The theory of generalized Dirichlet forms and its applications in analysis and stochastics

1999 ◽  
Vol 142 (678) ◽  
pp. 0-0 ◽  
Author(s):  
Wilhelm Stannat
Keyword(s):  
Dirichlet Forms ◽  
Generalized Dirichlet Forms ◽  
Generalized Dirichlet

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.

scholarly journals Recurrence Criteria for Generalized Dirichlet Forms

2017 ◽  
Vol 31 (4) ◽  
pp. 2129-2166 ◽  
Author(s):  
Minjung Gim ◽  
Gerald Trutnau
Keyword(s):  
Dirichlet Forms ◽  
Generalized Dirichlet Forms ◽  
Generalized Dirichlet

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.

scholarly journals BSDE and generalized Dirichlet forms: The infinite dimensional case

2015 ◽  
Vol 27 (1) ◽  
Author(s):  
Rong-Chan Zhu
Keyword(s):  
Banach Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Parabolic System ◽  
Dirichlet Forms ◽  
Dimensional Case ◽  
Partial Differential ◽  
Infinite Dimensional ◽  
Generalized Dirichlet Forms ◽  
Generalized Dirichlet

AbstractWe consider the following quasi-linear parabolic system of backward partial differential equations on a Banach space

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.

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