Narrow Results

We construct an infinite particle/infinite volume Langevin dynamics on the space of simple configurations in ℝ^d having velocities as marks. The construction is done via a limiting procedure using N-particle dynamics in cubes (-λ, λ]^d with periodic boundary condition. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of N-particle systems in (-λ, λ]^d with periodic boundary condition. After proving tightness of the laws of the finite particle dynamics, the identification of accumulation points as martingale solutions of the Langevin equation is based on a general study of properties of measures on configuration space fulfilling a uniform Ruelle bound (and their weak limits). Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for a wide class of repulsive interaction potentials ϕ (including, e.g., the Lennard–Jones potential) and all temperatures, densities and dimensions d ≥ 1.

We study two-faced families of non-commutative random variables having bi-free (additive) infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced families of random variables within a triangular array. As a corollary of our limit theorem, we get Voiculescu’s bi-free central limit theorem. Using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.

It has been recently discovered that some random processes may satisfy limit theorems even though they exhibit intermittency, namely an unusual growth of moments. In this paper, we provide a deeper understanding of these intricate limiting phenomena. We show that intermittent processes may exhibit a multiscale behavior involving growth at different rates. To these rates correspond different scales. In addition to a dominant scale, intermittent processes may exhibit secondary scales. The probability of these scales decreases to zero as a power function of time. For the analysis, we consider large deviations of the rate of growth of the processes. Our approach is quite general and covers different possible scenarios with special focus on the so-called supOU processes.

We show how the construction of t-transformation can be applied to the construction of a sequence of monotonically independent noncommutative random variables. We introduce the weakly monotone Fock space, on which these operators act. This space can be derived in a natural way from the papers by Pusz and Woronowicz on twisted second quantization. It was observed by Bożejko that, by taking μ = 0, for the μ-CAR relations one obtains the Muraki's monotone Fock space, while for the μ-CCR relations one obtains the weakly monotone Fock space. We show that the direct proof of the central limit theorem for these operators provides an interesting recurrence for the highest binomial coefficients. Moreover, we show the Poisson type theorem for these noncommutative random variables.

We define two families of deformations of probability measures depending on the second free cumulants and the corresponding new associative convolutions arising from the conditionally free convolution. These deformations do not commute with dilation of measures, which means that the limit theorems cannot be obtained as a direct application of the theorems for the conditionally free case. We calculate the general form of the central and Poisson limit theorems. We also find the explicit form for three important examples.

In Ref. 2 the authors introduced field operators in one-mode type Interacting Fock Spaces whose spectral measures have common symmetric Jacobi recurrence coefficients but differ in the nonsymmetric ones. We show that the convolution of measures arising from addition of such field operators is the universal convolution of Accardi and Bożejko. We also present the associated central limit theorem in a more general form than in Ref. 2 and give it a proof based on the properties of the convolution.

Infinite divisibility for the free additive convolution was studied in Ref. 20. A complete characterization of -infinitely divisible distributions was given, and it was explained in Ref. 21 that this characterization is an analogue of the classical Lévy–Khintchine characterization. In fact, the analogue of the Gaussian distribution appeared even earlier, when the central limit theorem for free additive convolution was proven in Ref. 19.

In this paper we define the notion of -infinitely divisibility and give the description of infinitely divisible compactly supported probability measures relative to the conditionally free convolution. We also show that the Lévy–Khintchine measures associated with a -infinitely divisible distribution μ can be calculated, as in the classical or free case, as a weak limit of measures related with the convolution semigroup generated by (μ, φ) for -infinitely divisible.

We study the properties of the (noncommutative) bm-independence of algebras, indexed by partially ordered sets. The index sets are given by positive cones, in particular the symmetric cones, which include the positive-definite Hermitian matrices with complex or quaternionic entries. We formulate and prove the general versions of the bm-Central Limit Theorems for bm-independent random variables, indexed by lattices in such positive cones. The limit measures we obtain are symmetric and compactly supported on the real line. Their (even) moment sequences (g_n)_n≥0 satisfy the generalized recurrence for the Catalan numbers: , where the coefficients γ(r) are computed by the Euler's beta-function of the first kind, related to the given positive cone. Example of a nonsymmetric cone, the Vinberg's cone, is also studied in this context.

Assuming that the fast motion in averaging is sufficiently well mixing we show that the slow motion can be approximated in the L²-sense by a diffusion solving Hasselmann's nonlinear stochastic differential equation and which provides a much better approximation than the one suggested by the averaging principle. Previously, only weak limit theorems in averaging were known which cannot justify, in principle, a nonlinear diffusion approximation of the slow motion.

We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet W^{α, β} with Hurst parameter (α, β) ∈ (0, 1)². When or a central limit theorem holds for the renormalized Hermite variations of order q ≥ 2, while for we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time (1, 1).

The central result of this paper is the existence of limiting distributions for two classes of critical homogeneous-in-space branching processes with heavy tails spatial dynamics in dimension d = 2. In dimension d ≥ 3, the same results are true without any special assumptions on the underlying (non-degenerated) stochastic dynamics.