Kinetic Brownian motion in the diffeomorphism group of a
closed Riemannian manifold
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. On the one hand the motion converges to
the straight line motion (= geodesic motion) when 'a' goes to 0. On the
other hand, when 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. 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.