# E. W. Lytvynov

Search this author in Google Scholar

Articles: 11

### Quasi-invariance of completely random measures

Methods Funct. Anal. Topology 24 (2018), no. 3, 207-239

Let $X$ be a locally compact Polish space. Let $\mathbb K(X)$ denote the space of discrete Radon measures on $X$. Let $\mu$ be a completely random discrete measure on $X$, i.e., $\mu$ is (the distribution of) a completely random measure on $X$ that is concentrated on $\mathbb K(X)$. We consider the multiplicative (current) group $C_0(X\to\mathbb R_+)$ consisting of functions on $X$ that take values in $\mathbb R_+=(0,\infty)$ and are equal to 1 outside a compact set. Each element $\theta\in C_0(X\to\mathbb R_+)$ maps $\mathbb K(X)$ onto itself; more precisely, $\theta$ sends a discrete Radon measure $\sum_i s_i\delta_{x_i}$ to $\sum_i \theta(s_i)s_i\delta_{x_i}$. Thus, elements of $C_0(X\to\mathbb R_+)$ transform the weights of discrete Radon measures. We study conditions under which the measure $\mu$ is quasi-invariant under the action of the current group $C_0(X\to\mathbb R_+)$ and consider several classes of examples. We further assume that $X=\mathbb R^d$ and consider the group of local diffeomorphisms $\operatorname{Diff}_0(X)$. Elements of this group also map $\mathbb K(X)$ onto itself. More precisely, a diffeomorphism $\varphi\in \operatorname{Diff}_0(X)$ sends a discrete Radon measure $\sum_i s_i\delta_{x_i}$ to $\sum_i s_i\delta_{\varphi(x_i)}$. Thus, diffeomorphisms from $\operatorname{Diff}_0(X)$ transform the atoms of discrete Radon measures. We study quasi-invariance of $\mu$ under the action of $\operatorname{Diff}_0(X)$. We finally consider the semidirect product $\mathfrak G:=\operatorname{Diff}_0(X)\times C_0(X\to \mathbb R_+)$ and study conditions of quasi-invariance and partial quasi-invariance of $\mu$ under the action of $\mathfrak G$.

### The projection spectral theorem and Jacobi fields

Eugene Lytvynov

Methods Funct. Anal. Topology 21 (2015), no. 2, 188–198

We review several applications of Berezansky's projection spectral theorem to Jacobi fields in a symmetric Fock space, which lead to L\'evy white noise measures.

### A note on equilibrium Glauber and Kawasaki dynamics for permanental point processes

Methods Funct. Anal. Topology 17 (2011), no. 1, 29-46

We construct two types of equilibrium dynamics of an infinite particle system in a locally compact metric space $X$ for which a permanental point process is a symmetrizing, and hence invariant measure. The Glauber dynamics is a birth-and-death process in $X$, while in the Kawasaki dynamics interacting particles randomly hop over $X$. In the case $X=\mathbb R^d$, we consider a diffusion approximation for the Kawasaki dynamics at the level of Dirichlet forms. This leads us to an equilibrium dynamics of interacting Brownian particles for which a permanental point process is a symmetrizing measure.

### A note on equilibrium Glauber and Kawasaki dynamics for fermion point processes

Methods Funct. Anal. Topology 14 (2008), no. 1, 67-80

We construct two types of equilibrium dynamics of infinite particle systems in a locally compact Polish space $X$, for which certain fermion point processes are invariant. The Glauber dynamics is a birth-and-death process in $X$, while in the case of the Kawasaki dynamics interacting particles randomly hop over $X$. We establish conditions on generators of both dynamics under which corresponding conservative Markov processes exist.

### An equivalent representation of the Jacobi field of a Lévy process

E. Lytvynov

Methods Funct. Anal. Topology 11 (2005), no. 2, 188-194

### On a spectral representation for correlation measures in configuration space analysis

Methods Funct. Anal. Topology 5 (1999), no. 4, 87-100

### Analysis and geometry on ${\mathbb R}_{+}$-marked configuration space

Methods Funct. Anal. Topology 5 (1999), no. 1, 29-64

### Euclidean Gibbs states for quantum continuous systems with Boltzmann statistics via cluster expansion

Methods Funct. Anal. Topology 3 (1997), no. 1, 62-81

### Dual Appell systems in non-Gaussian white noise calculus

Methods Funct. Anal. Topology 2 (1996), no. 2, 70-85

### A generalization of Gaussian white noise analysis

Methods Funct. Anal. Topology 1 (1995), no. 1, 28-55

### Multiple Wiener integrals and non-Gaussian white noises: a Jacobi field approach

E. W. Lytvynov

Methods Funct. Anal. Topology 1 (1995), no. 1, 61-85