The strong weak convergence of the quasi-EA

In this paper, we investigate the convergence of a novel simulation scheme to the target diffusion process. This scheme, the Quasi-EA, is closely related to the Exact Algorithm (EA) for diffusion processes, as it is obtained by neglecting the rejection step in EA. We prove the existence of a myopic coupling between the Quasi-EA and the diffusion. Moreover, an upper bound for the coupling probability is given. Consequently we establish the convergence of the Quasi-EA to the diffusion with respect to the total variation distance

On quasianalytic classes of Gelfand-Shilov type. Parametrix and convolution

We develop a convolution theory for quasianalytic ultradistributions of Gelfand-Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in the study of new topological and structural properties of several quasianalytic spaces of functions and ultradistributions. In particular, our results apply to Fourier hyperfunctions and Fourier ultra-hyperfunctions.Comment: 37 page

Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials

We investigate the construction of diffusions consisting of infinitely numerous Brownian particles moving in $\mathbb{R}^d$ and interacting via logarithmic functions (two-dimensional Coulomb potentials). These potentials are very strong and act over a long range in nature. The associated equilibrium states are no longer Gibbs measures. We present general results for the construction of such diffusions and, as applications thereof, construct two typical interacting Brownian motions with logarithmic interaction potentials, namely the Dyson model in infinite dimensions and Ginibre interacting Brownian motions. The former is a particle system in $\mathbb{R}$, while the latter is in $\mathbb{R}^2$. Both models are translation and rotation invariant in space, and as such, are prototypes of dimensions $d=1,2$, respectively. The equilibrium states of the former diffusion model are determinantal or Pfaffian random point fields with sine kernels. They appear in the thermodynamical limits of the spectrum of the ensembles of Gaussian random matrices such as GOE, GUE and GSE. The equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non-Hermitian Gaussian random matrices known as the Ginibre ensemble.Comment: Published in at http://dx.doi.org/10.1214/11-AOP736 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org