Graduate Student / Postdoc Seminar

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Sticky Brownian motion and its applications

Speaker: Miranda Holmes-Cerfon, Courant Institute of Mathematical Sciences

Location: Warren Weaver Hall 1302

Date: Friday, February 1, 2019, 1 p.m.

Synopsis:

Sticky diffusion processes are solutions to stochastic differential equations which can “stick” to, i.e. spend finite time on, a lower-dimensional boundary. The sticking is reversible, so the process can hit the boundary and leave again, and while on the boundary it can move according to dynamics that are different from those in the interior. Such processes are characterized by a second derivative appearing in the boundary condition for the forward and backward Kolmogorov equations. A canonical example is a (root-2) reflecting Brownian motion that is sticky at the origin, whose forward and backward equations are identically f_t = f_{xx} with sticky boundary condition f_{x} = kf_{xx} at x=0, where k>0. 

I will show how sticky diffusions arise when modeling systems of mesoscale particles (those with diameters around 1 micrometer), which form the building blocks for many of the materials around us. I will focus on sticky Brownian motion to build intuition into such processes, and will show that it arises in a natural asymptotic limit when a particle diffuses on a line with a deep but narrow potential well near the origin. I will introduce a numerical method to simulate this process, which gives insight into some of its peculiar properties, and also turns out to be orders of magnitude faster than traditional methods for simulating such short-ranged systems. Finally, I will sketch how generalizations of these ideas to higher-dimensional systems of particles can sometimes lead to cleaner and more accurate theories for their dynamics, and will illustrate this with our experiments on colloidal clusters.