arXiv subjects
Publications and source records attributed to Eugene W. Lytvynov.
We prove a characterization of the Dirichlet-Ferguson measure over an arbitrary finite diffuse measure space. We provide an interpretation of this characterization in analogy with the Mecke identity for Poisson point processes.
By definition, a Jacobi field $J=(J(ϕ))_{ϕ\in H_+}$ is a family of commuting selfadjoint three-diagonal operators in the Fock space $\mathcal F(H)$. The operators $J(ϕ)$ are indexed by the vectors of a real Hilbert space $H_+$. The spectral measure $ρ$ of the field $J$ is defined on the space $H_-$ of functionals over $H_+$. The image of the measure $ρ$ under a mapping $K^+:T_-\to H_-$ is a probability measure $ρ_K$ on $T_-$. We obtain a family $J_K$ of operators whose spectral measure is equal to $ρ_K$. We also obtain the chaotic decomposition for the space $L^2(T_-,dρ_K)$.
A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in $\mathbb{R}^d$ which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure $mu$ as invariant measure. We study a scaling limit of such a dynamics, derived through a scaling of the jump rate. Informally, we expect that, in the limit, only jumps of ``infinite length'' will survive, i.e., we expect to arrive at a Glauber dynamics in continuum (a birth-and-death process in $\mathbb{R}^d$). We prove that, in the low activity-high temperature regime, the generators of the Kawasaki dynamics converge to the generator of a Glauber dynamics. The convergence is on the set of exponential functions, in the $L^2(μ)$-norm. Furthermore, additionally assuming that the potential of pair interaction is positive, we prove the weak convergence of the finite-dimensional distributions of the processes.