arXiv subjects

James Nolen

Publications and source records attributed to James Nolen.

33 records · Page 2Linked to original sources

arXiv2014

Normal approximation for the net flux through a random conductor

We consider solutions to an elliptic partial differential equation in $\mathbb{R}^d$ with a stationary, random conductivity coefficient. The boundary condition on a square domain of width $L$ is chosen so that the solution has a macroscopic unit gradient. We then consider the average flux through the domain. It is known that in the limit $L \to \infty$, this quantity converges to a deterministic constant, almost surely. Our main result is about normal approximation for this flux when $L$ is large: we give an estimate of the Kantorovich-Wasserstein distance between the law of this random variable and that of a normal random variable. This extends a previous result of the author to a much larger class of random conductivity coefficients. The analysis relies on elliptic regularity, on bounds for the Green's function, and on a normal approximation method developed by S. Chatterjee based on Stein's method.

math.PR↗

arXiv2014

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

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 natural to ask, can one develop efficient numerical schemes to estimate the probability of an Erdős-Rényi graph containing an excessively large number of a fixed given subgraph? Using the large deviation principle we study an importance sampling scheme as a method to numerically compute the small probabilities of large triangle counts occurring within Erdős-Rényi graphs. We show that the exponential tilt suggested directly by the large deviation principle does not always yield an optimal scheme. The exponential tilt used in the importance sampling scheme comes from a generalized class of exponential random graphs. Asymptotic optimality, a measure of the efficiency of the importance sampling scheme, is achieved by a special choice of the parameters in the exponential random graph that makes it indistinguishable from an Erdős-Rényi graph conditioned to have many triangles in the large network limit. We show how this choice can be made for the conditioned Erdős-Rényi graphs both in the replica symmetric phase as well as in parts of the replica breaking phase to yield asymptotically optimal numerical schemes to estimate this rare event probability.

math.PR↗

arXiv2013

Sticky central limit theorems on open books

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fréchet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension $1$ and hence measure $0$) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine. We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky).

math.PR↗

arXiv2013

Reactive trajectories and the transition path process

We study the trajectories of a solution $X_t$ to an Itô stochastic differential equation in $\Rm^d$, as the process passes between two disjoint open sets, $A$ and $B$. These segments of the trajectory are called transition paths or reactive trajectories, and they are of interest in the study of chemical reactions and thermally activated processes. In that context, the sets $A$ and $B$ represent reactant and product states. Our main results describe the probability law of these transition paths in terms of a transition path process $Y_t$, which is a strong solution to an auxiliary SDE having a singular drift term. We also show that statistics of the transition path process may be recovered by empirical sampling of the original process $X_t$. As an application of these ideas, we prove various representation formulas for statistics of the transition paths. We also identify the density and current of transition paths. Our results fit into the framework of the transition path theory by E and Vanden-Eijnden.

math.PR↗

arXiv2012

The logarithmic delay of KPP fronts in a periodic medium

We consider solutions of the KPP-type equations with a periodically varying reaction rate, and compactly supported initial data. It has been shown by M. Bramson in the case of the constant reaction rate that the lag between the position of such solutions and that of the traveling waves grows as (3/2) log(t). We generalize this result to the periodic case

math.AP↗

arXiv2012

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

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 $d \geq 2$. It is known that homogenization occurs as $ε\to 0$, but little is known about the statistical fluctuations of $w^ε$. Our main result shows that the variance of the solution $w^ε$ is bounded by $O(ε/|\log ε|)$. The proof relies on a modified Poincaré inequality of Talagrand.

math.PR↗

arXiv2010

Existence and Non-existence of Fisher-KPP Transition Fronts

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 in time bump-like solution. This is the first example of a medium in which no reaction-diffusion transition front exists. A weaker localized inhomogeneity leads to existence of transition fronts but only in a finite range of speeds. These results are in contrast with both Fisher-KPP reactions in homogeneous media as well as ignition-type reactions in inhomogeneous media.

math.AP↗

arXiv2010

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

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 stationary and ergodic, we prove sufficient conditions for homogenization to hold with probability one. These conditions are expressed in terms of travel times for the associated control problem. When the spatial dimension is equal to two and the fluid velocity is divergence-free, we verify that these conditions hold under suitable assumptions about the growth of the random stream function.

