Browsing by Subject "math.PR"
Now showing items 120 of 20

A Variation on the DonskerVaradhan Inequality for the Principial Eigenvalue
(20170423)The purpose of this short note is to give a variation on the classical DonskerVaradhan inequality, which bounds the first eigenvalue of a secondorder elliptic operator on a bounded domain $\Omega$ by the largest mean first ... 
Asymptotic behavior of the Brownian frog model
We introduce an extension of the frog model to Euclidean space and prove properties for the spread of active particles. The new geometry introduces a phase transition that does not occur for the frog model on the lattice. ... 
Coupling and Decoupling to bound an approximating Markov Chain
(20170727)This simple note lays out a few observations which are well known in many ways but may not have been said in quite this way before. The basic idea is that when comparing two different Markov chains it is useful to couple ... 
Ergodicity and Lyapunov functions for Langevin dynamics with singular potentials
(20171130)We study Langevin dynamics of $N$ particles on $R^d$ interacting through a singular repulsive potential, e.g.~the wellknown LennardJones type, and show that the system converges to the unique invariant Gibbs measure ... 
Error bounds for Approximations of Markov chains
(20171130)The first part of this article gives error bounds for approximations of Markov kernels under FosterLyapunov conditions. The basic idea is that when both the approximating kernel and the original kernel satisfy a FosterLyapunov ... 
Fractional stochastic differential equations satisfying fluctuationdissipation theorem
(20170423)We consider in this work stochastic differential equation (SDE) model for particles in contact with a heat bath when the memory effects are nonnegligible. As a result of the fluctuationdissipation theorem, the differential ... 
Geometric ergodicity of Langevin dynamics with Coulomb interactions
This paper is concerned with the long time behavior of Langevin dynamics of {\em Coulomb gases} in $\mathbf{R}^d$ with $d\geq 2$, that is a second order system of Brownian particles driven by an external force and ... 
Learning interacting particle systems: diffusion parameter estimation for aggregation equations
(20180214)In this article, we study the parameter estimation of interacting particle systems subject to the Newtonian aggregation. Specifically, we construct an estimator $\widehat{\nu}$ with partial observed data to approximate the ... 
Limiting Behaviors of High Dimensional Stochastic Spin Ensemble
Lattice spin models in statistical physics are used to understand magnetism. Their Hamiltonians are a discrete form of a version of a Dirichlet energy, signifying a relationship to the Harmonic map heat flow equation. ... 
Moderate Deviation for Random Elliptic PDEs with Small Noise
(20170423)Partial differential equations with random inputs have become popular models to characterize physical systems with uncertainty coming from, e.g., imprecise measurement and intrinsic randomness. In this paper, we perform ... 
Multiscale integrators for stochastic differential equations and irreversible Langevin samplers
(20170423)We study multiscale integrator numerical schemes for a class of stiff stochastic differential equations (SDEs). We consider multiscale SDEs that behave as diffusions on graphs as the stiffness parameter goes to its limit. ... 
Numerical methods for stochastic differential equations based on Gaussian mixture
We develop in this work a numerical method for stochastic differential equations (SDEs) with weak second order accuracy based on Gaussian mixture. Unlike the conventional higher order schemes for SDEs based on It\^oTayl... 
Propagation of Fluctuations in Biochemical Systems, II: Nonlinear Chains
We consider biochemical reaction chains and investigate how random external fluctuations, as characterized by variance and coefficient of variation, propagate down the chains. We perform such a study under the assumption ... 
Scaling and Saturation in InfiniteDimensional Control Problems with Applications to Stochastic Partial Differential Equations
(20170727)We establish the dual notions of scaling and saturation from geometric control theory in an infinitedimensional setting. This generalization is applied to the lowmode control problem in a number of concrete nonlinear partial ... 
Scaling Limit: Exact and Tractable Analysis of Online Learning Algorithms with Applications to Regularized Regression and PCA
We present a framework for analyzing the exact dynamics of a class of online learning algorithms in the highdimensional scaling limit. Our results are applied to two concrete examples: online regularized linear regression ... 
Scaling limits of a model for selection at two scales
The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. ... 
Scaling limits of a model for selection at two scales
(2015)The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. ... 
Seemingly stable chemical kinetics can be stable, marginally stable, or unstable
We present three examples of chemical reaction networks whose ordinary differential equation scaling limit are almost identical and in all cases stable. Nevertheless, the Markov jump processes associated to these ... 
Smooth invariant densities for random switching on the torus
(20170830)We consider a random dynamical system obtained by switching between the flows generated by two smooth vector fields on the 2dtorus, with the random switchings happening according to a Poisson process. Assuming that the ... 
The strong Feller property for singular stochastic PDEs
(2016)We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $\Phi^4_3$ model. As a corollary, ...