# A critical drift-diffusion equation: intermittent behavior

# A critical drift-diffusion equation: intermittent behavior

This short course is about a simple but rich model problem at the cross section of stochastic homogenization and singular stochastic partial differential equation (PDE): We consider a drift-diffusion process with a time independent and divergence-free random drift b that is of white-noise character. As already realized in the physics literature, the critical case of two space dimensions is most interesting: The elliptic generator L requires a small-scale cut-off for well-posedness, and one has marginally super-diffusive behavior on large scales. We shall study the drift-diffusion equation tu Lu = 0 with initial data u(t = 0, x) = x, and establish intermittent behavior of the increments u(x) u(y). Inter-mittency means that p-th stochastic moments (w. r. t. to the environmental noise b) depend anomalously on p. Hence the Gaussian drift b generates a very non-Gaussian behavior in the solution of the linear PDE. Note that u(x) is the averaged position (w. r. t. to the thermal noise) of a particle started at x, and thus can be interpreted as Lagrangian coordinates. We establish this by approximating u through a stochastic exponential in terms of a large-scale cut off L = t. In fact, this stochastic exponential is a tensorial version of a Gaussian multiplicative chaos, driven by a log-log correlated field. It is the natural diffusion on the Lie group SL(2).

This is joint work with G. Chatzigeorgiou, P. Morfe, L.Wang, and with C. Wagner.