About

Professor Scott Armstrong is a faculty member in the Department of Mathematics at NYU. His research interests include partial differential equations, probability theory, and stochastic homogenization. As a professor, he contributes to the mathematical community through his focus on these areas, advancing understanding in the behavior of complex systems described by PDEs and probabilistic models.

Research topics

• Mathematics
• Applied mathematics
• Mathematical analysis
• Physics

Selected publications

• Issue Information ‐ TOC

  Communications on Pure and Applied Mathematics · 2024-04-09

  paratextOpen access

  Rest of World), €5,204.00(Europe)

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• Variational methods for the kineticFokker–Planck equation

  Analysis & PDE · 2024-07-19 · 15 citations

  articleOpen access

  We develop a functional-analytic approach to the study of the Kramers and kinetic Fokker-Planck equations which parallels the classical H 1 theory of uniformly elliptic equations.In particular, we identify a function space analogous to H 1 and develop a well-posedness theory for weak solutions in this space.In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly convex functional.We prove new functional inequalities of Poincar-and Hrmander-type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the C regularity of weak solutions.We also use the Poincar-type inequality to give an elementary proof of the exponential convergence to equilibrium for solutions of the kinetic Fokker-Planck equation which mirrors the classic dissipative estimate for the heat equation.Finally, we prove enhanced dissipation in a weakly collisional limit.which is often called the kinetic Fokker-Planck equation.

  PublisherOA PDFDOI

• Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift

  arXiv (Cornell University) · 2024-04-01

  preprintOpen access1st authorCorresponding

  We consider the long-time behavior of a diffusion process on $\mathbb{R}^d$ advected by a stationary random vector field which is assumed to be divergence-free, dihedrally symmetric in law and have a log-correlated potential. A special case includes $\nabla^\perp$ of the Gaussian free field in two dimensions. We show the variance of the diffusion process at a large time $t$ behaves like $2 c_* t (\log t)^{1/2}$, in a quenched sense and with a precisely determined, universal prefactor constant $c_*>0$. We also prove a quenched invariance principle under this superdiffusive scaling. The proof is based on a rigorous renormalization group argument in which we inductively analyze coarse-grained diffusivities, scale-by-scale. Our analysis leads to sharp homogenization and large-scale regularity estimates on the infinitesimal generator, which are subsequently transferred into quantitative information on the process.

  PublisherOA PDFDOI

• Issue Information ‐ TOC

  Communications on Pure and Applied Mathematics · 2023-09-08

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  00 (Rest of World), €5,003.00(Europe), £3

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• The scaling limit of the continuous solid-on-solid model

  arXiv (Cornell University) · 2023-10-20

  preprintOpen access1st authorCorresponding

  We prove that the scaling limit of the continuous solid-on-solid model in $\mathbb{Z}^d$ is a multiple of the Gaussian free field.

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• Thermal approximation of the equilibrium measure and obstacle problem

  Annales de la faculté des sciences de Toulouse Mathématiques · 2022-10-28 · 12 citations

  articleOpen access1st authorCorresponding

  We consider the probability measure minimizing a free energy functional equal to the sum of a Coulomb interaction, a confinement potential and an entropy term, which arises in the statistical mechanics of Coulomb gases. In the limit where the inverse temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> tends to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> the entropy term disappears and the measure, which we call the “thermal equilibrium measure” tends to the well-known equilibrium measure, which can also be interpreted as a solution to the classical obstacle problem. We provide quantitative estimates on the convergence of the thermal equilibrium measure to the equilibrium measure in strong norms in the bulk of the latter, with a sequence of explicit correction terms in powers of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>β</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> , as well as an analysis of the tail after the boundary layer of size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>β</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> .

