Infinite Dimensional Analysis, Quantum Probability and Related Topics

We prove that, replacing the left Jordan–Wigner $q$-embedding by the symmetric $q$-embedding described in Sec. 2, the result of the corresponding central limit theorem changes drastically with respect to those obtained in Ref. 5. In fact, in the former case, for any $q\in ℂ$, the limit space is precisely the $1$-mode Interacting Fock Space (IFS) that realizes the canonical quantum decomposition of the limit classical random variable. In the latter case, this happens if and only if $q=\pm 1$. Furthermore, as shown in Sec. 4, the limit classical random variable turns out to coincide with the $1$-mode version of the $q^{\mathrm{\Lambda }}$-deformed quantum Brownian introduced by Parthasarathy${}^{8,9}$, and extended to the general context of bi-algebras by Schürman${}^{10,11}$. The last section of the paper (Appendix) describes this continuous version in white noise language, leading to a simplification of the original proofs, based on quantum stochastic calculus.

We study the Lévy infinite-dimensional differential operators (differential operators defined by the analogy with the Lévy Laplacian) and their relationship to the Yang–Mills equations. We consider the parallel transport on the space of curves as an infinite-dimensional analogue of chiral fields and show that it is a solution to the system of differential equations if and only if the associated connection is a solution to the Yang–Mills equations. This system is an analogue of the equations of motion of chiral fields and contains the Lévy divergence. The systems of infinite-dimensional equations containing Lévy differential operators, that are equivalent to the Yang–Mills–Higgs equations and the Yang–Mills–Dirac equations (the equations of quantum chromodynamics), are obtained. The equivalence of two ways to define Lévy differential operators is shown.

By using the asymptotic coupling method, the asymptotic log-Harnack inequality is established for the transition semigroup associated to the 3D Leray-$\alpha$ model with fractional dissipation driven by highly degenerate noise. As applications, we derive the asymptotic strong Feller property and ergodicity for the stochastic 3D Leray-$\alpha$ model with fractional dissipation, which is the stochastic 3D Navier–Stokes equation regularized through a smoothing kernel of order ${𝜃}_1$ in the nonlinear term and a ${𝜃}_2$-fractional Laplacian. The main results can be applied to the classical stochastic 3D Leray-$\alpha$ model (${𝜃}_1={𝜃}_2=1$), stochastic 3D hyperviscous Navier–Stokes equation (${𝜃}_1=0,{𝜃}_2\ge \frac{5}{4}$) and stochastic 3D critical Leray-$\alpha$ model (${𝜃}_1=\frac{1}{4},{𝜃}_2=1$).

In this paper, we first introduce the notion of controlled weaving $K$-$g$-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving $K$-$g$-frames in separable Hilbert spaces. Also, a characterization of controlled weaving $K$-$g$-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled $K$-$g$-woven frames give ordinary weaving $K$-frames and vice-versa.

We introduce an infinite-dimensional version of the classical Laplace method, in its original form, relative to a canonical Gaussian measure associated with a Hilbert space, and for a general phase function. Particular attention is given to the case of a phase function with finite-dimensional degeneracy. Explicit results on expansions in the form of power series in the relevant parameter, with estimates on remainders, are provided.

Motivated by an observation of Namioka and Phelps on an approximation property of order unit spaces, we introduce the $p$-tensor product and the $m$-tensor product of two compact matrix convex sets. We define a new approximation property for operator systems, and give a characterization using the $p$- and $m$-tensor products in the spirit of Grothendieck. Thus, an operator system has the operator system approximation property if and only if it is $(p,m)$-nuclear in a natural sense.

In this paper, we study the stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise in three dimensions with a hereditary viscous term which depends on the past history. We establish the local solvability of the Cauchy problem for such systems. The local monotonicity property of the nonlinear term of the cutoff problem and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. Finally, we show that the global solvability results hold under smallness condition on the initial data and suitable assumptions on the noise coefficients.

A stochastic integral inequality of the Gronwall type is established. The result complements a stochastic Gronwall lemma proved by Scheutzow³.