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

Abstract

We present an introduction (also for non{experts) to a new framework for the analysis of (up to) second order di erential operators (with merely measurable coe cients and in possibly in nitely many variables) on L 2 {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 C 0 {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 di erential operators that can be treated within the new framework only, e.g. parabolic di erential operators with unbounded drifts satisfying no L p {conditions, singular and fractional di usion operators. Subsequently, we analyze the probabilistic counterpart. More precisely, we identify necessary and su cient analytic properties of a generalized Dirichlet form on an arbitrary topological state space to be associated with a nice strong Markov process, e.g. a di usion. These results extend previous results on the existence of associated processes in the elliptic as well as the parabolic case.

Markov processes associated with Lp-resolvents and applications to stochastic differential equations on Hilbert space

“…Those two general theorems are formulated for general C 0 -resolvents on L p (E, µ) for abstract (Lusin) topological spaces E with Borel σ-algebra B and σ-finite measures µ on (E, B). In particular, Theorem 1.3 generalizes the corresponding results in [13] and [16], [17] and is the first of its kind on L p -measure spaces for arbitrary p ≥ 1 in the theory of Markov processes giving conditions on the generator directly, which can be verified in many models for stochastic dynamics.…”

“…e.g. [9], [13] and [16]) and even a strong solution if C is invertible and F 0 ∈ L p loc (dx), where dx denotes Lebesgue measure, and…”

“…Finally, we would like to mention that the main motivation for this paper (apart from aiming to extend the results in [13] and [16], [17]) came from [7]. The entire Section 5 of this paper is devoted to extend the results therein to the present case drawing a lot of ideas from [7].…”

Mosco Type Convergence of Bilinear Forms and Weak Convergence of n-Particle Systems

“…In the particular case of sequences of L 2 spaces we would like to refer to [14] which has a documented history beginning 2005. Initiated by the two established generalizations of Dirichlet forms to the non-symmetric case, namely [17] and [22], also Mosco (type) convergence for non-symmetric Dirichlet forms has been investigated, cf. [23] and [4].…”

“…However the framework here is more sophisticated. As an application, the particle system considered in [15] doesn't seem to be compatible with [17] or [22]. Moreover the limiting initial distribution is no longer concentrated on one single state as in [14].…”