  PublisherOA PDFDOI

• Local laws and rigidity for Coulomb gases at any temperature

  The Annals of Probability · 2021-01-01 · 6 citations

  preprintOpen access1st authorCorresponding

  We study Coulomb gases in any dimension $d \geq 2$ and in a broad temperature regime. We prove local laws on the energy, separation and number of points down to the microscopic scale. These yield the existence of limiting point processes generalizing the Ginibre point process for arbitrary temperature and dimension. The local laws come together with a quantitative expansion of the free energy with a new explicit error rate in the case of a uniform background density. These estimates have explicit temperature dependence, allowing to treat regimes of very large or very small temperature, and exhibit a new minimal lengthscale for rigidity depending on the temperature. They apply as well to energy minimizers (formally zero temperature). The method is based on a bootstrap on scales and reveals the additivity of the energy modulo surface terms, via the introduction of subadditive and superadditive approximate energies.

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• <i>C</i>² Regularity of the Surface Tension for the ∇<i>ϕ</i> Interface Model

  Communications on Pure and Applied Mathematics · 2021-12-16 · 9 citations

  article1st authorCorresponding

  We consider the ∇ ϕ interface model with a uniformly convex interaction potential possessing Hölder continuous second derivatives. Combining ideas of Naddaf and Spencer with methods from quantitative homogenization, we show that the surface tension (or free energy) associated to the model is at least C 2,β for some β &gt; 0. We also prove a fluctuation‐dissipation relation by identifying its Hessian with the covariance matrix characterizing the scaling limit of the model. Finally, we obtain a quantitative rate of convergence for the Hessian of the finite‐volume surface tension to that of its infinite‐volume limit.

  PublisherDOI

• Optimal unique continuation for periodic elliptic equations on large scales

  arXiv (Cornell University) · 2021-07-29 · 1 citations

  preprintOpen access1st authorCorresponding

  We prove a quantitative, large-scale doubling inequality and large-scale three-ellipsoid inequality for solutions of uniformly elliptic equations with periodic coefficients. These estimates are optimal in terms of the minimal length scale on which they are valid, and are at least "almost" optimal in the prefactor constants--up to, at most, an iterated logarithm of the initial doubling ratio.

  PublisherOA PDFDOI

• Homogenization, Linearization, and <scp>Large‐Scale</scp> Regularity for Nonlinear Elliptic Equations

  Communications on Pure and Applied Mathematics · 2020 · 11 citations

  1st authorCorresponding
  • Mathematics
  • Mathematical analysis
  • Applied mathematics

  Abstract We consider nonlinear, uniformly elliptic equations with random, highly oscillating coefficients satisfying a finite range of dependence. We prove that homogenization and linearization commute in the sense that the linearized equation (linearized around an arbitrary solution) homogenizes to the linearization of the homogenized equation (linearized around the corresponding solution of the homogenized equation). We also obtain a quantitative estimate on the rate of this homogenization. These results lead to a better understanding of differences of solutions to the nonlinear equation, which is of fundamental importance in quantitative homogenization. In particular, we obtain a large‐scale C 0, 1 estimate for differences of solutions—with optimal stochastic integrability. Using this estimate, we prove a large‐scale C 1, 1 estimate for solutions, also with optimal stochastic integrability. Each of these regularity estimates are new even in the periodic setting. As a second consequence of the large‐scale regularity for differences, we improve the smoothness of the homogenized Lagrangian by showing that it has the same regularity as the heterogeneous Lagrangian, up to C 2, 1 . © 2020 Wiley Periodicals LLC

  PublisherDOI

Recent grants

• PostDoctoral Research Fellowship

  NSF · $135k · 2010–2014

Frequent coauthors

• Guofang Tokyo

  Walter de Gruyter (Germany)

  75 shared

• Parma Mingione

  Walter de Gruyter (Germany)

  75 shared

• Tristan Uppsala

  Friedrich-Alexander-Universität Erlangen-Nürnberg

  75 shared

• Luis Shen

  Friedrich-Alexander-Universität Erlangen-Nürnberg

  75 shared

• Giuseppe Martell

  Walter de Gruyter (Germany)

  75 shared

• Pavia Gianazza

  Friedrich-Alexander-Universität Erlangen-Nürnberg

  75 shared

• Bernard Dacorogna

  École Polytechnique Fédérale de Lausanne

  75 shared

• Yoshihiro Chicago

  Walter de Gruyter (Germany)

  75 shared

Education

• Ph.D.

  University of California, Berkeley

  2009

• B.S.

  Texas A&M University

  2002