scholarly journals EXISTENCE AND APPROXIMATION OF HUNT PROCESSES ASSOCIATED WITH GENERALIZED DIRICHLET FORMS

2011 ◽  
Vol 14 (04) ◽  
pp. 613-628 ◽  
Author(s):  
VITALI PEIL ◽  
GERALD TRUTNAU
Keyword(s):  
Dirichlet Form ◽  
Structural Condition ◽  
Dirichlet Forms ◽  
Poisson Processes ◽  
Hunt Process ◽  
Generalized Dirichlet Forms ◽  
Generalized Dirichlet

We show that any strictly quasi-regular generalized Dirichlet form that satisfies the mild structural condition D3 is associated to a Hunt process, and that the associated Hunt process can be approximated by a sequence of multivariate Poisson processes. This also gives a new proof for the existence of a Hunt process associated to a strictly quasi-regular generalized Dirichlet form that satisfies SD3 and extends all previous results.

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.

scholarly journals BSDE AND GENERALIZED DIRICHLET FORMS: THE FINITE-DIMENSIONAL CASE

2012 ◽  
Vol 15 (04) ◽  
pp. 1250022
Author(s):  
RONGCHAN ZHU
Keyword(s):  
Backward Stochastic Differential Equation ◽  
Generalized Solution ◽  
Dirichlet Forms ◽  
Stochastic Calculus ◽  
Order Differential Operator ◽  
Monotonicity Condition ◽  
Measurable Coefficients ◽  
Martingale Representation Theorem ◽  
Generalized Dirichlet Forms ◽  
Generalized Dirichlet

We consider the following quasi-linear parabolic system of backward partial differential equations [Formula: see text] where L is a possibly degenerate second-order differential operator with merely measurable coefficients. We solve this system in the framework of generalized Dirichlet forms and employ the stochastic calculus associated to the Markov process with generator L to obtain a probabilistic representation of the solution u by solving the corresponding backward stochastic differential equation. The solution satisfies the corresponding mild equation which is equivalent to being a generalized solution of the PDE. A further main result is the generalization of the martingale representation theorem using the stochastic calculus associated to the generalized Dirichlet form given by L. The nonlinear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to ∇u.

Symmetric Hunt Processes and Regular Dirichlet Forms

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Dirichlet Form ◽  
Dirichlet Forms ◽  
Hunt Process ◽  
Numerical Function ◽  
Locally Compact ◽  
Positive Radon Measure ◽  
The One ◽  
One Point Compactification ◽  
Separable Metric Space

This chapter studies a symmetric Hunt process associated with a regular Dirichlet form. Without loss of generality, the majority of the chapter assumes that E is a locally compact separable metric space, m is a positive Radon measure on E with supp[m] = E, and X = (Xₜ, Pₓ) is an m-symmetric Hunt process on (E,B(E)) whose Dirichlet form (E,F) is regular on L²(E; m). It adopts without any specific notices those potential theoretic terminologies and notations that are formulated in the previous chapter for the regular Dirichlet form (E,F). Furthermore, throughout this chapter, the convention that any numerical function on E is extended to the one-point compactification E ∂ = E ∪ {∂} by setting its value at δ‎ to be zero is adopted.

Dirichlet forms on partial *-algebras

1988 ◽  
Vol 104 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. O. S. Ekhaguere
Keyword(s):  
Markov Processes ◽  
Dirichlet Form ◽  
Dirichlet Forms ◽  
Function Spaces ◽  
Hunt Process ◽  
Partial Algebras ◽  
Markov Fields ◽  
Associated Function ◽  
Number Of Authors

Dirichlet forms and their associated function spaces have been studied by a number of authors [4, 6, 7, 12, 15–18, 22, 25, 26]. Important motivation for the study has been the connection of Dirichlet forms with Markov processes [16–18, 25, 26]: for example, to every regular symmetric Dirichlet form, there is an associated Hunt process [13, 20]. This makes the theory of Dirichlet forms a convenient source of examples of Hunt processes. In the non-commutative setting, Markov fields have been studied by several authors [1–3, 14, 19, 24, 28]. It is therefore interesting to develop a non-commutative extension of the theory of Dirichlet forms and to study their connection with non-commutative Markov processes.

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