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Alexander J Dunlap

Assistant Professor of Mathematics
Mathematics
120 Science Drive-Physics 117, Box 90320, Durham, NC 27710
120 Science Dr, Physics 297, Durham, NC 27708

Selected Publications


Fluctuation exponents of the KPZ equation on a large torus

Journal Article Communications on Pure and Applied Mathematics · November 2023 We study the one‐dimensional KPZ equation on a large torus, started at equilibrium. The main results are optimal variance bounds in the super‐relaxation regime and part of the relaxation regime. ... Full text Cite

Invariant measures for stochastic conservation laws on the line

Journal Article Nonlinearity · September 1, 2023 We consider a stochastic conservation law on the line with solution-dependent diffusivity, a super-linear, sub-quadratic Hamiltonian, and smooth, spatially-homogeneous kick-type random forcing. We show that this Markov process admits a unique ergodic spati ... Full text Cite

Localization length of the $1+1$ continuum directed random polymer

Journal Article Annales Henri Poincaré · July 2023 In this paper, we study the localization length of the continuum directed polymer, defined as the distance between the endpoints of two paths sampled independently from the quenched polymer measure. We show that the localization length converges in distrib ... Full text Cite

Local versions of sum-of-norms clustering

Journal Article SIAM Journal on Mathematics of Data Science · December 31, 2022 Sum-of-norms clustering is a convex optimization problem whose solution can be used for the clustering of multivariate data. We propose and study a localized version of this method, and show in particular that it can separate arbitrarily close balls in the ... Full text Cite

A forward-backward SDE from the 2D nonlinear stochastic heat equation

Journal Article Annals of Probability · May 1, 2022 We consider a nonlinear stochastic heat equation in spatial dimension $d = 2$, forced by a white-in-time multiplicative Gaussian noise with spatial correlation length $ε > 0$ but divided by a factor of $\sqrt{\log ε^{ −1}}$. We impose a condition on the Li ... Full text Cite

A quenched local limit theorem for stochastic flows

Journal Article Journal of Functional Analysis · March 15, 2022 We consider a particle undergoing Brownian motion in Euclidean space of any dimension, forced by a Gaussian random velocity field that is white in time and smooth in space. We show that conditional on the velocity field, the quenched density of the particl ... Full text Cite

Viscous shock solutions to the stochastic Burgers equation

Journal Article Archive for Rational Mechanics and Analysis · November 1, 2021 We define a notion of a viscous shock solution of the stochastic Burgers equation that connects “top” and “bottom” spatially stationary solutions of the same equation. Such shocks generally travel in space, but we show that they admit time-invariant measur ... Full text Cite

The random heat equation in dimensions three and higher: the homogenization viewpoint

Journal Article Archive for Rational Mechanics and Analysis · November 2021 Full text Cite

Stationary solutions to the stochastic burgers equation on the line

Journal Article Communications in Mathematical Physics · March 1, 2021 We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or extremal, if it can ... Full text Cite

Tightness of Liouville first passage percolation for $γ∈ (0 , 2)$

Journal Article Publications Mathématiques de l'IHÉS · December 1, 2020 We study Liouville first passage percolation metrics associated to a Gaussian free field $h$ mollified by the two-dimensional heat kernel $p_t$ in the bulk, and related star-scale invariant metrics. For $γ∈ (0 , 2)$ and $ξ=γ/d_γ$, where $d_γ$ is the Liouvi ... Full text Cite

Existence of stationary stochastic Burgers evolutions on $\mathbf{R}^2$ and $\mathbf{R}^3$

Journal Article Nonlinearity · October 1, 2020 We prove that the stochastic Burgers equation on $\mathbf{R}^d$, $d < 4$, forced by gradient noise that is white in time and smooth in space, admits spacetime-stationary solutions. These solutions are thus the gradients of solutions to the KPZ equation on ... Full text Cite

The continuum parabolic Anderson model with a half-Laplacian and periodic noise

Journal Article Electronic Communications in Probability · September 17, 2020 We construct solutions of a renormalized continuum fractional parabolic Anderson model, formally given by $∂_tu = −(−∆)^{1/2} u + ξu$, where $ξ$ is a periodic spatial white noise. To be precise, we construct limits as $ε → 0$ of solutions of $∂_tu_ε = −(−∆ ... Full text Cite

Subsequential scaling limits for Liouville graph distance

Journal Article Communications in Mathematical Physics · June 1, 2020 For $0 < γ< 2$ and $δ> 0$, we consider the Liouville graph distance, which is the minimal number of Euclidean balls of $γ$-Liouville quantum gravity measure at most $δ$ whose union contains a continuous path between two endpoints. In this paper, we show th ... Full text Cite

Fluctuation exponents of the KPZ equation on a large torus

Journal Article Communications on Pure and Applied Mathematics · November 2023 We study the one‐dimensional KPZ equation on a large torus, started at equilibrium. The main results are optimal variance bounds in the super‐relaxation regime and part of the relaxation regime. ... Full text Cite

