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James H. Nolen

James H. Nolen

Professor of Mathematics
Mathematics
james.nolen@duke.edu
Box 90320, Durham, NC 27708-0320
120 Science Drive, Durham, NC 27708
Office hours Mondays 10:30am-12:00
Wednesdays 2:00pm-3:30  

Selected Publications

USING BERNOULLI MAPS TO ACCELERATE MIXING OF A RANDOM WALK ON THE TORUS

Journal Article Quarterly of Applied Mathematics · January 1, 2024 Featured Publication We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone is O(1/ε2), where ε is the step size. Our main results show that for a ... Full text Cite

The Fleming-Viot Process with McKean-Vlasov Dynamics

Journal Article Electronic Journal of Probability · August 3, 2022 Featured Publication The Fleming-Viot particle system consists of N identical particles diffusing in a domain U⊂Rd. Whenever a particle hits the boundary ∂U, that particle jumps onto another particle in the interior. It is known that this system provides a particle representat ... Full text Link to item Cite

Brownian bees in the infinite swarm limit

Journal Article Annals of Probability · 2022 Featured Publication Full text Link to item Cite

A free boundary problem arising from branching Brownian motion with selection

Journal Article Transactions of the American Mathematical Society · May 18, 2021 Featured Publication We study a free boundary problem for a parabolic partial differential equation in which the solution is coupled to the moving boundary through an integral constraint. The problem arises as the hydrodynamic limit of an interacting particle system involvi ... Full text Cite

Anomalous diffusion in comb-shaped domains and graphs

Journal Article Communications in Mathematical Sciences · 2020 Full text Link to item Cite

Asymptotic behavior of branching diffusion processes in periodic media

Journal Article Electronic Journal of Probability · January 1, 2020 We study the asymptotic behavior of branching diffusion processes in periodic media. For a super-critical branching process, we distinguish two types of behavior for the normalized number of particles in a bounded domain, depending on the distance of the d ... Full text Open Access Cite

Quantitative propagation of chaos in a bimolecular chemical reaction-diffusion model

Journal Article SIAM Journal on Mathematical Analysis · January 1, 2020 Featured Publication We study a stochastic system of N interacting particles which models bimolecular chemical reaction-diffusion. In this model, each particle i carries two attributes: the spatial location Xit ∈ Td, and the type [I]it ∈ { 1, . . ., n} . While Xit is a standar ... Full text Cite

Refined long-time asymptotics for Fisher–KPP fronts

Journal Article Communications in Contemporary Mathematics · November 2019 Featured Publication We study the one-dimensional Fisher–KPP equation, with an initial condition [Formula: see text] that coincides with the step function except on a compact set. A well-known result of Bramson in [Maximal displacement of branching Brownian motion, Co ... Full text Cite

Ratiometric GPCR signaling enables directional sensing in yeast.

Journal Article PLoS Biol · October 2019 Featured Publication Accurate detection of extracellular chemical gradients is essential for many cellular behaviors. Gradient sensing is challenging for small cells, which can experience little difference in ligand concentrations on the up-gradient and down-gradient sides of ... Full text Open Access Link to item Cite

Scaling limit of the Stein variational gradient descent: The mean field regime

Journal Article SIAM Journal on Mathematical Analysis · January 1, 2019 Featured Publication We study an interacting particle system in Rd motivated by Stein variational gradient descent [Q. Liu and D. Wang, Proceedings of NIPS, 2016], a deterministic algorithm for approximating a given probability density with unknown normalization based on parti ... Full text Cite

Block size in Geometric(p)-biased permutations

Journal Article Electronic Communications in Probability · 2018 Full text Link to item Cite

Scaling limit of the corrector in stochastic homogenization

Journal Article Annals of Applied Probability · 2017 Full text Link to item Cite

Convergence to a single wave in the Fisher-KPP equation

Journal Article Chinese Annals of Mathematics, Series B · 2017 Full text Link to item Cite

The logarithmic delay of KPP fronts in a periodic medium

Journal Article Journal of the European Mathematical Society · January 1, 2016 We extend, to parabolic equations of the KPP type in periodic media, a result of Bramson which asserts that, in the case of a spatially homogeneous reaction rate, the time lag between the position of an initially compactly supported solution and that of a ... Full text Cite

