Grant: Discovery Grants Program - Individual

This proposal focuses on stochastic differential systems driven by fractional Brownian motions, stochastic heat equations driven by general Gaussian noises and Dirichlet processes. In particular, it is proposed the following projects. (1) Construct parameter estimators for nonlinear stochastic differential systems driven by fractional Brownian motions of Hurst parameter H1/2 to the case H1/2. It is planned to study the existence and uniqueness of this equation for Hurst parameter H<1/2. Global solution will also be studied. (4) Find the broadest condition on covariance structure of the noise so that the stochastic heat equation has a unique classical solution. Of course the best one is the necessary and sufficient condition and it is intended to search such condition. (5) For fixed t and x, u(t,x) is a random variable, which is of continuous type, namely, it has a density. It is planned to have a better understanding of this density, for example, to obtain the asymptotic behavior of this density. (6) Establish an Ito formula for stochastic heat equation so that the formula can be applied to the Cole-Hopf transformation and then study rigorously the relation between stochastic heat equation and KPZ equation. (7) Extend the Brox diffusion from one dimension to high dimension. Namely, study the high dimensional Brownian motion in fractional noisy environment and study the branching processes where each individual follows a Brox diffusion. (7) Study other stochastic partial differential equations such as stochastic elliptic equations with boundary conditions and fractional order stochastic partial differential equations. (8) Explore if the Gaussian analysis will be useful in machine learning or not. (9) Dirichlet processes play important role in Bayesian nonparametrics and have application in machine learning. It is proposed to bring more stochastic analysis including methodology from nonlinear filtering to the study of Dirichlet processes. (10) Study random matrix theory and seek if there is any relation with stochastic heat equation.