Alex Dunlap's home page
I am an assistant professor in the Department of Mathematics at Duke University. I study problems in probability theory, partial differential equations, and applied math. I am particularly interested in the asymptotic behavior of stochastic partial differential equations.
Before coming to Duke, from 2020 to 2023 I was an NSF postdoc at NYU Courant, sponsored by Jean-Christophe Mourrat and Yuri Bakhtin. I completed my Ph.D. at Stanford in 2020, advised by Lenya Ryzhik.
Along with the other probabilists in the department, I help organize the Duke Probability Seminar. Also, with colleagues at UNC and NC State, we organize the Triangle Probability Forum. Everyone is welcome to attend both series.
My research is supported in part by the National Science Foundation under grant no. DMS-2346915. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Math papers and preprints
(in reverse order of first arXiv posting)
- Invariant measures for the open KPZ equation: an analytic perspective (with Yu Gu and Tommaso Rosati). Submitted, 2025. [arXiv]
- A critical stochastic heat equation with long range noise (with Martin Hairer and Xue-Mei Li). Submitted, 2025. [arXiv]
- McKean–Vlasov limits of scaling-critical reaction-diffusion equations with random initial data (with Bryan Castillo). Submitted, 2025. [arXiv]
- Invariant measures and shocks in the KPZ fixed point (with Evan Sorensen). Submitted, 2025. [arXiv]
- Simultaneous global inviscid Burgers flows with periodic Poisson forcing. Annales Henri Lebesgue 8 (2025), pp. 873–923. [arXiv, doi, animation]
- Viscous shock fluctuations in KPZ (with Evan Sorensen). Submitted, 2024. [arXiv]
- Edwards-Wilkinson fluctuations in subcritical 2D stochastic heat equations (with Cole Graham). Electronic Communications in Probability 30 (2025), paper no. 96. [arXiv, doi]
- Additive-multiplicative stochastic heat equations, stationary solutions, and Cauchy statistics (with Chiranjib Mukherjee). Annals of Applied Probability 35 (2025), no. 4, pp. 2526–2543. [arXiv, doi]
- Branching Brownian motion with generation-dependent diffusivity and nonlocal partial differential equations (with Lenya Ryzhik). To appear in Israel Journal of Mathematics, 2025. [arXiv]
- Jointly stationary solutions of periodic Burgers flow (with Yu Gu). Journal of Functional Analysis 287 (2024), no. 12, 110656. [arXiv, doi]
- The 2D nonlinear stochastic heat equation: pointwise statistics and the decoupling function (with Cole Graham). Submitted, 2023. [arXiv]
- Uniqueness and root-Lipschitz regularity for a degenerate heat equation (with Cole Graham). SIAM Journal on Mathematical Analysis 57 (2025), no. 2, pp. 1866–1891. [arXiv, journal, doi]
- Localization length of the \(1+1\) continuum directed polymer (with Yu Gu and Liying Li). Annales Henri Poincaré 24 (2023), pp. 2537–2555. [arXiv, doi]
- Invariant measures for stochastic conservation laws on the line (with Theodore D. Drivas, Cole Graham, Joonhyun La, and Lenya Ryzhik). Nonlinearity 36 (2023), no. 9, pp. 4553–4584. [arXiv, doi]
- Fluctuation exponents of the KPZ equation on a large torus (with Yu Gu and Tomasz Komorowski). Communications on Pure and Applied Mathematics 76 (2023), no. 11, pp. 3104–3149. [arXiv, doi]
- Local versions of sum-of-norms clustering (with Jean-Christophe Mourrat). SIAM Journal on Mathematics of Data Science (SIMODS) 4 (2022), no. 4, pp. 1250–1271. [arXiv, doi]
- A quenched local limit theorem for stochastic flows (with Yu Gu). Journal of Functional Analysis 282 (2022), no. 6, 109372. [arXiv, doi]
- Sum-of-norms clustering does not separate nearby balls (with Jean-Christophe Mourrat). Journal of Machine Learning Research 25 (2024), no. 123, pp. 1–40. [arXiv, journal, code]
- A forward-backward SDE from the 2D nonlinear stochastic heat equation (with Yu Gu). Annals of Probability 50 (2022), no. 3, pp. 1204–1253. [arXiv, doi]
- Viscous shock solutions to the stochastic Burgers equation (with Lenya Ryzhik). Archive for Rational Mechanics and Analysis 242 (2021), no. 2, pp. 937–971. [arXiv, doi]
