## 60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX)

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The interdisciplinary workshop STOCHASTIC PROCESSES WITH APPLICATIONS IN THE NATURAL SCIENCES was held in Bogotá, at Universidad de los Andes from December 5 to December 9, 2016. It brought together researchers from Colombia, Germany, France, Italy, Ukraine, who communicated recent progress in the mathematical research related to stochastic processes with application in biophysics.
The present volume collects three of the four courses held at this meeting by Angelo Valleriani, Sylvie Rœlly and Alexei Kulik.
A particular aim of this collection is to inspire young scientists in setting up research goals within the wide scope of fields represented in this volume.
Angelo Valleriani, PhD in high energy physics, is group leader of the team "Stochastic processes in complex and biological systems" from the Max-Planck-Institute of Colloids and Interfaces, Potsdam.
Sylvie Rœlly, Docteur en Mathématiques, is the head of the chair of Probability at the University of Potsdam.
Alexei Kulik, Doctor of Sciences, is a Leading researcher at the Institute of Mathematics of Ukrainian National Academy of Sciences.

In this thesis, stochastic dynamics modelling collective motions of populations, one of the most mysterious type of biological phenomena, are considered. For a system of N particle-like individuals, two kinds of asymptotic behaviours are studied : ergodicity and flocking properties, in long time, and propagation of chaos, when the number N of agents goes to infinity. Cucker and Smale, deterministic, mean-field kinetic model for a population without a hierarchical structure is the starting point of our journey : the first two chapters are dedicated to the understanding of various stochastic dynamics it inspires, with random noise added in different ways. The third chapter, an attempt to improve those results, is built upon the cluster expansion method, a technique from statistical mechanics. Exponential ergodicity is obtained for a class of non-Markovian process with non-regular drift. In the final part, the focus shifts onto a stochastic system of interacting particles derived from Keller and Segel 2-D parabolicelliptic model for chemotaxis. Existence and weak uniqueness are proven.

In this thesis we study reciprocal classes of Markov chains. Given a continuous time Markov chain on a countable state space, acting as reference dynamics, the associated reciprocal class is the set of all probability measures on path space that can be written as a mixture of its bridges. These processes possess a conditional independence property that generalizes the Markov property, and evolved from an idea of Schrödinger, who wanted to obtain a probabilistic interpretation of quantum mechanics.
Associated to a reciprocal class is a set of reciprocal characteristics, which are space-time functions that determine the reciprocal class. We compute explicitly these characteristics, and divide them into two main families: arc characteristics and cycle characteristics. As a byproduct, we obtain an explicit criterion to check when two different Markov chains share their bridges.
Starting from the characteristics we offer two different descriptions of the reciprocal class, including its non-Markov probabilities.
The first one is based on a pathwise approach and the second one on short time asymptotic. With the first approach one produces a family of functional equations whose only solutions are precisely the elements of the reciprocal class. These equations are integration by parts on path space associated with derivative operators which perturb the paths by mean of the addition of random loops. Several geometrical tools are employed to construct such formulas. The problem of obtaining sharp characterizations is also considered, showing some interesting connections with discrete geometry. Examples of such formulas are given in the framework of counting processes and random walks on Abelian groups, where the set of loops has a group structure.
In addition to this global description, we propose a second approach by looking at the short time behavior of a reciprocal process. In the same way as the Markov property and short time expansions of transition probabilities characterize Markov chains, we show that a reciprocal class is characterized by imposing the reciprocal property and two families of short time expansions for the bridges. Such local approach is suitable to study reciprocal processes on general countable graphs. As application of our characterization, we considered several interesting graphs, such as lattices, planar
graphs, the complete graph, and the hypercube.
Finally, we obtain some first results about concentration of measure implied by lower bounds on the reciprocal characteristics.