The importance sampling technique for understanding rare events in Erdős-Rényi random graphs

Journal Article Electronic Journal of Probability · October 19, 2015 In dense Erdős-Rényi random graphs, we are interested in the events where large numbers of a given subgraph occur. The mean behavior of subgraph counts is known, and only recently were the related large deviations results discovered. Consequently, it is na ... Full text Cite

Power-Like Delay in Time Inhomogeneous Fisher-KPP Equations

Journal Article Communications in Partial Differential Equations · March 4, 2015 We consider solutions of the KPP equation with a time-dependent diffusivity of the form σ(t/T). For an initial condition that has sufficiently fast decay in x, we show that when σ(s) is increasing in time the front position at time T is (Formula presented. ... Full text Cite

Reactive trajectories and the transition path process

Journal Article Probability Theory and Related Fields · February 2015 Full text Open Access Cite

Normal approximation for the net flux through a random conductor

Journal Article Stochastic Partial Differential Equations: Analysis and Computations · 2015 Full text Cite

Sticky central limit theorems on open books

Journal Article The Annals of Applied Probability · 2013 Full text Open Access Cite

Normal approximation for a random elliptic equation

Journal Article Probability Theory and Related Fields · 2013 We consider solutions of an elliptic partial differential equation in {Mathematical expression} with a stationary, random conductivity coefficient that is also periodic with period {Mathematical expression}. Boundary conditions on a square domain of width ... Full text Cite

Multiscale modelling and inverse problems

Conference Lecture Notes in Computational Science and Engineering · January 1, 2012 The need to blend observational data and mathematical models arises in many applications and leads naturally to inverse problems. Parameters appearing in the model, such as constitutive tensors, initial conditions, boundary conditions, and forcing can be e ... Full text Cite

Capillary drops on a rough surface

Journal Article Interfaces and Free Boundaries · 2012 We study liquid drops lying on a rough planar surface. The drops are minimizers of an energy functional that includes a random adhesion energy. We prove the existence of minimizers and the regularity of the free boundary. When the length scale of the rando ... Full text Cite

Existence and Non-Existence of Fisher-KPP Transition Fronts

Journal Article Archive for Rational Mechanics and Analysis · 2012 We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global-in-time solutions while creating a global ... Full text Cite

A Sublinear Variance Bound for Solutions of a Random Hamilton-Jacobi Equation

Journal Article Journal of Statistical Physics · 2012 We estimate the variance of the value function for a random optimal control problem. The value function is the solution w ε of a Hamilton-Jacobi equation with random Hamiltonian H(p, x, ω)=K(p)-V(x/ε, ω) in dimension d ≥ 2. It is known that homogenization ... Full text Cite

Homogenization and Enhancement for the G-Equation

Journal Article Archive for Rational Mechanics and Analysis · 2011 We consider the so-called G-equation, a level set Hamilton-Jacobi equation used as a sharp interface model for flame propagation, perturbed by an oscillatory advection in a spatio-temporal periodic environment. Assuming that the advection has suitably smal ... Full text Cite

Homogenization of the G-equation with incompressible random drift in two dimensions

Journal Article Communications in Mathematical Sciences · January 1, 2011 We study the homogenization limit of solutions to the G-equation with random drift. This Hamilton-Jacobi equation is a model for flame propagation in a turbulent fluid in the regime of thin flames. For a fluid velocity field that is statistically stationar ... Full text Cite

An invariance principle for random traveling waves in one dimension

Journal Article SIAM Journal on Mathematical Analysis · 2011 We consider solutions to a nonlinear reaction diffusion equation when the reaction term varies randomly with respect to the spatial coordinate. The nonlinearity is either the ignition nonlinearity or the bistable nonlinearity, under suitable restrictions o ... Full text Cite

A central limit theorem for pulled fronts in a random medium

Journal Article Networks and Heterogeneous Media · 2011 We consider solutions to a nonlinear reaction diffusion equation when the reaction term varies randomly with respect to the spatial coordinate. The nonlinearity is the KPP type nonlinearity. For a stationary and ergodic medium, and for certain initial cond ... Full text Cite

Bounds on front speeds for inviscid and viscous G-equations

Journal Article Methods and Applications of Analysis · December 2009 Link to item Cite

