### Abstract

In its simplest instance, kinetic Brownian in R^{d} is a C^{1} random path (m_{t}, v_{t}) with unit velocity v_{t} a Brownian motion on the unit sphere run at speed a > 0. Properly time rescaled as a function of the parameter a, its position process converges to a Brownian motion in R^{d} as a tends to infinity. On the other side the motion converges to the straight line motion (= geodesic motion) when a goes to 0. Kinetic Brownian motion provides thus an interpolation between geodesic and Brownian flows in this setting. Think now about changing R^{d} for the diffeomorphism group of a fluid domain, with a velocity vector now a vector field on the domain. I will explain how one can prove in this setting an interpolation result similar to the previous one, giving an interpolation between Euler’s equations of incompressible flows and a Brownian-like flow on the diffeomorphism group.

### Our speaker

Ismaël Bailleul is an associate professor at the Université de Rennes.

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