math.AP↗

arXiv2010

Homogenization and enhancement for the G-equation

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 small spatial divergence, we prove that, as the size of the oscillations diminishes, the solutions homogenize (average out) and converge to the solution of an effective anisotropic first-order (spatio-temporal homogeneous) level set equation. Moreover we obtain a rate of convergence and show that, under certain conditions, the averaging enhances the velocity of the underlying front. We also prove that, at scale one, the level sets of the solutions of the oscillatory problem converge, at long times, to the Wulff shape associated with the effective Hamiltonian. Finally we also consider advection depending on position at the integral scale.

math.AP↗

arXiv2007

Traveling waves in a one-dimensional random medium

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 existence of generalized random traveling waves and of transition fronts in general heterogeneous media.

math.AP↗

arXiv2005

Variational Principle of KPP Front Speeds in Temporally Random Shear Flows

We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows inside an infinite cylinder, under suitable assumptions of the shear field. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. The variational principle then allows us to bound and compute the front speeds. We show the linear and quadratic laws of speed enhancement as well as a resonance-like dependence of front speed on the temporal shear correlation length. To prove the variational principle, we use the comparison principle of solutions, the path integral representation of solutions, and large deviation estimates of the associated stochastic flows.

math.AP↗

arXiv2005

Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds

We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type traveling fronts in space-time periodic and mean zero incompressible advection, and establish a variational (minimization) formula for the minimal speeds. We approach the existence by considering limit of a sequence of front solutions to a regularized traveling front equation where the nonlinearity is combustion type with ignition cut-off. The limiting front equation is degenerate parabolic and does not permit strong solutions, however, the necessary compactness follows from monotonicity of fronts and degenerate regularity. We apply a dynamic argument to justify that the constructed KPP traveling fronts propagate at minimal speeds, and derive the speed variational formula. The dynamic method avoids the degeneracy in traveling front equations, and utilizes the parabolic maximum principle of the governing reaction-diffusion-advection equation. The dynamic method does not rely on existence of traveling fronts.

math.AP↗

arXiv2005

Min-Max Variational Principle and Front Speeds in Random Shear Flows

Speed ensemble of bistable (combustion) fronts in mean zero stationary Gaussian shear flows inside two and three dimensional channels is studied with a min-max variational principle. In the small root mean square regime of shear flows, a new class of multi-scale test functions are found to yield speed asymptotics. The quadratic speed enhancement law holds with probability arbitrarily close to one under the almost sure continuity (dimension two) and mean square Hölder regularity (dimension three) of the shear flows. Remarks are made on the conditions for the linear growth of front speed expectation in the large root mean square regime.

math.AP↗

arXiv2005

A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears

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 out on the ensemble of KPP speeds through spatially stationary random shear flows inside infinite channel domains. In the regime of small root mean square (rms) shear amplitude, the enhancement of the ensemble averaged KPP front speeds is proved to obey the quadratic law under certain shear moment conditions. Similarly, in the large rms amplitude regime, the enhancement follows the linear law. In particular, both laws hold for the Ornstein-Uhlenbeck process in case of two dimensional channels. An asymptotic ensemble averaged speed formula is derived in the small rms regime and is explicit in case of the Ornstein-Uhlenbeck process of the shear. Variational principle based computation agrees with these analytical findings, and allows further study on the speed enhancement distributions as well as the dependence of enhancement on the shear covariance. Direct simulations in the small rms regime suggest quadratic speed enhancement law for non-KPP nonlinearities.

math.AP↗

arXiv2004

Existence of KPP Type Fronts in Space-Time Periodic Shear Flows and a Study of Minimal Speeds Based on Variational Principle

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 is characterized by a variational principle involving the principal eigenvalue of a space-time periodic parabolic operator. Analysis of the variational principle shows that adding a mean-zero space time periodic shear flow to an existing mean zero space-periodic shear flow leads to speed enhancement. Computation of KPP minimal speeds is performed based on the variational principle and a spectrally accurate discretization of the principal eigenvalue problem. It shows that the enhancement is monotone decreasing in temporal shear frequency, and that the total enhancement from pure reaction-diffusion obeys quadratic and linear laws at small and large shear amplitudes.

math.AP↗