Infinite Dimensional Analysis, Quantum Probability and Related Topics

In this note we study the 2D stochastic quasi-geostrophic equation in 𝕋² for general parameter α ∈ (0, 1) and multiplicative noise. We prove the existence of martingale solutions and pathwise uniqueness under some condition in the general case, i.e. for all α ∈ (0, 1). In the subcritical case α > 1/2, we prove existence and uniqueness of (probabilistically) strong solutions and construct a Markov family of solutions. In particular, it is uniquely ergodic for α > 2/3 provided the noise is non-degenerate. In this case, the convergence to the (unique) invariant measure is exponentially fast. In the general case, we prove the existence of Markov selections.

In this paper we consider two related stochastic models. The first one is a branching system consisting of particles moving according to a Markov family in ℝ^d and undergoing subcritical branching with a constant rate of V > 0. New particles immigrate to the system according to a homogeneous space-time Poisson random field. The second model is the superprocess corresponding to the branching particle system. We study rescaled occupation time process and the process of its fluctuations under mild assumptions on the Markov family. In the general setting a functional central limit theorem as well as large and moderate deviation principles are proved. The subcriticality of the branching law determines the behaviour in large time scales and it "overwhelms" the properties of the particles' motion. For this reason the results are the same for all dimensions and can be obtained for a wide class of Markov processes (both properties are unusual for systems with critical branching).

Let g : ℝ → ℂ be a C^∞-function with all derivatives bounded and let tr_n denote the normalized trace on the n × n matrices. In Ref. 3 Ercolani and McLaughlin established asymptotic expansions of the mean value 𝔼{tr_n(g(X_n))} for a rather general class of random matrices X_n, including the Gaussian Unitary Ensemble (GUE). Using an analytical approach, we provide in the present paper an alternative proof of this asymptotic expansion in the GUE case. Specifically we derive for a random matrix X_n that

In this paper the quantum white noise (QWN)-Euler operator is defined as the sum , where and N^Q stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that has an integral representation in terms of the QWN-derivatives as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to the QWN-Euler operator is worked out in the basis of the QWN coordinate system.

We analyse degenerate infinite-dimensional sub-elliptic generators, and obtain estimates on the long-time behaviour of the corresponding Markov semigroups that describe a model of heat conduction. In particular, we establish ergodicity of the system for a family of invariant measures, and show that the optimal rate of convergence to equilibrium is polynomial. Consequently, there is no spectral gap, but a Liggett–Nash-type inequality is shown to hold.