Path integrals and non-Markovian Langevin equations

The Langevin equation is a paradigmatic tool for studying the dynamics of particles subject to stochastic and dissipative forces. In the open-quantum system picture, it arises as the classical limit of a system linearly coupled to an equilibrium thermal bath. However, when the bath responds nonlinearly or is itself driven far from equilibrium by thermal gradients or other mechanisms, the resulting effective Langevin description becomes non-Markovian and might exhibit memory kernels and colored noise. These effects often lead to anomalous diffusion and the emergence of prethermal dynamical phases marked by weak ergodicity breaking, including time crystals and time glasses. In this talk, I will present recent results on the fractional Langevin equation for out-of-equilibrium systems involving white and colored noises and introduce a generalized path-integral formalism for these non-Markovian settings. The results recover the conventional Langevin and Klein–Kramers equations as limiting cases and extend them to regimes where memory and non-equilibrium fluctuations play a dominant role.