Let 𝒫2(ℝd) be the space of probability measures on ℝd with finite second moment. The path independence of additive functionals of McKean–Vlasov SDEs is characterized by PDEs on the product space ℝd×𝒫2(ℝd) equipped with the usual derivative in space variable and Lions’ derivative in distribution. These PDEs are solved by using probabilistic arguments developed from Ref. 2. As a consequence, the path

We present a new more flexible definition of interacting Fock space that allows to resolve in full generality the problem of embeddability. We show that the same is not possible for regularity. We apply embeddability to classify interacting Fock spaces by squeezings. We give necessary and sufficient criteria for when an interacting Fock space has only bounded creators, giving thus rise to new classes

In this paper, we consider Quantum Markov States (QMS) corresponding to the Ising model with competing interactions on the Cayley tree of order two. Earlier, some algebraic properties of these states were investigated. In this paper, we prove that if the competing interaction is rational then the von Neumann algebra, corresponding to the QMS associated with disordered phase of the model, has type IIIλ

Strassen’s theorem circa 1965 gives necessary and sufficient conditions on the existence of a probability measure on two product spaces with given support and two marginals. In the case where each product space is finite, Strassen’s theorem is reduced to a linear programming problem which can be solved using flow theory. A density matrix of bipartite quantum system is a quantum analog of a probability

In this paper, we will consider the minima of an exponentially growing number of independent time-inhomogeneous random walks, where the first- and second-order limit behaviors for the minima have been obtained.

In the framework of the Hudson–Parthasarathy quantum stochastic calculus, we employ a recent generalization of the Michael selection results in the present noncommutative settings to prove that the function space of the matrix elements of solutions to discontinuous quantum stochastic differential inclusion (DQSDI) is arcwise connected.

In this paper, we establish a Freidlin–Wentzell-type large deviation principle for stochastic models of two-dimensional second grade fluids driven by Lévy noise. The weak convergence method introduced by Budhiraja, Dupuis and Maroulas plays a key role.

Let Φ be a locally convex space and let Φ′ denote its strong dual. In this paper, we introduce sufficient conditions for the existence of a continuous or a càdlàg Φ′-valued version to a cylindrical process defined on Φ′. Our result generalizes many other known results on the literature and their different connections will be discussed. As an application, we provide sufficient conditions for the existence

In this paper, we introduce a new white noise delta function based on the Kubo–Yokoi delta function and an infinite-dimensional Brownian motion. We also give a white noise differential equation induced by the delta function through the Itô formula introducing a differential operator directed by the time derivative of the infinite-dimensional Brownian motion and an extension of the definition of the

In this paper, with the help of a variant of Schauder fixed point theorem in the real Banach algebra together with the finite difference method (FDM), we take a brief look at the p-adic analog of Richards’ equation derived by Khrennikov et al. [Application of p-adic wavelets to model reaction–diffusion dynamics in random porous media, J. Fourier Anal. Appl.22 (2016) 809–822], and study the solvability

We prove a stochastic Gronwall lemma of the convolution type. Our results extend that of Scheutzow [A stochastic Gronwall lemma, Infin. Dimens. Anal. Quantum Probab. Relat. Top.16 (2013) 1350019], and the related results established in the non-convolution case. The proofs of the present results are essentially based on the Métivier–Pellaumail inequality for semimartingales.

The equivalence of the anti-selfduality Yang–Mills equations on the four-dimensional orientable Riemannian manifold and the Laplace equations for some infinite-dimensional Laplacians is proved. A class of modified Lévy Laplacians parameterized by the choice of a curve in the group SO(4) is introduced. It is shown that a connection is an instanton (a solution of the anti-selfduality Yang–Mills equations)

It is shown that if a Quantum Markov semigroup is “multiplicatively” perturbed under suitable conditions, the resulting minimal semigroup remains Markov (or conservative).

We consider a stochastic Cahn–Hilliard equation driven by a space–time white noise. We prove that the law of the solution satisfies a large deviations principle in the Hölder norm. Our proof is based on the weak convergence approach for large deviations.

The importance of a law of large numbers is well known. In this paper, conditional law of large numbers for Markov processes is proved, which can be used in computing quantities related to sub-Markov sequences. A variational interpretation for this limit is given, which shows that typically the quantities of interests tend to minimize a certain entropy. A quasi-compact operator argument is involved

We introduce the notion of acceptable noncommutative random variables and investigate their essential properties. More precisely, we provide several efficient estimation of tail probabilities of sums of noncommutative random variables under some mild conditions. Moreover, we investigate the complete convergence of a sequence of the form 1λn∑i=1nxj in which (xn) is a sequence of acceptable noncommutative

We show that ergodic flows in the noncommutative L1-space (associated with a semifinite von Neumann algebra) generated by continuous semigroups of positive Dunford–Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly. The corresponding local ergodic theorem is also proved. We then extend these results to arbitrary noncommutative fully symmetric

