On Poisson multiple stochastic integrals and associated equilibrium Markov processes

Сургайлис, Донатас

doi:10.1007/bfb0044696

Cited by 95 publications

(169 citation statements)

References 9 publications

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“…In the case where the interaction between the particles is absent (i.e., φ = 0 and, therefore, µ is the Poisson measure π z with intensity z), the Markov process corresponding to the Dirichlet form (1.1) was explicitly constructed and studied by D. Surgailis [24,25].…”

Section: Introductionmentioning

confidence: 99%

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Glauber dynamics of continuous particle systems

Kondratiev

Lytvynov

2005

Annales de l'Institut Henri Poincare (B) Probability and Statis

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This paper is devoted to the construction and study of an equilibrium Glauber-type dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in R d , we fix a Gibbs measure µ corresponding to a general pair potential φ and activity z > 0. We consider a Dirichlet form E on L 2 (Γ, µ) which corresponds to the generator H of the Glauber dynamics. We prove the existence of a Markov process M on Γ that is properly associated with E. In the case of a positive potential φ which satisfies δ:= R d (1 − e −φ(x) ) z dx < 1, we also prove that the generator H has a spectral gap ≥ 1 − δ. Furthermore, for any pure Gibbs state µ, we derive a Poincaré inequality. The results about the spectral gap and the Poincaré inequality are a generalization and a refinement of a recent result from [6]. MSC: 60K35, 60J75, 60J80, 82C21, 82C22

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Section: Introductionmentioning

confidence: 99%

“…[24]. Under this isomorphism, the operator H goes over into the number operator N in the Fock space, see [1,Theorem 5.1].…”

Section: Introductionmentioning

confidence: 99%

Glauber dynamics of continuous particle systems

Kondratiev

Lytvynov

2005

Annales de l'Institut Henri Poincare (B) Probability and Statis

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“…The version of (1.6) with a ≡ 0 is known as the Surgailis model, see [14] and the discussion in [3]. This model is exactly soluble.…”

Section: Without Competitionmentioning

confidence: 99%

Evolution of states in a continuum migration model

Kondratiev

Kozitsky

2017

Anal.Math.Phys.

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The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in R d in which the constituents appear (immigrate) with rate b(x) and disappear, also due to competition. For this model, we prove the existence of the evolution of states μ 0 → μ t such that the moments μ t (N n ), n ∈ N, of the number of entities in compact ⊂ R d remain bounded for all t > 0. Under an additional condition, we prove that the density of entities and the second correlation function remain point-wise bounded globally in time.

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“…It is well known that in the Wiener case we have the relation P ( n) t H n (X t , t), where H n (x, y) y na2 h n (xa p y) and h n is the Hermite polynomial of degree n. Also in the Poisson case, we have P ( n) t C n (X t , t), where fC n (x, y)g are the Charlier polynomials (Meyer 1976;Surgailis 1984).…”

Section: Existence Of a Family Of Orthogonal Polynomials Associated Wmentioning

confidence: 99%

“…Our ®rst goal in this paper is to relate the product formulae of Kabanov (1975), Russo and Vallois (1998) and Surgailis (1984) to their counterparts in quantum probability; see Section 3. With these tools we compute the chaotic expansion of the product of a multiple stochastic integral with a single stochastic integral, and refer to this formula as the chaotic Kabanov formula.…”

Section: Introductionmentioning

confidence: 99%

Chaotic Kabanov Formula for the Azéma Martingales

2000

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We derive the chaotic expansion of the product of nth-and ®rst-order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulae for multiple stochastic integrals. Our approach extends existing results on chaotic calculus for normal martingales and exhibits properties, relative to multiple stochastic integrals, polynomials and Wick products, that characterize the Wiener and Poisson processes.

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On Poisson multiple stochastic integrals and associated equilibrium Markov processes

Cited by 95 publications

References 9 publications

Glauber dynamics of continuous particle systems

Glauber dynamics of continuous particle systems

Evolution of states in a continuum migration model

Chaotic Kabanov Formula for the Azéma Martingales

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