semi-Dirichlet Forms Research Articles

Variational Inequalities and Optimization for Markov Processes Associated with Semi-Dirichlet Forms

We consider an optimal stopping problem and an impulsive control for Markov processes associated with semi-Dirichlet forms. We show that the value functions are the maximum solutions of certain variational inequalities. Examples both in finite dimensions and infinite dimensions are given.

Maximum Principles for Subharmonic Functions Via Local Semi-Dirichlet Forms

AbstractMaximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

Extensions of Lévy–Khintchine formula and Beurling–Deny formula in semi-Dirichlet forms setting

The Lévy–Khintchine formula or, more generally, Courrège's theorem characterizes the infinitesimal generator of a Lévy process or a Feller process on R d . For more general Markov processes, the formula that comes closest to such a characterization is the Beurling–Deny formula for symmetric Dirichlet forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet form. We start with decomposing a regular semi-Dirichlet form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet forms. Finally, we extend the formulae of Lévy–Khintchine and Beurling–Deny in semi-Dirichlet forms setting through introducing a quasi-compatible metric.

Beurling–Deny formula of semi-Dirichlet forms

In this Note we announce a structure result for non-symmetric Dirichlet forms and semi-Dirichlet forms. Our result is regarded as an extension of the celebrated Beurling–Deny formula which is up to now available only for symmetric Dirichlet forms. The result can also be regarded as an extension of Lévy–Khinchine formula or more generally, an extension of Courrège's Theorem in the semi-Dirichlet forms setting. To cite this article: Z.-C. Hu, Z.-M. Ma, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Fine Densities for Excessive Measures and the Revuz Correspondence

Suppose that U is the resolvent of a Borel right process on a Lusin space X. If ξ is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure η with η≪ξ the Radon–Nikodym derivative dη/dξ possesses a finely continuous version. (Fitzsimmons and Fitzsimmons and Getoor gave a probabilistic approach for this result.) We extend essentially a technique initiated by Mokobodzki and deepened by Feyel. The result allows us to establish a Revuz type formula involving the fine versions, and to study the Revuz correspondence between the σ-finite measures charging no set that is both ξ-polar and ρ-negligible (ρ○U being the potential component of ξ) and the strongly supermedian kernels on X. This is an analytic version of a result of Azema, Fitzsimmons and Dellacherie, Maisonneuve and Meyer, in terms of additive functionals or homogeneous random measures. Finally we give an application to the context of the semi-Dirichlet forms, covering a recent result of Fitzsimmons.

On the Quasi-regularity of Semi-Dirichlet Forms

We prove that if a right Markov process is associated with a semi-Dirichlet form, then the form is necessarily quasi-regular. As applications, we develop the theory of Revuz measures in the semi-Dirichlet context and we show that quasi-regularity is invariant with respect to time change.

Markov processes associated with semi-Dirichlet forms

Markov processes associated with semi-Dirichlet forms

Characterization of (non-symmetric) semi-Dirichlet forms associated with Hunt processes

Article Characterization of (non-symmetric) semi-Dirichlet forms associated with Hunt processes was published on January 1, 1995 in the journal Random Operators and Stochastic Equations (volume 3, issue 2).