@article{ConfortiKosenkovaRoelly2019, author = {Conforti, Giovanni and Kosenkova, Tetiana and Roelly, Sylvie}, title = {Conditioned Point Processes with Application to Levy Bridges}, series = {Journal of theoretical probability}, volume = {32}, journal = {Journal of theoretical probability}, number = {4}, publisher = {Springer}, address = {New York}, issn = {0894-9840}, doi = {10.1007/s10959-018-0863-8}, pages = {2111 -- 2134}, year = {2019}, abstract = {Our first result concerns a characterization by means of a functional equation of Poisson point processes conditioned by the value of their first moment. It leads to a generalized version of Mecke's formula. En passant, it also allows us to gain quantitative results about stochastic domination for Poisson point processes under linear constraints. Since bridges of a pure jump L{\´e}vy process in Rd with a height a can be interpreted as a Poisson point process on space-time conditioned by pinning its first moment to a, our approach allows us to characterize bridges of L{\´e}vy processes by means of a functional equation. The latter result has two direct applications: First, we obtain a constructive and simple way to sample L{\´e}vy bridge dynamics; second, it allows us to estimate the number of jumps for such bridges. We finally show that our method remains valid for linearly perturbed L{\´e}vy processes like periodic Ornstein-Uhlenbeck processes driven by L{\´e}vy noise.}, language = {en} } @unpublished{Roelly2013, author = {Roelly, Sylvie}, title = {Reciprocal processes : a stochastic analysis approach}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64588}, year = {2013}, abstract = {Reciprocal processes, whose concept can be traced back to E. Schr{\"o}dinger, 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 This presentation is based on joint works with M. Thieullen, R. Murr and C. L{\´e}onard.}, language = {en} } @unpublished{MeleardRoelly2013, author = {M{\´e}l{\´e}ard, Sylvie and Roelly, Sylvie}, title = {Evolutive two-level population process and large population approximations}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64604}, year = {2013}, abstract = {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.}, language = {en} } @unpublished{RoellyVallois2016, author = {Roelly, Sylvie and Vallois, Pierre}, title = {Convoluted Brownian motion}, volume = {5}, number = {9}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-96339}, pages = {37}, year = {2016}, abstract = {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.}, language = {en} } @book{ChampagnatRoelly2007, author = {Champagnat, Nicolas and Roelly, Sylvie}, title = {Multitype Dawson-Watanabe superprocesses conditioned by remote survival}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Mathematische Statistik un}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Mathematische Statistik un}, publisher = {Univ.}, address = {Potsdam}, issn = {1613-3307}, pages = {39 S.}, year = {2007}, language = {en} } @unpublished{RoellyFradon2006, author = {Roelly, Sylvie and Fradon, Myriam}, title = {Infinite system of Brownian balls : equilibrium measures are canonical Gibbs}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6720}, year = {2006}, abstract = {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.}, language = {en} } @unpublished{CattiauxFradonKuliketal.2013, author = {Cattiaux, Patrick and Fradon, Myriam and Kulik, Alexei Michajlovič and Roelly, Sylvie}, title = {Long time behavior of stochastic hard ball systems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-68388}, year = {2013}, abstract = {We study the long time behavior of a system of two or three Brownian hard balls living in the Euclidean space of dimension at least two, submitted to a mutual attraction and to elastic collisions.}, language = {en} } @unpublished{ConfortiRoelly2015, author = {Conforti, Giovanni and Roelly, Sylvie}, title = {Reciprocal class of random walks on an Abelian group}, volume = {4}, number = {1}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72604}, pages = {22}, year = {2015}, abstract = {Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, we compute explicitly its reciprocal characteristics and we present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group.}, language = {en} } @misc{RoellySortais2004, author = {Roelly, Sylvie and Sortais, Michel}, title = {Space-time asymptotics of an infinite-dimensional diffusion having a long- range memory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6700}, year = {2004}, abstract = {We develop a cluster expansion in space-time for an infinite-dimensional system of interacting diffusions where the drift term of each diffusion depends on the whole past of the trajectory; these interacting diffusions arise when considering the Langevin dynamics of a ferromagnetic system submitted to a disordered external magnetic field.}, language = {en} } @misc{RoellyThieullen2005, author = {Roelly, Sylvie and Thieullen, Mich{\`e}le}, title = {Duality formula for the bridges of a Brownian diffusion : application to gradient drifts}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6710}, year = {2005}, abstract = {In this paper, we consider families of time Markov fields (or reciprocal classes) which have the same bridges as a Brownian diffusion. We characterize each class as the set of solutions of an integration by parts formula on the space of continuous paths C[0; 1]; R-d) Our techniques provide a characterization of gradient diffusions by a duality formula and, in case of reversibility, a generalization of a result of Kolmogorov.}, language = {en} }