Invariant measures for stochastic conservation laws on the line

Journal Article Nonlinearity · September 1, 2023 We consider a stochastic conservation law on the line with solution-dependent diffusivity, a super-linear, sub-quadratic Hamiltonian, and smooth, spatially-homogeneous kick-type random forcing. We show that this Markov process admits a unique ergodic spati ... Full text Cite

Localization length of the $1+1$ continuum directed random polymer

Journal Article Annales Henri Poincaré · July 2023 In this paper, we study the localization length of the continuum directed polymer, defined as the distance between the endpoints of two paths sampled independently from the quenched polymer measure. We show that the localization length converges in distrib ... Full text Cite

Local versions of sum-of-norms clustering

Journal Article SIAM Journal on Mathematics of Data Science · December 31, 2022 Sum-of-norms clustering is a convex optimization problem whose solution can be used for the clustering of multivariate data. We propose and study a localized version of this method, and show in particular that it can separate arbitrarily close balls in the ... Full text Cite

A forward-backward SDE from the 2D nonlinear stochastic heat equation

Journal Article Annals of Probability · May 1, 2022 We consider a nonlinear stochastic heat equation in spatial dimension $d = 2$, forced by a white-in-time multiplicative Gaussian noise with spatial correlation length $ε > 0$ but divided by a factor of $\sqrt{\log ε^{ −1}}$. We impose a condition on the Li ... Full text Cite

A quenched local limit theorem for stochastic flows

Journal Article Journal of Functional Analysis · March 15, 2022 We consider a particle undergoing Brownian motion in Euclidean space of any dimension, forced by a Gaussian random velocity field that is white in time and smooth in space. We show that conditional on the velocity field, the quenched density of the particl ... Full text Cite

Viscous shock solutions to the stochastic Burgers equation

Journal Article Archive for Rational Mechanics and Analysis · November 1, 2021 We define a notion of a viscous shock solution of the stochastic Burgers equation that connects “top” and “bottom” spatially stationary solutions of the same equation. Such shocks generally travel in space, but we show that they admit time-invariant measur ... Full text Cite

The random heat equation in dimensions three and higher: the homogenization viewpoint

Journal Article Archive for Rational Mechanics and Analysis · November 2021 Full text Cite

Stationary solutions to the stochastic burgers equation on the line

Journal Article Communications in Mathematical Physics · March 1, 2021 We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or extremal, if it can ... Full text Cite

Tightness of Liouville first passage percolation for $γ∈ (0 , 2)$

Journal Article Publications Mathématiques de l'IHÉS · December 1, 2020 We study Liouville first passage percolation metrics associated to a Gaussian free field $h$ mollified by the two-dimensional heat kernel $p_t$ in the bulk, and related star-scale invariant metrics. For $γ∈ (0 , 2)$ and $ξ=γ/d_γ$, where $d_γ$ is the Liouvi ... Full text Cite

Existence of stationary stochastic Burgers evolutions on $\mathbf{R}^2$ and $\mathbf{R}^3$

Journal Article Nonlinearity · October 1, 2020 We prove that the stochastic Burgers equation on $\mathbf{R}^d$, $d < 4$, forced by gradient noise that is white in time and smooth in space, admits spacetime-stationary solutions. These solutions are thus the gradients of solutions to the KPZ equation on ... Full text Cite

The continuum parabolic Anderson model with a half-Laplacian and periodic noise

Journal Article Electronic Communications in Probability · September 17, 2020 We construct solutions of a renormalized continuum fractional parabolic Anderson model, formally given by $∂_tu = −(−∆)^{1/2} u + ξu$, where $ξ$ is a periodic spatial white noise. To be precise, we construct limits as $ε → 0$ of solutions of $∂_tu_ε = −(−∆ ... Full text Cite

Subsequential scaling limits for Liouville graph distance

Journal Article Communications in Mathematical Physics · June 1, 2020 For $0 < γ< 2$ and $δ> 0$, we consider the Liouville graph distance, which is the minimal number of Euclidean balls of $γ$-Liouville quantum gravity measure at most $δ$ whose union contains a continuous path between two endpoints. In this paper, we show th ... Full text Cite

Fluctuations of the solutions to the KPZ equation in dimensions three and higher

Journal Article Probability Theory and Related Fields · April 1, 2020 We prove, using probabilistic techniques and analysis on the Wiener space, that the large scale fluctuations of the KPZ equation in $d≥ 3$ with a small coupling constant, driven by a white in time and colored in space noise, are given by the Edwards-Wilkin ... Full text Cite

Constructing a solution of the $(2+1)$-dimensional KPZ equation

Journal Article Annals of Probability · March 1, 2020 The $(d+ 1)$-dimensional KPZ equation is the canonical model for the growth of rough $d$-dimensional random surfaces. A deep mathematical understanding of the KPZ equation for $d= 1$ has been achieved in recent years, and the case $d≥ 3$ has also seen some ... Full text Cite

Liouville first-passage percolation: Subsequential scaling limits at high temperature

Journal Article Annals of Probability · January 1, 2019 Let $\{Y_{\mathfrak{B}}(x): x ∈ \mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which ... Full text Cite