We prove the existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we

After introducing g-frames and fusion frames by Sun and Casazza, respectively, combining these frames together is an interesting topic for research. In this paper, we introduce the generalized fusion frames or g-fusion frames for Hilbert spaces and give characterizations of these frames from the viewpoint of closed range and g-fusion frame sequences. Also, the canonical dual g-fusion frames are presented

In this paper, we study a unified approach for quantum Markov chains (QMCs). A new quantum Markov property that generalizes the old one, is discussed. We introduce Markov states and chains on general local algebras, possessing a generic algebraic property. We stress that this kind of algebras includes both Boson and Fermi algebras. Our main results concern two reconstruction theorems for quantum Markov

In this paper, we consider the estimation of the reliability in a stress–strength model by the maximum likelihood and Bayesian methods under generalized exponential distribution. We provide the estimation of the reliability with simple random sampling and ranked set sampling methods. Lindley’s algorithm is used to obtain the approximate Bayesian estimation of the reliability with gamma priors. The

We introduce an infinite-dimensional p-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However, it is possible to define its action on some classes of functions.

We prove that every stationary state in the annihilator of all Kraus operators of a weak coupling limit-type Markov generator consists of two pieces, one of them supported on the interaction-free subspace and the second one on its orthogonal complement. In particular, we apply the previous result to describe in detail the structure of a slightly modified quantum transport model due to Arefeva et al

Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure μ. The associated Cameron–Martin space is denoted by H. Consider two sufficiently regular convex functions U:X→ℝ and G:X→ℝ. We let ν=e−Uμ and Ω=G−1(−∞,0]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form (ψ,φ)↦∫Ω∇Hψ,∇HφHdνψ,φ∈W1,2(Ω,ν),(0.1) and we give sharp embedding

In this paper we prove, for small Hurst parameters, the higher-order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis-based variational calculus techniques

For non-anticipative functionals, differentiable in Chitashvili’s sense, the Itô formula for cadlag semimartingales is proved. Relations between different notions of functional derivatives are established.

In this paper, we study the self-intersection local times of multifractional Brownian motion (mBm) in higher dimensions in the framework of white noise analysis. We show that when a suitable number of kernel functions of self-intersection local times of mBm are truncated then we obtain a Hida distribution. In addition, we present the expansion of the self-intersection local times in terms of Wick powers

Paschke’s version of Stinespring’s theorem associates a Hilbert C∗-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a C∗-algebra 𝒜 one may associate an inclusion system E=(Et) of Hilbert 𝒜-𝒜-modules with a generating unit ξ=(ξt). Suppose ℬ is a von Neumann algebra, consider M2(ℬ), the von Neumann algebra of 2×2

We reduce the conditionally monotone (c-monotone) independence of Hasebe to tensor independence on suitably constructed larger algebras. For that purpose, we use the approach developed for a reduction of similar type for boolean, free and monotone independences. We apply the tensor product realization of c-monotone random variables to introduce the c-comb (loop) product of birooted graphs, a generalization

We introduce the quadratic analog of the tensor Bogolyubov representation of the CCR. Our main result is the determination of the structure of these maps: each of them is uniquely determined by two arbitrary complex-valued Borel functions of modulus 1 and two maps of ℝd into itself whose inverses induce transformations that map the Lebesgue measure λ into measures λc,λs absolutely continuous with respect

This paper combines probabilistic and algebraic techniques for computing quantum expectations of operator exponentials (and their products) of quadratic forms of quantum variables in Gaussian states. Such quadratic-exponential functionals (QEFs) resemble quantum statistical mechanical partition functions with quadratic Hamiltonians and are also used as performance criteria in quantum risk-sensitive

We analyze the solution to the linear stochastic heat equation driven by a multiparameter Hermite process of order q≥1. This solution is an element of the qth Wiener chaos. We discuss various properties of the solution, such as the necessary and sufficient condition for its existence, self-similarity, α-variation and regularity of its sample paths. We will also focus on the probability distribution

Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean independent entries. More precisely, we show that, under some moment conditions, such random matrices are asymptotically η-diagonal and Boolean independent from

We review results on GKSL-type equations with multi-modal generators which are quadratic in bosonic or fermionic creation and annihilation operators. General forms of such equations are presented. The Gaussian solutions are obtained in terms of equations for the first and the second moments. Different approaches for their solutions are discussed.

In this paper, we develop the notion of free-Boolean independence in an amalgamated setting. We construct free-Boolean cumulants and show that the vanishing of mixed free-Boolean cumulants is equivalent to our free-Boolean independence with amalgamation. We also provide a characterization of free-Boolean independence by conditions in terms of mixed moments. In addition, we study free-Boolean independence

We first develop a theory of conditional expectations for random variables with values in a complete metric space M equipped with a contractive barycentric map β, and then give convergence theorems for martingales of β-conditional expectations. We give the Birkhoff ergodic theorem for β-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the β-conditional

In this paper, weak Poincaré inequalities are obtained for convolution probabilities with explicit rate functions by constructing suitable Lyapunov functions. Here, one of these Lyapunov functions is introduced for the first time and can be used to improve parts of results on the estimate of rate functions for super Poincaré inequalities13 as well. In addition, these weak Poincaré inequalities are

We investigate parameter identifiability of spectral distributions of random matrices. In particular, we treat compound Wishart type and signal-plus-noise type. We show that each model is identifiable up to some kind of rotation of parameter space. Our method is based on free probability theory.

