Generating constrained stochastic processes

I will review recent progress on generating constrained stochastic processes. Starting from constrained Brownian trajectories, I will highlight the notion of an effective Langevin equation and generalise it to the case of discrete-time processes, including the case of Lévy flights. I will then extend these results to the case of non-Markovian processes, including the run-and-tumble particles which are commonly used as a toy model of active matter. Finally, I will show how this method not only allows to generate efficiently constrained multi-particles dynamics, such as nonintersecting Brownian bridges, but also provides a useful framework to obtain exact analytical results for such interacting particles systems.