Scaling limits of interacting diffusions in domains

We survey recent effort in establishing the hydrodynamic limits and the fluctuation limits for a class of interacting diffusions in domains. These systems are introduced to model the transport of positive and negative charges in solar cells. They are general microscopic models that can be used to describe macroscopic phenomena with coupled boundary conditions, such as the popula- tion dynamics of two segregated species under competition. Proving these two types of limits represents establishing the functional law of large numbers and the functional central limit theorem, respectively, for the empirical measures of the spatial positions of the particles. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation.     

CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

Let x =(xn)n≥1 be a martingale on a noncommutative probability space (M,T) and (wn)n≥1 a sequence of positive numbers such that Wn =n∑k=1wk → ∞ as n → ∞.We prove that x =(xn)n≥1 converges in E(M) if an...     

Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommutative martingale Hardy space ${\mathsf{h}}_p$ for all $0<p<2$ by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of ${\mathsf{h}}_p$ for all $0<p<1$, and provide a constructive proof of the atomic decomposition for $p=1$ which resolves a main problem on the subject left open for the last twelve years. We also study ${(p,\infty )}_c$-atoms, and show that every ${(p,2)}_c$-atom can be decomposed into a sum of ${(p,\infty )}_c$-atoms; consequently, for every $0<p\le 1$, the ${(p,q)}_c$-atoms lead to the same atomic space for all $2\le q\le \infty$. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space ${\mathsf{h}}_p$ ($0<p<1$) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to prove some sharp martingale inequalities.