Meixner class of non-commutative generalized stochastic processes with freely independent values II. The generating function

Let $T$ be an underlying space with a non-atomic measure $\sigma$ on it. In [{\it Comm.\ Math.\ Phys.}\ {\bf 292} (2009), 99--129] the Meixner class of non-commutative generalized stochastic processes with freely independent values, $\omega=(\omega(t))_{t\in T}$, was characterized through the continuity of the corresponding orthogonal polynomials. In this paper, we derive a generating function for these orthogonal polynomials. The first question we have to answer is: What should serve as a generating function for a system of polynomials of infinitely many non-commuting variables? We construct a class of operator-valued functions $Z=(Z(t))_{t\in T}$ such that $Z(t)$ commutes with $\omega(s)$ for any $s,t\in T$. Then a generating function can be understood as $G(Z,\omega)=\sum_{n=0}^\infty \int_{T^n}P^{(n)}(\omega(t_1),...,\omega(t_n))Z(t_1)...Z(t_n)\sigma(dt_1)...\sigma(dt_n)$, where $P^{(n)}(\omega(t_1),...,\omega(t_n))$ is (the kernel of the) $n$-th orthogonal polynomial. We derive an explicit form of $G(Z,\omega)$, which has a resolvent form and resembles the generating function in the classical case, albeit it involves integrals of non-commuting operators. We finally discuss a related problem of the action of the annihilation operators $\partial_t$, $t\in T$. In contrast to the classical case, we prove that the operators \di_t related to the free Gaussian and Poisson processes have a property of globality. This result is genuinely infinite-dimensional, since in one dimension one loses the notion of globality

Non-equilibrium stochastic dynamics in continuum: The free case

We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process $M$ on a Riemannian manifold $X$. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in $X$ such that, with probability one, infinitely many particles will arrive at this set at some time $t>0$. We assume that $X$ has infinite volume and, for each $\alpha\ge1$, we consider the set $\Theta_\alpha$ of all infinite configurations in $X$ for which the number of particles in a compact set is bounded by a constant times the $\alpha$-th power of the volume of the set. We find quite general conditions on the process $M$ which guarantee that the corresponding infinite particle process can start at each configuration from $\Theta_\alpha$, will never leave $\Theta_\alpha$, and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in $X$), and free Kawasaki dynamics on the configuration space. We also show that if $X=\mathbb R^d$, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics

On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum

We deal with two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions, in the $L^2$-norm with respect to the invariant measure of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair interaction, in the low activity-high temperature regime. Our result generalizes that of [Finkelshtein D.L. et al., to appear in Random Oper. Stochastic Equations], which was proved for a special Glauber (Kawasaki, respectively) dynamics

Binary jumps in continuum. II. Non-equilibrium process and a Vlasov-type scaling limit

Let $\Gamma$ denote the space of all locally finite subsets (configurations) in $\mathbb R^d$. A stochastic dynamics of binary jumps in continuum is a Markov process on $\Gamma$ in which pairs of particles simultaneously hop over $\mathbb R^d$. We discuss a non-equilibrium dynamics of binary jumps. We prove the existence of an evolution of correlation functions on a finite time interval. We also show that a Vlasov-type mesoscopic scaling for such a dynamics leads to a generalized Boltzmann non-linear equation for the particle density