The Ornstein-Zernike (OZ) equations [1] can be used to find probability distributions of atoms in fluid states where the primary unknowns are the radial pair correlation function and the direct correlation function. In order to specify conditions of the state, certain parameters such as density and temperature are chosen. While one can find a single solution to the OZ equations corresponding to...

Recently, there has been increasing interest in continuous-time stochastic models with jumps, a class of models which has applications in the fields of finance, insurance mathematics and storage theory, to name just a few. In this chapter we shall collect known results about a prominent class of these continuoustime models with jumps, namely the class of Lévy-driven Ornstein–Uhlenbeck processes...

An Ornstein-Uhlenbeck process is the most basic mean-reversion model and has been used in various fields such as finance and biology. In some instances, reflecting boundary conditions are needed to restrict the state space of this process. We study an Ornstein-Uhlenbeck diffusion process with a reflecting boundary and its application to finance and neuroscience. In the financial application, th...

– We rigorously derive the Ornstein-Zernike asymptotics of the pair-correlation functions for finite-range Ising ferromagnets in any dimensions and at any temperature above critical. The celebrated heuristic argument by Ornstein and Zernike [1] implies that the asymptotic form of the truncated two-point density correlation function of simple fluids away from the critical region is given by G(~r...

We extend to infinite dimensions an explicit formula of Chill, Fašangová, Metafune, and Pallara [2] for the optimal angle of analyticity of analytic Ornstein-Uhlenbeck semigroups. The main ingredient is an abstract representation of the Ornstein-Uhlenbeck operator in divergence form.

In this paper we study first passage times of (reflected) Ornstein–Uhlenbeck processes over compound Poisson-type boundaries. In fact, we extend the results of first rendezvous times of (reflected) Brownian motion and compound Poisson-type processes in Perry, Stadje and Zacks (2004) to the (reflected) Ornstein–Uhlenbeck case.

In this short note, the identity in law, which was obtained by P. Salminen [6], between on one hand, the Ornstein-Uhlenbeck process with parameter γ, killed when it reaches 0, and on the other hand, the 3-dimensional radial Ornstein-Uhlenbeck process killed exponentially at rate γ and conditioned to hit 0, is derived from a simple absolute continuity relationship.

We study the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup in an open convex subset of an infinite dimensional separable Banach space X. This is done by finite dimensional approximation. In particular we prove Logarithmic-Sobolev and Poincaré inequalities, and thanks to these inequalities we deduce spectral properties of the OrnsteinUhlenbeck operator. 2010 Mathematics Subjec...