**Seminars and Conferences**

**Archived**

**28 November 2022 at 4,00 pm**

# Spectral content of a single random trajectory: exact results for gaussian processes

In this talk will concentrate on **power spectral densities** (PSD) of **Gaussian centered stochastic processes**. Will first introduce the standard textbook definition which involves the integrated covariance function of the process taken in the limit of an infinitely long observation time T. On examples of fractional **Brownian motion** and a **Brownian gyrator model**, will highlight some shortcomings of the standard definition and thus a necessity of going beyond it considering instead the PSD of a single trajectory with finite T. For such a random variable I will prove an inequality obeyed by its noise-to-signal ratio and moreover derive its full probability density function for an arbitrary Gaussian process, arbitrary frequency and T. Eventually, will discuss a remarkable behavior of the frequency-frequency correlations of a **single-trajectory PSD**.**BiographyProfessor Oshanin's career** started in Moscow, where he got a PhD in Theoretical and Mathematical Physics under the supervision of Professor Burlatsky and Professor Ovchinnikov. Then, after a period spent at Institute of Chemical Physics in Moscow, he moved abroad working for a period at University of Freiburg (

**Germany**), University of Mons-Hainaut (

**Belgium**) and Paris, at University Pierre & Marie Curie and CNRS; then, in 1997 he obtained a position as Research Director at Sorbonne University - CNRS. He is currently long term visiting professor at Department of Mathematical Science (

**DISMA**) in Politecnico. His scientific activity spans between many different topics; among all, we can highlight some main subjects: since the beginning of his careeer he is working on tracer diffusion in quiescent lattice gases of hard-core particles with stochastic dynamics; in 2018 he has initiated a collaborative project on the analysis of spectral densities of individual trajectories of experimentally relevant stochastic processes; recently, he got interested in extreme events characterizing diverse stochastic processes in biophysical systems, such as, e.g., binding of ligands to receptors on cellular membranes. He also worked on fluctuation phenomena in reaction-diffusion systems and on dynamics of wetting and capillarity on a microscopic scale with experimentalists from laboratory "P.G. de Gennes" in

**Paris**.