Fine scale uncertainty in parameter estimation for elliptic equations

Journal Article Inverse Problems · November 26, 2009 We study the problem of estimating the coefficients in an elliptic partial differential equation using noisy measurements of a solution to the equation. Although the unknown coefficients may vary on many scales, we aim only at estimating their slowly varyi ... Full text Cite

KPP Fronts in 1D Random Drift

Journal Article Discrete and Continuous Dynamical Systems B · 2009 Link to item Cite

Stability of generalized transition fronts

Journal Article Communications in Partial Differential Equations · 2009 We study the qualitative properties of the generalized transition fronts for the reaction-diffusion equations with the spatially inhomogeneous nonlinearity of the ignition type. We show that transition fronts are unique up to translation in time and are gl ... Full text Cite

Traveling waves in a one-dimensional heterogeneous medium

Journal Article Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis · 2009 We consider solutions of a scalar reaction-diffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We also establish ... Full text Cite

Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows

Journal Article Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis · 2009 We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in space-time random incompressible flows in dimension d > 1. We prove that if the flow field is stationary, ergodic, and obeys a suitable moment condition, the large time f ... Full text Cite

KPP fronts in a one-dimensional random drift

Journal Article Discrete and Continuous Dynamical Systems - Series B · 2009 We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in a one dimensional random drift which is a mean zero stationary ergodic process with mixing property and local Lipschitz continuity. To prove the variational princ ... Full text Cite

Variational principle and reaction-diffusion front speeds in random flows

Journal Article ICIAM07-Proceedings · December 2008 Cite

A framework for adaptive multiscale methods for elliptic problems

Journal Article Multiscale Modeling and Simulation · 2008 We describe a projection framework for developing adaptive multiscale methods for computing approximate solutions to elliptic boundary value problems. The framework is consistent with homogenization when there is scale separation. We introduce an adaptive ... Full text Cite

Computing reactive front speeds in random flows by variational principle

Journal Article Physica D: Nonlinear Phenomena · 2008 We study reactive front speeds in randomly perturbed cellular flows using a variational representation for the front speed. We develop this representation into a computational tool for computing the front speeds without resorting to closure approximations. ... Full text Cite

A variational principle based study of KPP minimal front speeds in random shears

Journal Article Nonlinearity · 2005 The variational principle for Kolmogorov-Petrovsky-Piskunov (KPP) minimal front speeds provides an efficient tool for statistical speed analysis, as well as a fast and accurate method for speed computation. A variational principle based analysis is carried ... Full text Cite

Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle

Journal Article Discrete and Continuous Dynamical Systems · January 1, 2005 We prove the existence of reaction-diffusion traveling fronts in mean zero space-time periodic shear flows for nonnegative reactions including the classical KPP (Kolmogorov-Petrovsky-Piskunov) nonlinearity. For the KPP nonlinearity, the minimal front speed ... Full text Cite

Transient and persistent spectral hole burning in Eu3+-doped sol-gel produced SiO2 glass

Journal Article Journal of Luminescence · June 1, 2004 Transient and persistent spectral hole burning (TSHB and PSHB) experiments were performed on Eu3+ ions in sol-gel SiO2 glasses with aluminum co-doping. Differences in the hole burning behavior were observed among samples made from two organosilicate precur ... Full text Cite

Min-Max Variational Principles and Fronts Speeds in Random Shear Flows

Journal Article Methods and Applications of Analysis · 2004 Link to item Cite

Reaction-diffusion front speeds in spatially-temporally periodic shear flows

Journal Article Multiscale Modeling and Simulation · January 1, 2003 We study the asymptotics of two space dimensional reaction-diffusion front speeds through mean zero space-time periodic shears using both analytical and numerical methods. The analysis hinges on traveling fronts and their estimates based on qualitative pro ... Full text Cite

Red-to-green up-conversion in Er-doped SiO2 and SiO2-TiO2 sol-gel silicate glasses

Journal Article Journal of Luminescence · December 1, 2001 Monolithic Er-doped SiO2-TiO2 binary glasses of high optical quality were used in an investigation of the effects of different annealing conditions and titanium content on fluorescence yields and decay times of the 4S3/2 level of Er3+. In addition, the cha ... Full text Cite