Abstract: We prove differentiability of the effective Lagrangian for continuous time multidimensional directed variational problems in random dynamic environments with positive dependence range in time. This implies that limiting fundamental solutions in the associated homogenization problems for HJB equations are classical.
Abstract: For directed polymers, the shape function computes the limiting average energy accrued by paths with a given average slope. We prove that, for a large family of directed polymer models in discrete time and continuous space in dimension 1+1, for positive and zero temperature, the shape function is differentiable with respect to the slope on the entire real line.
Abstract: We consider (1+1)-dimensional directed polymers in a random potential and provide sufficient conditions guaranteeing joint localization. Joint localization means that for typical realizations of the environment, and for polymers started at different starting points, all the associated endpoint distributions localize in a common random region that does not grow with the length of the polymer. In particular, we prove that joint localization holds when the reference random walk of the polymer model is either a simple symmetric lattice walk or a Gaussian random walk. We also prove that the very strong disorder property holds for a large class of space-continuous polymer models, implying the usual single polymer localization.
Abstract: Dynamical spectral estimation is a well-established numerical approach for estimating eigenvalues and eigenfunctions of the Markov transition operator from trajectory data. Although the approach has been widely applied in biomolecular simulations, its error properties remain poorly understood. Here we analyze the error of a dynamical spectral estimation method called “the variational approach to conformational dynamics" (VAC). We bound the approximation error and estimation error for VAC estimates. Our analysis establishes VAC's convergence properties and suggests new strategies for tuning VAC to improve accuracy.
Abstract: In this paper we seek to present a detailed and relatively self-contained proof of the existence of area minimizing surfaces diffeomorphic to a disk that spans closed curves in $\mathbb{R}^3.$ The proof will require machinery from PDE theory and differential geometry. We seek to develop the basic tools needed, namely Sobolev Spaces and basic riemannian geometry, assuming knowledge of $L^p$ spaces and some entry level familiarity with differentiable 2-manifolds. The key steps to solving this Plateau Problem will be: 1) identify the relationship between the area and energy functionals, 2) prove existence and regularity of energy minimizing functions with given boundary values, and 3) properly adjust our minimizing sequence of boundary values so as to extract a uniformly convergent subsequence. Along the way we will prove some interesting and useful facts about harmonic functions.