32 citations · 115 across the 22 of their papers we have counts for
25 papers
Quantitative positivity of transition densities for random perturbations of Hamiltonian systems
Shimaa Elesaely, David P. Herzog, Kyle L. Liss
We study a class of diffusion processes arising from random perturbations of conservative Hamiltonian systems. Under a set of abstract hypotheses -- including basic structural assu…
Small-time asymptotics for hypoelliptic diffusions
Juraj Földes, David P. Herzog
An inductive procedure is developed to calculate the asymptotic behavior at time zero of a diffusion with polynomial drift and degenerate, additive noise. The procedure gives rise…
Phase space contraction of degenerately damped random splittings
David P. Herzog, Jonathan C. Mattingly
When studying out-of-equilibrium systems, one often excites the dynamics in some degrees of freedom while removing the excitation in others through damping. In order for the system…
Functional inequalities for a family of infinite-dimensional diffusions with degenerate noise
Fabrice Baudoin, Maria Gordina, David Herzog +2
For a family of infinite-dimensional diffusions with degenerate noise, we develop a modified calculus on finite-dimensional projections of the equation in order to produce expl…
Cutoff in the Bernoulli-Laplace Model With Unequal Colors and Urn Sizes
Thomas Griffin, Bailey Hall, Jackson Hebner +4
We consider a generalization of the Bernoulli-Laplace model in which there are two urns and total balls, of which are red and white, and where the left urn holds $m…
Cutoff in the Bernoulli-Laplace model with swaps
Joseph S. Alameda, Caroline Bang, Zachary Brennan +3
This paper considers the -Bernoulli--Laplace model in the case when there are two urns, the total number of red and white balls is the same, and the number of selections …
The method of stochastic characteristics for linear second-order hypoelliptic equations
Juraj Foldes, David Herzog
We study hypoelliptic stochastic differential equations (SDEs) and their connection to degenerate-elliptic boundary value problems on bounded or unbounded domains. In particular, w…
Gibbsian dynamics and the generalized Langevin equation
David P. Herzog, Jonathan C. Mattingly, Hung D. Nguyen
We study the statistically invariant structures of the nonlinear generalized Langevin equation (GLE) with a power-law memory kernel. For a broad class of memory kernels, including…
A functional Law of the Iterated Logarithm for weakly hypoelliptic diffusions at time zero
Marco Carfagnini, Juraj Foldes, David P. Herzog
We study the almost sure behavior of solutions of stochastic differential equations (SDEs) as time goes to zero. Our main general result establishes a functional law of the iterate…
Weighted -contractivity of Langevin dynamics with singular potentials
Evan Camrud, David P. Herzog, Gabriel Stoltz +1
Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential allowing for singularities. By modifying the direct approach t…
Stability and invariant measure asymptotics in a model for heavy particles in rough turbulent flows
David P. Herzog, Hung D. Nguyen
We study a system of Skorokhod stochastic differential equations (SDEs) modeling the pairwise dispersion (in spatial dimension ) of heavy particles transported by a rough self…
Sensitivity of steady states in a degenerately-damped stochastic Lorenz system
Juraj Foldes, Nathan E. Glatt-Holtz, David P. Herzog
We study stability of solutions for a randomly driven and degenerately damped version of the Lorenz '63 model. Specifically, we prove that when damping is absent in one of the temp…
Gamma calculus beyond Villani and explicit convergence estimates for Langevin dynamics with singular potentials
Fabrice Baudoin, Maria Gordina, David P. Herzog
This paper studies convergence to equilibrium for second-order Langevin dynamics under general growth conditions on the potential. Although we are principally motivated by the case…
Exponential relaxation of the Nosé-Hoover equation under Brownian heating
David P. Herzog
We study a stochastic perturbation of the Nosé-Hoover equation (called the Nosé-Hoover equation under Brownian heating) and show that the dynamics converges at a geometric rate to…
The Generalized Langevin Equation with a power-law memory in a nonlinear potential well
Nathan Glatt-Holtz, David Herzog, Scott McKinley +1
The generalized Langevin equation (GLE) is a stochastic integro-differential equation that has been used to describe the velocity of microparticles in viscoelastic fluids. In this…
Ergodicity and Lyapunov functions for Langevin dynamics with singular potentials
David P. Herzog, Jonathan C. Mattingly
We study Langevin dynamics of particles on interacting through a singular repulsive potential, e.g.~the well-known Lennard-Jones type, and show that the system converges…
Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations
Nathan E. Glatt-Holtz, David P. Herzog, Jonathan C. Mattingly
We establish the dual notions of scaling and saturation from geometric control theory in an infinite-dimensional setting. This generalization is applied to the low-mode control pro…
The Small-Mass Limit for Langevin Dynamics with Unbounded Coefficients and positive friction
David P. Herzog, Scott Hottovy, Giovanni Volpe
A class of Langevin stochastic differential equations is shown to converge in the small-mass limit under very weak assumptions on the coefficients defining the equation. The conver…
A practical criterion for positivity of transition densities
David P. Herzog, Jonathan C. Mattingly
We establish a simple criterion for locating points where the transition density of a degenerate diffusion is strictly positive. Throughout, we assume that the diffusion satisfies…
Noise-Induced Stabilization of Planar Flows I
David P. Herzog, Jonathan C. Mattingly
We show that the complex-valued ODE \begin{equation*} \dot z_t = a_{n+1} z^{n+1} + a_n z^n+\cdots+a_0, \end{equation*} which necessarily has trajectories along which the dynamics b…
Noise-Induced Stabilization of Planar Flows II
David P. Herzog, Jonathan C. Mattingly
We continue the work started in Part I of this article, showing how the addition of noise can stabilize an otherwise unstable system. The analysis makes use of nearly optimal Lyapu…
An extension of Hörmander's hypoellipticity theorem
David P. Herzog, Nathan Totz
Motivated by applications to stochastic differential equations, an extension of Hörmander's hypoellipticity theorem is proved for second-order degenerate elliptic operators with no…
Transition from ergodic to explosive behavior in a family of stochastic differential equations
Jeremiah Birrell, David P. Herzog, Jan Wehr
We study a family of quadratic stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, we fin…
Geometric Ergodicity of Two--dimensional Hamiltonian systems with a Lennard--Jones--like Repulsive Potential
Ben Cooke, David P. Herzog, Jonathan C. Mattingly +2
In this paper we establish the ergodicity of Langevin dynamics for simple two-particle system involving a Lennard-Jones type potential. To the best of our knowledge, this is the fi…