Mathematical Colloquium Convection-enhanced diffusion, and Brownian motion on the Lie group SL_n

This talk draws a connection between a well-known phenomenon in fluid dynamics and an object from differential geometry. On the one hand, the ubiquitous phenomenon is that advection by a turbulent divergence-free drift effectively (and dramatically) enhances the diffusion of particles in n-dimensional Euclidean space. On the other hand, the object from differential geometry is a natural notion of Brownian motion on SL_n, the Lie group and Riemannian manifold of matrices of unit determinant; it is a tensorial version of geometric Brownian motion. The connection is given by monitoring the relative position of a pair of particles, as a function of their original position, and involves a change of time variables: The “time” variable for the Brownian motion on SL_n is the logarithm of the effective diffusivity for the particle, which is an increasing function of physical time.

This is joint work with Peter Morfe and Christian Wagner.