Hi, I'm Applied Math And Computational Sciences Postdoc at the University of Pennsylvania. 

I have worked on applying a dimension reduction technique for stochastic ODEs to a stochastic PDE to compute the stochastic effects on wave speed. The presence of stochastic noise induces not only a mean zero wobble in its speed, but also an additional small positive or negative net drift depending on the properties of the noise. On the left is a pulse wave solution in a stochastic FitzHugh-Nagumo equation.  Such models are used to study voltage propagation in Neuroscience. Using singular perturbation theory, we derive a kinematic model to which we apply the so called Isochronal Phase Reduction.  This allows us to compute the average net drift induced by the noise completely deterministically, given the correlation structure of the noise. This means the computation is more efficient because we don't need to sample realizations of the wave itself to estimate whether the noise makes it move faster or slower than it would without noise. 

Ion channels embedded in a cell 's membrane (in an axon say) are known to open and close depending upon the voltage potential across the membrane. However, their behavior is also stochastic. Thus, it is their average behavior over some number of neighboring ion channels which can give rise to propagation of an action potential as demonstrated by the toy model simulated in the figure to the left.  In the figure we simulate the upstroke of an action potential with a hybrid system.

The work below focuses on the development, simulation, and homogenization of such models. 

Sound waves of large wavelength can more easily penetrate walls or other barriers.  In much the same way,  a compression wave with a long wavelength traveling through a mass-spring lattice (masses connected via springs) can easily overcome the discreteness of the lattice as well as microscopic heterogeneities in its material properties.  If the springs are linear (Hook's law), dispersion eventually leads to the vanishing of the wave; however, if the springs are non-linear, certain waves may propagate unaffected for a much longer time (and distance). My research as has focused on the stochastic effects (or sometimes lack thereof) of such waves traveling through heterogeneous lattices.