MATJ5112 MA3: Stochastic Modelling and Numerical Tools around the Physics of Complex Flows (JSS31) (2 cr)

(0) Introduction, motivation
(1) A short introduction to stochastic differential equations (SDE). Existence and uniqueness (strong & weak solutions). Fokker-Planck equation. Old and new examples from transport models.
(2) A visit to the mean field approximation. An introduction to McKean-Vlasov SDEs, motivated by turbulent flow models.
(3) Numerical approximation for SDEs. Time integration schemes for SDEs; Sampling algorithms for McKean-Vlasov SDEs. An introduction to the main numerical analysis tools and results.

If time permits:
(4) Modelling strong interactions between objects immersed in a flow, particles and walls. Introduction to SDEs with boundaries : reflected & confined SDEs.
(5) Introduction to the ergodic theory for SDEs. Stationary phenomena and equilibrium. Fast and slow variables.

The main objective of these lectures is to give a concise overview of the theory of stochastic differential equations (SDE), as modelling and numerical tools. Starting form the basic properties of SDEs, the lectures will present different aspects of the theory, motivated and illustrated by their use in turbulent transport and its simulation. Stochastic differential equation are used in physics of fluids and in many related engineering approaches for industrial and environmental applications. SDEs' theory and turbulent transport have a long common history. Yet the design of predictive simulation tools for pollutant dispersion, sedimentation in the ocean, or many energy production processes, greatly challenge researches in the field of SDEs and their simulation, combining together all the aspects presented in this course.

Usual notions on measures, integration and probability theory. Brownian motion, Itô’s formula, notion on continuous time martingales and Markov processes.