Infinite Dimensional Analysis, Quantum Probability and Related Topics

Let be a Fock space and, for any non-negative integer k, let be the sum of all homogeneous chaos spaces of order at most k. For all non-negative integers m and n, the Wick product is a bounded bilinear operator from Γ (H_c)_m × Γ (H_c)_n into Γ (H_c)_{m +n} with norm greater than or equal to . In the monograph¹ S. Janson conjectured that this lower bound is exact. In this paper we prove this conjecture. In addition, we prove that a pair of nonzero vectors (ϕ, ψ)∈ Γ (H_c)_m × Γ (H_c)_n achieve this bound if and only if both vectors are multiples of the homogeneous products, i.e. ϕ =αu^⊗m, ψ =βu^⊗n, with u, v∈ H_c and α, β ∈ ℂ.

We introduce a continuous version of the contact model which is well known and widely studied in the lattice case. Under certain general assumptions on the infection spreading characteristics, we construct the contact process as a Markov process in the configuration space of the system.

Gaussian random field on general ultrametric space is introduced as a solution of pseudodifferential stochastic equation. Covariation of the introduced random field is computed with the help of wavelet analysis on ultrametric spaces.

Notion of ultrametric Markovianity, which describes independence of contributions to random field from different ultrametric balls is introduced. We show that the random field under investigation satisfies this property.

In this paper, a theory of stochastic processes generated by quantum extensions of Laplacians is developed. Representations of the associated heat semigroups are discussed by means of suitable time shifts. In particular the quantum Brownian motion associated to the Lévy–Laplacian is obtained as the usual Volterra–Gross Laplacian using the Cesàro Hilbert space as initial space of our process as well as multiplicity space of the associated white noise.

In this paper we study the Cauchy problem for the wave equation with spacetime Lévy noise initial data in the Kondratiev space of stochastic distributions. We prove that this problem has a strong and unique C²-solution, which takes an explicit form. Our approach is based on the use of the Hermite transform.

We consider a general type of perturbed stochastic partial differential equations:

A quantum central limit theorem for a continuous-time quantum walk on a homogeneous tree is derived from quantum probability theory. As a consequence, a new type of limit theorems for another continuous-time walk introduced by the walk is presented. The limit density is similar to that given by a continuous-time quantum walk on the one-dimensional lattice.

Konno's idea of quantum walk on trees remains applicable to a wider family of graphs by means of quantum probabilistic machinery for asymptotic spectral analysis of graphs.

With every E₀-semigroup (acting on the algebra of of bounded operators on a separable infinite-dimensional Hilbert space) there is an associated Arveson system. One of the most important results about Arveson systems is that every Arveson system is the one associated with an E₀-semigroup. In these notes we give a new proof of this result that is considerably simpler than the existing ones and allows for a generalization to product systems of Hilbert module (to be published elsewhere).

Product systems are the classifying structures for semigroups of endomorphisms of , in that two E₀-semigroups are cocycle conjugate iff their product systems are isomorphic. Thus it is important to know that every abstract product system is associated with an E₀-semigroup. This was first proved more than 15 years ago by rather indirect methods. Recently, Skeide has given a more direct proof. In this note we give yet another proof by an elementary construction.

In this paper, we give a simple proof of log-Sobolev inequalities on an infinite volume path space C (ℝ, ℝ^d) with Gibbs measures. We introduce a parabolic stochastic partial differential equation which is reversible with respect to the Gibbs measures. In the proof, the gradient estimate for the diffusion semigroup which is derived from the stochastic flow plays a central role.