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Stochastic processes
Leticia F. Cugliandolo, Gregory Schehr
Stochastic processes are an essential tool for understanding the behaviour of many physical systems. They are often useful to describe more complex systems such as in the study of phase transitions, coarsening phenomena or active matter. Our group has focused on relatively simple systems, like Brownian motion, random walks and Lévy flights or run-and-tumble particles where detailed analytical calculations, e.g., for first-passage time and extreme value statistics, can be carried out explicitly.
More recently, we addressed the question of how to generate efficiently constrained stochastic processes. For instance, how can one generate a (one-dimensional) Brownian trajectory constrained to start and end up at the same point after a given fixed time T - this is called a bridge configuration? For such processes, we have developed a quite general method based on an exact effective Langevin dynamics that allows one to generate such bridges in a very efficient way. We have also extended this method to more complex systems, such as non-intersecting Brownian motions, for which the effective Langevin equation turns out to be an efficient approach to obtain analytical results for such interacting particle systems.
Our group has also worked on other fundamental aspects, namely the construction of path-integral methods and stochastic calculus to describe stochastic processes.
