[Paper Review] Fukushima type decomposition for semi-Dirichlet forms

Zhi-Ming Ma, Wei Sun+1 more|arXiv (Cornell University)|Feb 18, 2014

Analytic Number Theory Research12 references3 citations

TL;DR

This paper establishes a Fukushima-type decomposition for quasi-regular semi-Dirichlet forms under a weakened assumption (Assumption 2.3), proving that a function $ u \in D(\mathcal{E})_{\text{loc}} $ admits such a decomposition if and only if it satisfies Condition (S), with the decomposition being unique. The result extends previous work by relaxing the need for sub-Markovian dual forms or local control conditions.

ABSTRACT

We present a Fukushima type decomposition in the setting of general quasi-regular semi-Dirichlet forms. The decomposition is then employed to give a transformation formula for martingale additive functionals. Applications of the results to some concrete examples of semi-Dirichlet forms are given at the end of the paper. We discuss also the uniqueness question about Doob-Meyer decomposition on optional sets of interval type.

Motivation & Objective

• To extend Fukushima-type decomposition from symmetric Dirichlet forms to general quasi-regular semi-Dirichlet forms without requiring the dual form to be sub-Markovian.
• To establish a necessary and sufficient condition (Condition S) for the existence of the decomposition in the semi-Dirichlet setting.
• To provide a transformation formula for martingale additive functionals using the decomposition.
• To investigate the uniqueness of the Doob-Meyer decomposition on optional sets of interval type.
• To verify the applicability of the results to concrete examples of semi-Dirichlet forms, including non-local and non-symmetric cases.

Proposed method

• Utilizes the semi-$ h $-transform method to associate a quasi-regular semi-Dirichlet form with a sub-Markovian dual form.
• Employs a localization and pasting technique to handle the decomposition on local sets, particularly on $ I(\zeta) = [\![0,\zeta[\![ \cup [\![\zeta_i]\!] $, where $ \zeta $ is the lifetime and $ \zeta_i $ the totally inaccessible part.
• Introduces Assumption 2.3, a weaker condition than local control (in [14]) or condition $ (\mathcal{E}.5) $ (in [19]), to ensure the existence of the decomposition.
• Applies the concept of $ \mathcal{E} $-quasi-continuous versions of functions to define the decomposition pathwise on the sample paths of the associated Markov process.
• Derives a transformation formula for martingale additive functionals using the decomposition, generalizing Itô’s formula in the Dirichlet form framework.
• Uses energy estimates and weak convergence arguments to prove the convergence of energy norms, particularly in the limit $ t \downarrow 0 $, to show the vanishing of cross-variation terms.

Experimental results

Research questions

• RQ1 Under what conditions does a Fukushima-type decomposition exist for functions in the local domain of a quasi-regular semi-Dirichlet form?
• RQ2 Is the decomposition unique when it exists, and what is the role of Condition (S) in ensuring this?
• RQ3 How does the decomposition extend to non-local and non-symmetric semi-Dirichlet forms?
• RQ4 Can a transformation formula for martingale additive functionals be derived from the decomposition?
• RQ5 What is the relationship between the Doob-Meyer decomposition and the Fukushima-type decomposition on optional sets of interval type?

Key findings

• A function $ u \in D(\mathcal{E})_{\text{loc}} $ admits a Fukushima-type decomposition if and only if it satisfies Condition (S), and the decomposition is unique under Assumption 2.3.
• The decomposition takes the form $ \tilde{u}(X_t) - \tilde{u}(X_0) = M^{[u]}_t + N^{[u]}_t $, where $ M^{[u]} $ is a locally square-integrable martingale additive functional and $ N^{[u]} $ is a continuous additive functional of zero quadratic variation on $ I(\zeta) $.
• Assumption 2.3 is strictly weaker than the local control condition in [14] and condition $ (\mathcal{E}.5) $ in [19], broadening the applicability of the decomposition.
• The transformation formula for martingale additive functionals is derived, generalizing classical results from the Dirichlet form framework.
• The results are applied to two concrete examples: a non-local semi-Dirichlet form with jump component and a non-symmetric diffusion-jump generator, both satisfying the required conditions.
• For both examples, the theorems hold for all $ u \in D(\mathcal{E})_{\text{loc}} $ satisfying Condition (S), and in particular for all $ u \in D(\mathcal{E}) $, due to the inequality $ |k_a(x,y)| \leq k_s(x,y) $.