Infinite Dimensional Analysis, Quantum Probability and Related Topics

We consider a locally regulated spatial population model introduced by Bolker and Pacala. Based on the deterministic approximation studied by Fournier and Méléard, we prove that the fluctuation theorem holds under some mild moment conditions. The limiting process is shown to be an infinite-dimensional Gaussian process solving a generalized Langevin equation. In particular, we further consider its properties in one dimension case, which is characterized as a time-inhomogeneous Ornstein–Uhlenbeck process.

The concepts of Hamiltonian Feynman integrals in white noise analysis are used to realize as the first velocity-dependent potential of the Hamiltonian Feynman integrand for a charged particle in a constant magnetic field in coordinate space as a Hida distribution. For this purpose the velocity-dependent potential gives rise to a generalized Gauss kernel. Besides the propagators, the generating functionals are obtained.

We study the relations between (tight) logarithmic Sobolev inequalities, entropy decay and spectral gap inequalities for Markov evolutions on von Neumann algebras. We prove that log-Sobolev inequalities (in the non-commutative form defined by Olkiewicz and Zegarlinski in Ref. 25) imply spectral gap inequalities, with optimal relation between the constants. Furthermore, we show that a uniform exponential decay of a proper relative entropy is equivalent to a modified version of log-Sobolev inequalities. The relations among the mentioned inequalities are investigated and often depend on some regularity conditions, which are also discussed. With regard to this aspect, we provide an example of a positive identity-preserving semigroup not verifying the usually requested regularity conditions (which are always fulfilled for reversible classical Markov processes).

The additive monotone (respectively boolean) unitary Brownian motion is a non-commutative stochastic process with monotone (respectively boolean) independent and stationary increments which are distributed according to the arcsine law (respectively Bernoulli law). We introduce the monotone and boolean unitary Brownian motions and derive a closed formula for their associated moments. This provides a description of their spectral measures. We prove that, in the monotone case, the multiplicative analog of the arcsine distribution is absolutely continuous with respect to the Haar measure on the unit circle, whereas in the boolean case the multiplicative analog of the Bernoulli distribution is discrete. Finally, we use quantum stochastic calculus to provide a realization of these processes as the stochastic exponential of the correspending additive Brownian motions.

We discuss the approach to investigate molecular machines using systems of integro–differential ultrametric (p-adic) reaction–diffusion equations with drift. This approach combines the features of continuous and discrete dynamic models. We apply this model to investigation of actomyosin molecular motor.

The introduced system of equations is solved analytically using p-adic wavelet theory. We find explicit stationary solutions and behavior in the relaxation regime.

This paper develops information geometric representations for nonlinear filters in continuous time. The posterior distribution associated with an abstract nonlinear filtering problem is shown to satisfy a stochastic differential equation on a Hilbert information manifold. This supports the Fisher metric as a pseudo-Riemannian metric. Flows of Shannon information are shown to be connected with the quadratic variation of the process of posterior distributions in this metric. Apart from providing a suitable setting in which to study such information-theoretic properties, the Hilbert manifold has an appropriate topology from the point of view of multi-objective filter approximations. A general class of finite-dimensional exponential filters is shown to fit within this framework, and an intrinsic evolution equation, involving Amari's -1-covariant derivative, is developed for such filters. Three example systems, one of infinite dimension, are developed in detail.

We define a notion of semi–stability in the conditionally free probability and explain that the semi–stable measures are infinitely divisible. We also show that in the conditionally free probability stable measures are semi–stable, and that semi–stability for all r implies stability.