Kategorientheoretische Behandlung der Zustandsraumtransformation von Markoffprozessen

Zessin HN. Kategorientheoretische Behandlung der Zustandsraumtransformation von Markoffprozessen. Journal für die reine und angewandte Mathematik . 1973;260:84-102

Point Processes in General Position

Zessin HN. Point Processes in General Position. Journal of Contemporary Mathematical Analysis. 2008;43(1):59-65.The aim of this note is to introduce for point processes in R-d the notions general position and reinforced general position, and to characterize these processes. As a consequence we show that Poisson processes P-rho with an infinite intensity measures. are in general position iff rho is diffuse in the sense that any affine subspace of dimension d - 1 is rho-nullset. Moreover, P-rho is in reinforced general position iff in addition any (d - 1)- sphere is a rho-nullset

A Theorem of Michael Murmann Revisited

Zessin HN. A Theorem of Michael Murmann Revisited. Journal of Contemporary Mathematical Analysis. 2008;43(1):50-58.A fundamental theorem of Murmann [2] characterizing equilibrium distributions of physical clusters is reconsidered. We recover this result by means of the integration by parts formula approach to Gibbs processes due to Nguyen Xuan Xanh and Hans Zessin [4]

On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit

Zessin HN. On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit. Probability Theory and Related Fields . 1988;77(4):605-622.We consider the time evolved statesP¯t=P¯∘θ−1t of the free motionθ t (q, v)=(q+tv,v),q,v∈ℝd, starting in some non-equilibrium stateP¯ and look at the associated processX ε t of fluctuations of the actual numberθ t/ε (μ)(1εA×B) of particles of the realization μ in1ε.A with velocities inB at timet/ε around its mean as ε→0 (i.e., in the hydrodynamic limit). It is shown that under natural conditions on the initial stateP¯, especially a mixing condition in the space variables, for eacht the laws of the fluctuations become Gaussian in the hydrodynamic limit in the following sense:P¯∘(Xεt)−1⇒Q¯t as ε→0, where ⇒ denotes weak convergence andQ¯t is a centered Gaussian state, which is translation invariant in the space variables. Furthermore the time evolution(Q¯t)t is also given by the free motion in the sense thatQ¯t=Q¯0∘θ−1t On the other hand we shall see thatP¯t⇒Pz⋅λ×σ ast→∞, whereP zηλ×τ is the Poisson process with intensity measurez·λ×τ, i.e., the equilibrium state for the free motion with particle densityz and velocity distribution τ. In the hydrodynamic limit this behaviour corresponds to the ergodic theorem for the fluctuation process:Q¯t⇒Q¯ ast→∞. HereQ¯ is a centered Gaussian state describing the equilibrium fluctuations, i.e., the fluctuations ofP zηλ×τ . Thus we prove the central limit theorem for the ideal gas: fluctuations are Gaussian even in non-equilibrium. The proofs rest on an adaption of the method of moments for sequences of generalized fields

On cluster dynamics of infinite Newtonian dynamics

Zessin HN. On cluster dynamics of infinite Newtonian dynamics. Journal of Mathematical Physics. 2023;64(1): 012706.On the basis of an evolution theorem for the infinite Newtonian dynamics, we give a rigorous construction of the associated cluster dynamics as a measurable flow. The latter is isomorphic to the former. This is enabled by a theorem of Murmann

The method of moments for random measures

Zessin HN. The method of moments for random measures. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete . 1983;62(3):395-409

Une caractérisation des diffusions par le calcul des variations stochastiques

Roelly S, Zessin HN. Une caractérisation des diffusions par le calcul des variations stochastiques. Comptes Rendus de l'Académie des Sciences - Series I - Mathematics. 1991;313(5):309-312.The Wiener measure is characterized as the unique diffusion law solution of the "equilibrium condition" which expresses that the Skorohod integral is equivalent to the dual of the derivative operator on Wiener space. This result is extended to the characterization of a large class of Wiener measures with drift

Cluster Representations of Classical and Quantum Processes

Poghosyan S, Zessin HN. Cluster Representations of Classical and Quantum Processes. Moscow Mathematical Journal. 2019;19(1):133-151.A cluster representation of projections onto bounded regions of limiting measures due to Minlos and Malyshev is developed under weaker conditions and in a general abstract way which covers not only classical but also permanental and determinantal processes

Cluster Representations of Classical and Quantum Processes

Poghosyan S, Zessin HN. Cluster Representations of Classical and Quantum Processes. Moscow Mathematical Journal. 2019;19(1):133-151.A cluster representation of projections onto bounded regions of limiting measures due to Minlos and Malyshev is developed under weaker conditions and in a general abstract way which covers not only classical but also permanental and determinantal processes

Une caractérisation des mesures de Gibbs sur C(0,1)Zd par le calcul des variations stochastiques

Roelly S, Zessin HN. Une caractérisation des mesures de Gibbs sur C(0,1)Zd par le calcul des variations stochastiques . Annales de l'Insitut Henri Poincaré - Probabilités et statistiques. 1993;29(3):327-338.Gibbs measures on C (0, 1)Z(d) associated to a given potential are characterized as the unique probability measures for which an equilibrium equation like (2. 6) holds, where appear a stochastic integral as operator, a derivative operator on path space and a Gibbsian density which takes into account the interaction between the particles. When the interaction desappears (free system) our result then gives a characterization of the infinite product of Wiener measures as the unique probability measure on C(0, 1)Z(d) under which the infinite dimensional Ito integral and the derivative operator are in duality