We introduce and develop a pathwise description of the dissipation of general convex entropies for continuous time Markov processes, based on simple backward martingales and convergence theorems with respect to the tail sigma field. The entropy is in this setting the expected value of a backward submartingale. In the case of (non necessarily reversible) Markov diffusion processes, we use Girsan...

That Birkhoff's individual ergodic theorem provides a strong law of large numbers for stationary stochastic processes has been known since its applications to this end by Doob (cf. [3; 4]) and Hopf [9]. The conclusion of the ergodic theorem as it applies to a sequence of random variables, {x " }, is from one point of view considerably stronger than that of the strong law of large numbers. The l...

If X(t) is a stochastic process taking values in some space 9C and g is a real-valued function on 9C then we may consider the process g(X(t)). Often from some knowledge about the process X(t) and the function g one can deduce properties of the process g(X(t)). For example: if X(t) is Brownian motion in R" and g is a subharmonic function on Rn that doesn't grow too fast then g(X(t)) is a semimar...

To each coherent configuration (scheme) C and positive integer m we associate a natural scheme Ĉ(m) on the m-fold Cartesian product of the point set of C having the same automorphism group as C. Using this construction we define and study two positive integers: the separability number s(C) and the Schurity number t(C) of C. It turns out that s(C) ≤ m iff C is uniquely determined up to isomorphi...

It is possible to represent each of a number of Markov chains as an evolving sequence of connected subsets of a directed acyclic graph that grow in the following way: initially, all vertices of the graph are unoccupied, particles are fed in one-byone at a distinguished source vertex, successive particles proceed along directed edges according to an appropriate stochastic mechanism, and each par...

Tyrone Duncan has made seminal contributions in the field of Filtering, Stochastic Control and the interface of probability theory and geometry. In his pioneering thesis, he presented a fundamental theory of nonlinear filtering and was instrumental in introducing the modern theory of martingales as developed by Doob, Meyer, Kunita-Watanabe and others in the study of nonlinear filtering. As he s...

This paper concentrates on a particular first-order coupled PDE system. It provides both a detailed treatment of the existence and uniqueness of monotone travelling waves to various equilibria, by differential-equation theory and by probability theory and a treatment of the corresponding hyperbolic initial-value problem, by analytic methods. The initial-value problem is studied using characteri...

In this paper we consider decompositions of processes of the form Y = f(t,Xt) where X is a one dimensional semimartingale, but f is not required to be differentiable so Ito’s formula does not apply. First, in the case where f(t, x) is independent of t, we show that requiring it to be locally Lipschitz continuous in x is enough for an Ito style decomposition to apply. This decomposes Y into a st...

We shall first consider the classical neutral Moran model with two alleles whose fate is either to become extinct or to reach fixation. We will study an ergodic version of the Moran model obtained by conditioning it to never hit the boundaries, making use of a Doob transform. We shall call it the recurrent Moran model. We will show that the Siegmund dual of the recurrent Moran process exists an...

In the present work, we consider spectrally positive Lévy processes (Xt, t ≥ 0) not drifting to +∞ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with X) before hitting 0. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law Px of this conditioned process is defined as ...