TY - INPR A1 - Roelly, Sylvie A1 - Vallois, Pierre T1 - Convoluted Brownian motion BT - a semimartingale approach N2 - In this paper we analyse semimartingale properties of a class of Gaussian periodic processes, called convoluted Brownian motions, obtained by convolution between a deterministic function and a Brownian motion. A classical example in this class is the periodic Ornstein-Uhlenbeck process. We compute their characteristics and show that in general, they are neither Markovian nor satisfy a time-Markov field property. Nevertheless, by enlargement of filtration and/or addition of a one-dimensional component, one can in some case recover the Markovianity. We treat exhaustively the case of the bidimensional trigonometric convoluted Brownian motion and the higher-dimensional monomial convoluted Brownian motion. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 9 KW - periodic Gaussian process KW - periodic Ornstein-Uhlenbeck process KW - Markov-field property KW - enlargement of filtration Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-96339 SN - 2193-6943 VL - 5 IS - 9 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Roelly, Sylvie A1 - Ruszel, Wioletta M. T1 - Propagation of Gibbsianness for infinite-dimensional diffusions with space-time interaction N2 - We consider infinite-dimensional diffusions where the interaction between the coordinates has a finite extent both in space and time. In particular, it is not supposed to be smooth or Markov. The initial state of the system is Gibbs, given by a strong summable interaction. If the strongness of this initial interaction is lower than a suitable level, and if the dynamical interaction is bounded from above in a right way, we prove that the law of the diffusion at any time t is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion in space uniformly in time of the Girsanov factor coming from the dynamics and exponential ergodicity of the free dynamics to an equilibrium product measure. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)18 KW - infinite-dimensional diffusion KW - cluster expansion KW - non-Markov drift KW - Girsanov formula KW - ultracontractivity Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69014 ER - TY - INPR A1 - Roelly, Sylvie A1 - Fradon, Myriam T1 - Infinite system of Brownian balls : equilibrium measures are canonical Gibbs N2 - We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with a local time term. We prove that the set of all equilibrium measures, solution of a detailed balance equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential. KW - Stochastic Differential Equation KW - hard core potential KW - Canonical Gibbs measure KW - detailed balance equation KW - reversible measure Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6720 ER - TY - INPR A1 - Roelly, Sylvie T1 - Reciprocal processes : a stochastic analysis approach N2 - Reciprocal processes, whose concept can be traced back to E. Schrödinger, form a class of stochastic processes constructed as mixture of bridges, that satisfy a time Markov field property. We discuss here a new unifying approach to characterize several types of reciprocal processes via duality formulae on path spaces: The case of reciprocal processes with continuous paths associated to Brownian diffusions and the case of pure jump reciprocal processes associated to counting processes are treated. This presentation is based on joint works with M. Thieullen, R. Murr and C. Léonard. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 6 KW - Reciprocal process KW - Brownian bridge KW - Poisson bridge KW - duality formula Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-64588 SN - 2193-6943 ER - TY - INPR A1 - Roelly, Sylvie T1 - Unas propiedades basicas de procesos de ramificación : Lectures held at ICIMAF La Habana, Cuba, 2009 and 2010 N2 - Aus dem Inhalt: 1. Unas propiedades de los procesos de Bienaymé-Galton-Watson de tiempo dis- creto (BGW) 2. Unas propiedades del proceso BGW de tiempo continuo 3. Limites de procesos de BGW cuando la población es numerosa T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2010, 07 Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49620 ER - TY - INPR A1 - Redig, Frank A1 - Roelly, Sylvie A1 - Ruszel, Wioletta T1 - Short-time Gibbsianness for infinite-dimensional diffusions with space-time interaction N2 - We consider a class of infinite-dimensional diffusions where the interaction between the components is both spatial and temporal. We start the system from a Gibbs measure with finiterange uniformly bounded interaction. Under suitable conditions on the drift, we prove that there exists t0 > 0 such that the distribution at time t = t0 is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion of both the initial interaction and certain time-reversed Girsanov factors coming from the dynamics. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2009, 04 Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49514 ER - TY - INPR A1 - Méléard, Sylvie A1 - Roelly, Sylvie T1 - Evolutive two-level population process and large population approximations N2 - We are interested in modeling the Darwinian evolution of a population described by two levels of biological parameters: individuals characterized by an heritable phenotypic trait submitted to mutation and natural selection and cells in these individuals influencing their ability to consume resources and to reproduce. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We are looking for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 8 KW - Two-level interacting process KW - birth-death-mutation-competition point process KW - non-linear integro-differential equations Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-64604 SN - 2193-6943 ER - TY - INPR A1 - Méléard, Sylvie A1 - Roelly, Sylvie T1 - A host-parasite multilevel interacting process and continuous approximations N2 - We are interested in modeling some two-level population dynamics, resulting from the interplay of ecological interactions and phenotypic variation of individuals (or hosts) and the evolution of cells (or parasites) of two types living in these individuals. The ecological parameters of the individual dynamics depend on the number of cells of each type contained by the individual and the cell dynamics depends on the trait of the invaded individual. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We look for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses. The study of the long time behavior of these processes seems very hard and we only develop some simple cases enlightening the difficulties involved. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2011, 01 KW - two-level interacting processes KW - birth-death-mutation-competition point process KW - host-parasite stochastic particle system Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51694 ER - TY - INPR A1 - Léonard, Christian A1 - Roelly, Sylvie A1 - Zambrini, Jean-Claude T1 - Temporal symmetry of some classes of stochastic processes N2 - In this article we analyse the structure of Markov processes and reciprocal processes to underline their time symmetrical properties, and to compare them. Our originality consists in adopting a unifying approach of reciprocal processes, independently of special frameworks in which the theory was developped till now (diffusions, or pure jump processes). This leads to some new results, too. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 7 KW - Markov processes KW - reciprocal processes KW - time symmetry Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-64599 SN - 2193-6943 ER - TY - INPR A1 - Keller, Peter A1 - Roelly, Sylvie A1 - Valleriani, Angelo T1 - A quasi-random-walk to model a biological transport process N2 - Transport Molecules play a crucial role for cell viability. Amongst others, linear motors transport cargos along rope-like structures from one location of the cell to another in a stochastic fashion. Thereby each step of the motor, either forwards or backwards, bridges a fixed distance. While moving along the rope the motor can also detach and is lost. We give here a mathematical formalization of such dynamics as a random process which is an extension of Random Walks, to which we add an absorbing state to model the detachment of the motor from the rope. We derive particular properties of such processes that have not been available before. Our results include description of the maximal distance reached from the starting point and the position from which detachment takes place. Finally, we apply our theoretical results to a concrete established model of the transport molecule Kinesin V. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 3 KW - Markov chain KW - random walk KW - molecular motor KW - step process Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63582 ER -