We present an introduction (also for non-experts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²-spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, so-called generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 --semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p --conditions, singular and fractional diffusion operators. Subsequently, we analyz...

We prove that for a certain class of bilinear forms satisfying some regularity conditions which include quasi-regular Dirichlet forms (cf. [3]) and time dependent Dirichlet forms (cf. [5]) as particular cases there exists an associated strong Markov process having nice sample path properties. These forms, called generalized Dirichlet forms, are the sum of a coercive part and a perturbation (e.g. the time derivative in the time dependent case), so that in general neither the sector condition is fulfilled by the sum nor is the associated L 2 -semigroup analytic. A wide variety of new examples can be treated in this extended framework of Dirichlet forms including fractional diffusion operators and transformations of time dependent Dirichlet forms by ff-excessive functions h (h-transformations). Formes de Dirichlet g'en'eralis'ees et processus de Markov associ'es R'esum'e - Nous construisons des processus standard sp'eciaux associ'es `a certaines formes bilin'eaires qui satisfont `a quel...

We extend the framework of classical Dirichlet forms to a class of bilinear forms, called generalized Dirichlet forms, which are the sum of a coercive part and a linear unbounded operator as a perturbation. The class of generalized Dirichlet forms, in particular, includes symmetric and coercive Dirichlet forms (cf. [Fu2], [M/R]) as well as time dependent Dirichlet forms (cf. [O1]) as special cases and also many new examples. Among these are, e.g. transformations of time dependent Dirichlet forms by ff-excessive functions h (h-transformations), Dirichlet forms with time dependent linear drift and fractional diffusion operators. One of the main results is that we identify an analytic property of these forms which ensures the existence of associated strong Markov processes with nice sample path properties, and give an explicit construction for such processes. This construction extends previous constructions of the processes in the elliptic and the parabolic cases, is, in particular, c...

We extend the main result in [A/M/R], which is a complete characterization of all Dirichlet forms defined on some L²-space L²(E; m) associated with m-tight special standard processes, to the framework of generalized Dirichlet forms.

Abstract. We present recent advances on Dirichlet forms methods either to extend financial models beyond the usual stochastic calculus or to study stochastic models with less classical tools. In this spirit, we interpret the asymptotic error on the solution of an sde due to the Euler scheme (Kurtz and Protter [Ku-Pr-91a]) in terms of a Dirichlet form on the Wiener space, what allows to propagate this error thanks to functional calculus.

Weak Dirichlet processes with a stochastic control perspective.