Infinite Dimensional Analysis, Quantum Probability and Related Topics

Diffusions on the pinned loop and pathgroup of a general (non-compact) Lie group are constructed. The laws of the endpoint processes satisfy heat equations related to second order differential operators associated to subspaces of the Cameron–Martin space.

We consider the forward integral with respect to fractional Brownian motion B^(H)(t) and relate this to the Wick–Itô–Skorohod integral by using the M-operator introduced by Ref. 10 and the Malliavin derivative . Using this connection we obtain a general Itô formula for the Wick–Itô–Skorohod integralswith respect to B^(H)(t), valid for .

We introduce and study a class of nonlinear jump type Markov semigroups for systems with infinite dimensional configuration space.

Quantum Bernoulli random walks can be realized as random walks on the dual of SU(2). We use this realization in order to study a model of dynamic quantum Bernoulli random walk with time-dependent transitions. For the corresponding dynamic random walk on the dual of SU(2), we prove several limit theorems (local limit theorem, central limit theorem, law of large numbers, large deviation principle). In addition, we characterize a large class of transient dynamic random walks.

We study the continuous version of the contact model. Using an analytic approach, we construct the non-equilibrium contact process as a Markov process on configuration space. The construction is based on the analysis of correlation functions evolution. The problem concerning invariant measures as well as asymptotics of correlation functions are also studied.

We identify a class of conditional entrance laws for superprocesses with dependent spatial motion (SDSM). Those entrance laws are used to characterize some conditional excursion laws. As an application of the results, we give a sample path decomposition of the SDSM and that of a related immigration superprocess. The main tool used here is the conditional log-Laplace functional technique that handles the difficulty of the loss of the multiplicative property due to the interactions in the spatial motions.

We prove essential self-adjointness of a class of Dirichlet operators in ℝⁿ using the hyperbolic equation approach. This method allows one to prove essential self-adjointness under minimal conditions on the logarithmic derivative of the density and a condition of Muckenhoupt type on the density itself.

We examine the free entropy and free entropy dimension for projections, and obtain a sufficient condition for the factoriality of the von Neumann algebra generated by projections in terms of their free entropy dimension. This corresponds to Voiculescu's result for self-adjoint elements.

In White Noise Analysis (WNA), various random quantities are analyzed as elements of (S)*, the space of Hida distributions.¹ Hida distributions are generalized functions of white noise, which is to be naturally viewed as the derivative of the Brownian motion. On (S)*, the Wick product is defined in terms of the -transform. We have found such a remarkable property that the Wick product has no zero divisors among Hida distributions. This result is a WNA version of Titchmarsh's theorem and is expected to play fundamental roles in developing the "operational calculus" in WNA along the line of Mikusiński's version for solving differential equations.