ABSTRACT: A harmonic function of the Brownian path is a local martingale. Is the converse true? We show that the class of local martingale functions of Brownian motion is co-extensive with the class of finely harmonic functions, and then use a results of Fuglede and Gardiner to answer this question in the negative, in dimensions bigger than 2.

With the demand from modeling systems level cellular biochemistry as a reaction network, different applied mathematical approaches are now being pursued. I will discuss three approaches based on (1) systems of ODE with nonlinearity, (1) stochastic processes with irreversibility, and (3) constraint-based optimization suggesting an oriented matroid. A unifying theme of these approaches is the nonequilibrium thermodynamics of living (open) systems.

We consider the edge Hall conductance and show it is invariant under perturbations located in a strip along the edge. This enables us to prove for the edge conductances a general sum rule relating currents due to the presence of two different media located respectively on the left and on the
right half plane. As a particular interesting case we put forward a general quantization formula for the difference of edge Hall conductances in semi-infinite samples with and without a confining wall. Applications to disordered Hall systems with and without a confining potential are discussed.