Georgia Institute of Technology College of Sciences

The G-equation is a Hamilton-Jacobi level-set equation, that is used in turbulent combustion theory. Level sets of the solution represent a flame surface which moves with normal velocity that is the sum of the laminar flame velocity and the fluid velocity. In this work I will discuss the large-scale long-time asymptotics of these solutions when the fluid velocity is modeled as a stationary incompressible random field. The main challenge of this work comes from the fact that our Hamiltonian is noncoercive. This is a joint work with J.Nolen.

We study a problem of estimation of a large Hermitian nonnegatively definite matrix S of unit trace based on n independent measurements Y_j = tr(SX_j ) + Z_j , j = 1, . . . , n, where {X_j} are i.i.d. Hermitian matrices and {Z_j } are i.i.d. mean zero random variables independent of {X_j}. Problems of this nature are of interest in quantum state tomography, where S is an unknown density matrix of a quantum system. The estimator is based on penalized least squares method with complexity penalty defined in terms of von Neumann entropy. We derive oracle inequalities showing how the estimation error depends on the accuracy of approximation of the unknown state S by low-rank matrices. We will discuss these results as well as some of the tools used in their proofs (such as generic chaining bounds for empirical processes and noncommutative Bernstein type inequalities).

We provide a sufficient condition for the continuity of real valued permanental processes. When applied to the subclass of permanental processes which consists of squares of Gaussian processes, we obtain the sufficient condition for continuity which is also known to be necessary. Using an isomorphism theorem of Eisenbaum and Kaspi which relates Markov local times and permanental processes we obtain a general sufficient condition for the joint continuity of the local times.

In this talk we shall discuss the problem of fitting a distribution function to the marginal distribution of a long memory process. It is observed that unlike in the i.i.d. set up, classical tests based on empirical process are relatively easy to implement. More importantly, we discuss fitting the marginal distribution of the error process in location, scale and linear regression models. An interesting observation is that the first order difference between the residual empirical process and the null model can not be used to asymptotically to distinguish between the two marginal distributions that differ only in their means. This finding is in sharp contrast to a recent claim of Chan and Ling to appear in the Ann. Statist. that such a process has a Gaussian weak limit. We shall also proposes some tests based on the second order difference in this case and analyze some of their properties. Another interesting finding is that residual empirical process tests in the scale problem are robust against not knowing the scale parameter. The third finding is that in linear regression models with a non-zero intercept parameter the first order difference between the empirical d.f. of residuals and the null d.f. can not be used to fit an error d.f. This talk is based on ongoing joint work with Donatas Surgailis.