Potential Theory and Right Processes

Beznea, Lucian; Boboc, Nicu

doi:10.1007/978-1-4020-2497-9

Cited by 44 publications

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“…(2.2) By Sections 1.7 and 1.8 in [2] and Theorem 1.3 in [4] we get that the following assertions are equivalent: (2.2a) Every U β -excessive measure dominated by a potential is also a potential.…”

Section: Path Regularity and Compact Lyapunov Functionsmentioning

confidence: 94%

“…For the proof of the equivalence between (i) and (ii) see [2] and [3], while for the proof of (i) ⇐⇒ (iii) see [8] and Remark 3.3 from [4].…”

Section: Compact Lyapunov Functions and Tightness Of Capacitiesmentioning

confidence: 99%

“…Any Ray topology is smaller than the fine topology and the resolvent U 1 is always the resolvent of a right process with state space E 1 endowed with a Ray topology; see [2].…”

Section: Path Regularity and Compact Lyapunov Functionsmentioning

confidence: 99%

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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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Section: Path Regularity and Compact Lyapunov Functionsmentioning

confidence: 94%

“…For the proof of the equivalence between (i) and (ii) see [2] and [3], while for the proof of (i) ⇐⇒ (iii) see [8] and Remark 3.3 from [4].…”

Section: Compact Lyapunov Functions and Tightness Of Capacitiesmentioning

confidence: 99%

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

Beznea

Röckner

2010

Complex Anal. Oper. Theory

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“…We recall that by Proposition 1.4 the process in Theorem 1.1 satisfies the martingale problem determined by L, hence is a martingale solution for (3).…”

Section: Applicationsmentioning

confidence: 96%

“…Assertion (a) follows now by ( * 3) and since there exists a µ-nest of T 0 -compact sets (see e.g. [3]). …”

Section: It Is Known That There Exists a Countable Setmentioning

confidence: 98%

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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Preliminaries on Transition Probabilities

Zaharopol

2014

Probability and Its Applications

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Potential Theory and Right Processes

Cited by 44 publications

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

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

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

Preliminaries on Transition Probabilities

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