Infinite Dimensional Analysis, Quantum Probability and Related Topics

In this paper we prove the existence and uniqueness for the solution to a stochastic reaction–diffusion equation, defined on a network, and subjected to nonlocal dynamic stochastic boundary conditions. The result is obtained by deriving a Gaussian-type estimate for the related leading semigroup, under rather mild regularity assumptions on the coefficients. An application of the latter to a stochastic optimal control problem on graphs, is also provided.

We study the exponential ergodicity of diffusions generated by a multivalued stochastic differential equation with Lévy jumps when the coefficients are non-Lipschitz continuous by proving that the transition semigroup is strongly Feller and irreducible, and that it admits a unique invariant measure. This is obtained through an $L^2$-convergence result, Girsanov’s theorem, coupling method combined and a stopping argument.

In this paper, we first introduce the concept of fractional quantum integral with general kernels, which generalizes several types of fractional integrals known from the literature. Then we give more general versions of some integral inequalities for this operator, thus generalizing some previous results obtained by many researchers.^{2,8,25,29,30,36}

The identification mentioned in the title allows a formulation of the multidimensional Favard lemma different from the ones currently used in the literature and which parallels the original 1-dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the real Jacobi sequence by a sequence of positive definite kernels. The above result opens the way to the program of a purely algebraic classification of probability measures on ${ℝ}^d$ with moments of any order and more generally of states on the polynomial algebra on ${ℝ}^d$.

The quantum decomposition of classical real-valued random variables with all moments is one of the main ingredients in the proof.

In this paper, we give a product formula of Hermite polynomials and a relation between real Wiener–Itô chaos and complex Wiener–Itô chaos is shown. By this relation and the known multivariate extension of the fourth moment theorem for real multiple integrals, a fourth moment theorem (the Nualart–Peccati criterion) for complex Wiener–Itô multiple integrals is obtained.

In this paper, we point out the relationship between the two Ramanujan’s formulas for Beta integrals and normal distribution. Some of the applications are also given.