Infinite Dimensional Analysis, Quantum Probability and Related Topics

We consider free bosons hopping on a network (infinite graph). The condition for Bose–Einstein condensation is given in terms of the random walk on a graph. In case of periodic lattices, we also consider boson moving in an external periodic potential and obatin the criterion for Bose–Einstein condensation.

We show that the right (Markov) process associated with a quasi-regular semi-Dirichlet form has a dual process, proving that the weak duality hypothesis is satisfied.

We present a short and elegant proof of the complete theory of strict representations of the algebra of all adjointable operators on a Hilbert -module E by operators on a Hilbert -module F. An analogue for W*-modules and normal representations is also proved. As an application we furnish a new proof of Blecher's Eilenberg–Watts theorem.

The aim of this paper is twofold. First, we continue the research of Accardi and Smolyanov^5,8 and investigate the eigenvalue problem for the Lévy–Laplacian. We show that the spectrum of the Lévy–Laplacian coincides with the whole complex plane and that the Lévy–Laplacian admits various non-self-adjoint extensions — the latter result is due to the structure of the rigged Hilbert space, on which the Lévy–Laplacian is defined. Second, we obtain solutions of the Lévy–Laplace equation in a wide set of functions defined on the rigged Hilbert space. This result is especially interesting due to the recent finding of the relation between Yang–Mills gauge field and Lévy harmonic functions.⁴

We construct a sequence of states called m-monotone product states which give a discrete interpolation between the monotone product of states of Muraki in monotone probability and the free product of states of Avitzour and Voiculescu in free probability. We derive the associated basic limit theorems and develop the combinatorics based on non-crossing ordered partitions with monotone order starting from depth m. We deduce an explicit formula for the Cauchy transforms of the m-monotone central limit measures and for the associated Jacobi coefficients. A new type of combinatorics of inner blocks in non-crossing partitions leads to explicit formulas for the mixed moments of m-monotone Gaussian operators, which are new even in the case of monotone independent Gaussian operators with arcsine distributions.

Some sufficient and necessary conditions are presented for two Dawson–Watanabe superprocesses to preserve the natural stochastic order. As applications, the critical phenomenon is described for some extinction and ergodicity properties of a class of superprocesses.

We prove some no-go theorems on the existence of a Fock representation of the *-Lie algebra generated by , where , b_s are the Hida white noise densities. In particular we prove the nonexistence of such a representation for any *-Lie algebra containing . This drastic difference with the quadratic case proves the necessity of investigating different renormalization rules for the case of higher powers of white noise.

An integration by parts formula for the measure μ of the Euclidean -quantum field model is proven. It is also proven that the regularized logarithmic derivative of μ when the regularization is removed does not converge in L²(μ).

We prove an existence and uniqueness result for a class of backward stochastic differential equations (BSDE) with dissipative drift in Hilbert spaces. We also give examples of stochastic partial differential equations which can be solved with our result.