- The continuum parabolic Anderson model with a half-Laplacian and periodic noise. Electronic Communications in Probability 25 (2020), paper no. 64. [arXiv, doi]
- Existence of stationary stochastic Burgers evolutions on \(\mathbf{R}^2\) and \(\mathbf{R}^3\). Nonlinearity 33 (2020), no. 12, pp. 6480–6501. [arXiv, doi]
- Stationary solutions to the stochastic Burgers equation on the line (with Cole Graham and Lenya Ryzhik). Communications in Mathematical Physics 382 (2021), no. 2, pp. 875–949. [arXiv, doi]
- Tightness of Liouville first passage percolation for \(\gamma\in (0,2)\) (with Jian Ding, Julien Dubédat, and Hugo Falconet). Publications Mathématiques de l'IHÉS 132 (2020), pp. 353–403. [arXiv, doi]
- Subsequential scaling limits for Liouville graph distance (with Jian Ding). Communications in Mathematical Physics 376 (2020), pp. 1499–1572. [arXiv, doi]
- Fluctuations of the solutions to the KPZ equation in dimensions three and higher (with Yu Gu, Lenya Ryzhik, and Ofer Zeitouni). Probability Theory and Related Fields 176 (2020), no. 3–4, pp. 1217–1258. [arXiv, doi]
- Constructing a solution of the \((2+1)\)-dimensional KPZ equation (with Sourav Chatterjee). Annals of Probability 48 (2020), no. 2, pp. 1014–1055. [arXiv, doi]
- The random heat equation in dimensions three and higher: the homogenization viewpoint (with Yu Gu, Lenya Ryzhik, and Ofer Zeitouni). Archive for Rational Mechanics and Analysis 242 (2021), no. 2, pp. 827–873. [arXiv, doi]
- Liouville first-passage percolation: subsequential scaling limit at high temperature (with Jian Ding). Annals of Probability 47 (2019), no. 2, pp. 690–742. [arXiv, doi]
- Expected regularized total variation of Brownian motion. Unpublished. [arXiv]
Published outreach report
- Curiosity-Based Biophysics Projects in a High School Setting with Graduate Student Mentorship (Cooper J Galvin, Katherine N. Liu, Andrew S. Kennard, Sahil K. Tembulkar, Alexander Dunlap, Tao A. G. Large, Thao Pham, Derek Le, Aurora Alvarez-Buylla, Helen Nguyen, Ezequiel Ponce, Sophia Tran, Nikki Nguyen, Christina Ngo, Christina Tran, Gabriela Huynh, Patrick Allamandola, Zev Bryant). The Biophysicist 2 (2021), no. 1. [doi]
Recorded talks
- Stationary measure for the open KPZ equation: an analytic viewpoint at the BIRS workshop Emerging Synergies between Stochastic Analysis and Statistical Mechanics, October 2025
- Viscous shock solutions in KPZ at the SLMath Pathways Workshop: Kinetic Theory & Stochastic Partial Differential Equations, August 2025
- Nonlinear weak-noise stochastic heat equations in two dimensions at the Probability and the City Seminar, April 2022
- A forward-backward SDE from the 2D nonlinear stochastic heat equation at the Junior Integrable Probability Seminar, December 2020
- Stationary and shock solutions for the stochastic Burgers equation at the Stanford Probability Seminar, June 2020
- Constructing (2+1)-dimensional KPZ evolutions at the BIRS workshop Interacting Particle Systems and Parabolic PDEs, August 2018
Fun stuff
Some mathematical visualizations (require a reasonably modern [in 2016] browser, e.g. Firefox or Chrome).
Teaching
- Spring 2026: Math 340: Advanced Introduction to Probability
- Spring 2026: Math 790: Minicourse on Regularity Structures and the KPZ Equation
- Notes from a June 2025 minicourse in Münster (updated each day)
- Spring 2025: Math 641 Probability
- Fall 2024 (September 27–October 25): Minicourse on Directed Polymers in Random Environments
- Spring 2024 (March 20–April 17): Minicourse on Stochastic PDE
- Fall 2023: Math 541 Applied Stochastic Processes
- Spring 2023: MATH-GA 2500 Partial Differential Equations
- Fall 2022: MATH-UA 262 Ordinary Differential Equations
- Spring 2022: MATH-UA 233 Theory of Probability
- Fall 2021: MATH-UA 120 Discrete Mathematics
Contact
- Email: alexander.dunlap@nospam.duke.edu
- Office: Physics 297
- ORCID
- Pronouns: he/him