1. Einstein relation for a tracer particle in simple exclusion processes: Einstein relation relates the mobility of a charged particle in the presence of an external field to its self-diffusion coefficient. In my Ph.D. thesis I prove that such a relation is valid for a tracer particle in symmetric simple exclusion processes in 3 or more dimensions. We would like to investigate its validity in dimensions 1 and 2, as well as in mean zero asymmetric simple exclusion processes.
2. The motion of a tracer particle in simple exclusion with reflecting boundary : The diffusively rescaled position of a tracer particle in simple exclusion converges to a Brownian motion. When the translation invariance symmetry is broken the question remains open . It is expected that when we consider a symmetric simple exclusion process on the positive integers with reflecting boundary at zero the position of the particle will converge to a reflected Brownian motion. This is a question we would like to investigate.
3. Asymptotic density in a coalescing driven system: Consider a particle system which consists of particles of two types, say A and B. Particles A (ambient gas molecules) perform simple exclusion and they swap places with particles B (aerosol particles), which also coagulate at a certain rate when they come close. We are interested in the asymptotic behavior in time of the density of particles B.