
Finite Measure Space ◽  
Markovian Semigroups

This chapter studies the concepts of Dirichlet form and Dirichlet space by first working with a σ‎-finite measure space (E,B(E),m) without any topological assumption on E and establish the correspondence of the above-mentioned notions to the semigroups of symmetric Markovian linear operators. Later on the chapter assumes that E is a Hausdorff topological space and considers the semigroups and Dirichlet forms generated by symmetric Markovian transition kernels on E. The chapter also considers quasi-regular Dirichlet forms and the quasi-homeomorphism of Dirichlet spaces. From here, the chapter shows that there is a nice Markov process called an m-tight special Borel standard process associated with every quasi-regular Dirichlet form.

scholarly journals Reverse integral Hardy inequality on metric measure spaces

2021 ◽  
Vol 47 (1) ◽  
pp. 39-55
Author(s):  
Aidyn Kassymov ◽  
Michael Ruzhansky ◽  
Durvudkhan Suragan
Keyword(s):  
Hardy Inequality ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Minkowski Inequality ◽  
Hyperbolic Spaces ◽  
Metric Measure Spaces ◽  
Continuous Version ◽  
Homogeneous Groups ◽  
Measure Spaces ◽  
Necessary And Sufficient

In this note, we obtain a reverse version of the integral Hardy inequality on metric measure spaces. Moreover, we give necessary and sufficient conditions for the weighted reverse Hardy inequality to be true. The main tool in our proof is a continuous version of the reverse Minkowski inequality. In addition, we present some consequences of the obtained reverse Hardy inequality on the homogeneous groups, hyperbolic spaces and Cartan-Hadamard manifolds.  

ON HUNT PROCESSES AND STRICT CAPACITIES ASSOCIATED WITH GENERALIZED DIRICHLET FORMS

2005 ◽  
Vol 08 (03) ◽  
pp. 357-382 ◽  
Author(s):  
GERALD TRUTNAU
Keyword(s):  
Dirichlet Form ◽  
Sufficient Conditions ◽  
Dirichlet Forms ◽  
Hunt Process ◽  
Capacity Zero ◽  
Smooth Measures ◽  
Generalized Dirichlet Forms ◽  
Necessary And Sufficient ◽  
Borel Measurable ◽  
Generalized Dirichlet

Introducing the corresponding strict capacity, we give necessary and sufficient conditions for a generalized Dirichlet form to be associated with a Hunt process. We also show that Borel measurable sets with strict capacity zero can be checked-out by an appropriate subclass of smooth measures. In the last part of this paper we present applications to three classes of examples.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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