Quasi-regular Dirichlet Forms and $L^p$ -resolvents on Measurable Spaces

Beznea, Lucian; Boboc, Nicu; Röckner, Michael

doi:10.1007/s11118-006-9016-2

Cited by 27 publications

(22 citation statements)

References 18 publications

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“…dm for all f, g ∈ pB and α > 0; see Corollary 2.4 in [4] for the existence of such a resolvent. Notice that if g ∈ pB is such that U * q g < ∞ m-a.e.…”

Section: Preliminariesmentioning

confidence: 99%

Feynman-Kac Formula for Left Continuous Additive Functionals and Extended Kato Class Measures

Beznea

Boboc

2008

Potential Anal

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The perturbation of the generator of a Borel right process by a signed measure is investigated, using probabilistic and analytic potential theoretical methods. We establish a Feynman-Kac formula associated with measures charging no polar set and belonging to an extended Kato class. A main tool of this approach is the validity of a Khas'minskii Lemma for Stieltjes exponentials of positive left continuous additive functionals.Keywords Feynman-Kac formula · Extended Kato class · Positive left additive functional · Khas'minskii lemma · L p -resolvent · Borel right process Mathematics Subject Classifications (2000) 60J45 · 47D08 · 60J40 · 60J35 · 47D07

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“…dm for all f, g ∈ pB and α > 0; see Corollary 2.4 in [4] for the existence of such a resolvent. Notice that if g ∈ pB is such that U * q g < ∞ m-a.e.…”

Section: Preliminariesmentioning

confidence: 99%

Feynman-Kac Formula for Left Continuous Additive Functionals and Extended Kato Class Measures

Beznea

Boboc

2008

Potential Anal

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“…By Remark 2.3 in [4] there exists a sub-Markovian resolvent of kernels U = (U α ) α>0 on (E, B) such that U α = V α as operators on L p (E, µ) for all α > 0, and the following condition is satisfied for one (and therefore for all) β > 0:…”

Section: Resultsmentioning

confidence: 98%

“…We have U α (1 M ) = 0 µ-a.e., because µ(M ) = 0. By Lemma 2.1 in [4] there exists a set F ∈ B such that U α (1 F ) = 0 on E \ F for all α > 0, more precisely we have F = F n where F n+1 = F n ∪ U α (1 Fn ) > 0 for all n ≥ 0, with F 0 = M . Since the function U α (1 Fn ) is µ-quasi continuous and U α (1 Fn ) = 0 µ-a.e., it follows that the set U α (1 Fn ) > 0 is µ-exceptional for all n, hence F is also µ-exceptional.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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

2006

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Abstract. We give general conditions on a generator of a C 0 -semigroup (resp. of a C 0 -resolvent) on L p (E, µ), p ≥ 1, where E is an arbitrary (Lusin) topological space and µ a σ-finite measure on its Borel σ-algebra, so that it generates a sufficiently regular Markov process on E. We present a general method how these conditions can be checked in many situations. Applications to solve stochastic differential equations on Hilbert space in the sense of a martingale problem are given. IntroductionIn this paper we study stochastic differential equations (SDE) in infinite dimensions, e.g. on a Hilbert space H, with possibly very singular coefficients. For such equations, strong or mild solutions (cf. [8]) do not exist in general, but it is only possible to construct weak solutions or even only martingale solutions, i.e. a Markov process that solves the martingale problem for the (partial) differential operator ("Kolmogorov operator") associated with the SDE (cf. Proposition 1.4 below and [18] for the general theory in finite dimensions). The latter notion of solution we shall briefly call "martingale solutions". The notions of weak and martingale solutions are only equivalent in finite dimensions (cf. [18]), but not in infinite dimensions. Under additional hypotheses, however, one can construct weak solutions from martingale solutions also in the infinite dimensional case. We refer to [2] and [14] for a detailed discussion.Given an SDE on a Hilbert space H (by heuristically applying Itô's formula, see Section 5 below) one can always write down the corresponding Kolmogorov operator L 0 on a space of nice test functions. If one can prove that its closure L generates a C 0 -semigroup P t = e tL , t ≥ 0, on L p (H, µ) for some suitably chosen measure µ (see Section 5) and if this semigroup is sufficiently regular, then one can prove that there exists a Markov process with transition probabilities given by P t , t ≥ 0. This process then automatically solves the martingale problem for L, and thus is a martingale solution to the SDE.The main results of this paper give general conditions, which are checkable in applications, on a given Kolmogorov operator L, more precisely, on the generator of a C 0 -semigroup (or C 0 -resolvent) on L p (H, µ), p ≥ 1, so that an associated sufficiently regular Markov process (namely, a µ-standard right process) exists giving the desired solution to the martingale problem determined 1

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“…Then there exists sub-Markovian resolvent of kernels U = (U α ) α>0 on (E, B) which satisfies (2.1) and such that U α = V α as operators on L p (E, m) for all α > 0; see Remark 2.3 in [5].…”

Section: The General Framementioning

confidence: 99%

Applications of Compact Superharmonic Functions: Path Regularity and Tightness of Capacities

Beznea

Röckner

2010

Complex Anal. Oper. Theory

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Abstract. We establish relations between the existence of the L-superharmonic functions that have compact level sets (L being the generator of a right Markov process), the path regularity of the process, and the tightness of the induced capacities. We present several examples in infinite dimensional situations, like the case when L is the Gross-Laplace operator on an abstract Wiener space and a class of measure-valued branching process associated with a nonlinear perturbation of L. Mathematics Subject Classification (2000). Primary 60J45, 31C15; Secondary 47D07, 60J40, 60J35.

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Quasi-regular Dirichlet Forms and $L^p$ -resolvents on Measurable Spaces

Cited by 27 publications

References 18 publications

Feynman-Kac Formula for Left Continuous Additive Functionals and Extended Kato Class Measures

Feynman-Kac Formula for Left Continuous Additive Functionals and Extended Kato Class Measures

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

Applications of Compact Superharmonic Functions: Path Regularity and Tightness of Capacities

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