Let G=U(∞)⋉H+(∞) be the infinite semigroup, inductive limit of the increasing sequence of the semigroups G(n)=U(n)⋉H+(n), where U(n) is the unitary group of matrices and H+(n) is the semigroup of positive hermitian matrices. The main purpose of this work is twofold. First, we give a complete classification of spherical functions defined on G, by following a general approach introduced by Olshanski

We study a family of quantum Markov semigroups with circulant structure. We obtain a complete description of the spectral representation for the Lindbladian and, putting it together with some purely probabilistic properties of a classical associated process, we can study asymptotic properties, invariant states, quantum-detailed balance conditions and reducibility. In particular, in the reducible case

Let X be a separable Banach space endowed with a nondegenerate centered Gaussian measure μ and let w be a positive function on X such that w∈W1,s(X,μ) and logw∈W1,t(X,μ) for some s>1 and t>s′. In this paper, we introduce and study Sobolev spaces with respect to the weighted Gaussian measure ν:=wμ. We obtain results regarding the divergence operator (i.e. the adjoint in L2 of the gradient operator along

In this paper, a new family of polyno-expo-trigonometric distributions is presented and investigated. A special case using the Weibull distribution, with three parameters, is considered as statistical model for lifetime data. The estimation of the parameters is performed with the maximum likelihood method. A numerical simulation study verifies that the bias and the mean squared error of the maximum

The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding

We characterize generators of quantum Markov semigroups leaving invariant a maximal abelian purely atomic algebra and certain operator subspaces associated with it in a natural way. From this result, we also establish a characterization of generators of quantum Markov semigroups of weak coupling limit type associated with a nondegenerate Hamiltonian.

In this work, we propose a method to investigate the factorization property of a adjontable Markov operator between two algebraic probability spaces without using the dilation theory. Assuming the existence of an anti-unitary operator on Hilbert space related to Stinespring representations of our Markov operator, which satisfy some particular modular relations, we prove that it admits a factorization

In this paper, we study compound bi-free Poisson distributions for two-faced families of random variables. We prove a Poisson limit theorem for compound bi-free Poisson distributions. Furthermore, a bi-free infinitely divisible distribution for a two-faced family of self-adjoint random variables can be realized as the limit of a sequence of compound bi-free Poisson distributions of two-faced families

In this work, we establish Titchmarsh–Weyl theory for singular q-Dirac systems. Thus, we extend classical Titchmarsh–Weyl theory for Dirac systems to q-analogue of this system. We show that it does not occur for the limit-circle case for the q-Dirac system.

We study the analytic characterization of S-transform in a general setting of white noise functionals. Then, the measurability condition of the norms generating the underlining locally convex space is a necessary and sufficient condition for the analytic characterization of the S-transform in terms of analytic and growth conditions.

In this work, we study the spectral statistics for Anderson model on ℓ2(ℕ) with decaying randomness whose single-site distribution has unbounded support. Here, we consider the operator Hω given by (Hωu)n=un+1+un−1+anωnun, an∼n−α and {ωn} are real i.i.d random variables following symmetric distribution μ with fat tail, i.e. μ((−R,R)c)12, we are able to show that the eigenvalue process in (−2,2) is the

A canonical definition of the joint semi-quantum operators of a finite family of random variables, having finite moments of all orders, is given first in terms of an existence and uniqueness theorem. Then two characterizations, one for the polynomially symmetric, and another for the polynomially factorizable probability measures, having finite moments of all orders, are presented.

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

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∈ℂ, the limit space is precisely the 1-mode Interacting Fock Space (IFS) that realizes the canonical quantum decomposition of the limit classical

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

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

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

By using the asymptotic coupling method, the asymptotic log-Harnack inequality is established for the transition semigroup associated to the 3D Leray-α 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-α model with fractional dissipation, which is the stochastic 3D Navier–Stokes

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

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

Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group On+, the so-called easy quantum groups, introduced by Banica and Speicher in 2009. This correspondence was generalized to two-colored set partitions, which, in addition, assign a black or white color to each point of a set. Globally colorized categories of

In this paper, we unify techniques of Pascal white noise analysis and harmonic analysis on configuration spaces establishing relations between the main structures of both ones. Fix a Random measure σ on a Riemannian manifold X, we construct on the space of finite compound configuration space Ω0 the so-called Lebesgue–Pascal measure λσ̂ and as a consequence we obtain the Pascal measure μσ̂ on the compound