Infinite Dimensional Analysis, Quantum Probability and Related Topics

We prove that the Kolmogorov operator L associated with a 2D-stochastic Navier–Stokes equation with periodic boundary conditions and space-time white noise is m-dissipative for sufficiently large viscosity on finitely based cylindrical test functions in the space L¹ w.r.t. the Gaussian measure induced by the enstrophy. The proof is based on a new a priori estimate for the solution of the resolvent equation λF - LF = H.

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.

In this paper we prove Kunita-type stochastic differential equation (SDE) , t > s, driven by a spatial continuous semimartingale F (x, t) =(F¹ (x, t), …, F^m (x, t)), x ∈ ℜ^m, with local characteristic (a, b), in the sense of Kunita (Chap. 3 of Ref. 10), can produce a stochastic flow of homeomorphisms of ℜ^m into itself almost surely under the (a, b) satisfies non-Lipschitz conditions. This result bases on recent works¹² by the author.

Both direct and reverse log-Sobolev inequalities, relating the Shannon entropy with a μ-deformed energy, are shown to hold in a family of μ-deformed Segal–Bargmann spaces. This shows that the μ-deformed energy of a state is finite if and only if its Shannon entropy is finite. The direct inequality is a new result, while the reverse inequality has already been shown by the authors but using different methods. Next the μ-deformed energy of a state is shown to be finite if and only if its Dirichlet form energy is finite. This leads to both direct and reverse log-Sobolev inequalities that relate the Shannon entropy with the Dirichlet energy. We obtain that the Dirichlet energy of a state is finite if and only if its Shannon entropy is finite. The main method used here is based on a study of the reproducing kernel function of these spaces and the associated integral kernel transform.

We examine classical Bogoliubov's model of a particle coupled to a heat bath which consists of infinitely many stochastic oscillators. Bogoliubov's result¹ suggests that, in the stochastic limit, the model exhibits convergence to thermodynamical equilibrium. It has recently been shown that the system does attain the equilibrium if the coupling constant is small enough.¹² We show that in the case of the large coupling constant, the distribution function ρ_S (q, p, t) → 0 pointwise as t → ∞. This implies that if there is convergence to equilibrium, then the limit measure has no finite momenta. Besides, the probability to find the particle in any finite domain of phase space tends to zero. This is also true for domains in the coordinate space and in the momentum space.

Let a⁰, a^- and a⁺ be the preservation, annihilation, and creation operators of a probability measure μ on ℝ^d, respectively. The operators a⁰ and [a^-, a⁺] are proven to uniquely determine the moments of μ. We discuss the question: "What conditions must two families of operators satisfy, in order to ensure the existence of a probability measure, having finite moments of any order, so that, its associated preservation operators and commutators between the annihilation and creation operators are the given families of operators?" For the case d = 1, a satisfactory answer to this question is obtained as a simple condition in terms of the Szegö–Jacobi parameters. For the multidimensional case, we give some necessary conditions for the answer to this question. We also give a table with the associated preservation and commutator between the annihilation and creation operators, for some of the classic probability measures on ℝ.

Starting with a symmetric Markov diffusion process X (with symmetry measure m and L² (m) infinitesimal generator A) and a suitable core for the Dirichlet form of X, we describe a class of derivations defined on . Associated with each such derivation B is a drift transformation of X, obtained through Girsanov's theorem. The transformed process X^B is typically non-symmetric, but we are able to show that if the "divergence" of B is positive, then m is an excessive measure for X^B, and the L² (m) infinitesimal generator of X^B is an extension of f ↦ Af + B (f). The methods used are mainly probabilistic, and involve the notions of even and odd continuous additive functionals, and Nakao's stochastic divergence. These methods yield a probabilistic approach to the adjoint of the semigroup of X^B, and in particular lead to a solution of a problem of W. Stannat.

Given a right-continuous Markov process (X_t)_{t ≥ 0} on a second countable metrizable space E with transition semigroup (p_t)_{t ≥ 0}, we prove that there exists a σ-finite Borel measure μ with full support on E, and a closed and densely defined linear operator generating (p_t)_{t ≥ 0} on L^p (E; μ). In particular, we solve the corresponding Cauchy problem in L^p (E; μ) for any initial condition . Furthermore, for any real β > 0 we show that there exists a generalized Dirichlet form which is associated to (e^-βt p_t)_{t ≥ 0}. If the β-subprocess of (X_t)_{t ≥ 0} corresponding to (e^-βt p_t)_{t ≥ 0}, β > 0, is μ-special standard then all results from generalized Dirichlet form theory become available, and Fukushima's decomposition holds for . If (X_t)_{t ≥ 0} is transient, then β can be chosen to be zero.

Cauchy problems for Schrödinger equations with matrix-valued potentials are explicitly solved under following assumptions:.

— equations are written in momentum form;.

— the potentials are Fourier transformed matrix-valued measures with, in general, noncommuting values;.

— initial Cauchy data are good enough.

The solutions at time t are presented in form of integrals over some spaces of piecewise continuous mappings of the segment [0, t] to a finite-dimensional momentum space, and measures of the integration are countably additive but matrix-valued (resulting in matrices of ordinary Lebesgue integrals).

Known results gave solutions either when the measure had commuting (complex) values, or when the integration over infinite dimensional spaces was quite symbolic, or when such integrating was of chronological type and hence more complicated.

The method used below is based on technique of matrix-valued transition amplitudes.