Journal articleCommunications in Mathematical Physics · August 1, 2026
The first identity in (5.6) of the referenced article is incorrect: the correct version would have E^X0,00,tE^X~0,00,t on the right side rather than E^X0,00,qE^X~0,00,q, but with this change the rest of the proof does not go through. Therefore, we retract ...
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Journal articleCommunications in Mathematical Physics · April 1, 2026
We study “V-shaped” solutions to the KPZ equation, those having opposite asymptotic slopes θ and -θ, with θ>0, at positive and negative infinity, respectively. Answering a question of Janjigian, Rassoul-Agha, and Seppäläinen, we show that the spatial incre ...
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Journal articleElectronic Journal of Probability · January 1, 2026
We construct a family of invariant measures from the perspective of a shock in the KPZ fixed point. These measures are parameterized by a positive number θ>0, and are supported on functions f satisfying lim|x|→∞f(x) |x|=2θ. Each can be described ...
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Journal articleAnnales Henri Lebesgue · December 12, 2025
We study the inviscid Burgers equation on the circle T := R/Z forced by the spatial derivative of a Poisson point process on R × T. We construct global solutions with mean θ simultaneously for all θ ∈ R, and in addition construct their associated global sh ...
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Journal articleAnnals of Applied Probability · August 1, 2025
We study long-term behavior and stationary distributions for stochastic heat equations forced simultaneously by a multiplicative noise and an independent additive noise with the same distribution. We prove that nontrivial space-time translation-invariant m ...
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Journal articleSIAM Journal on Mathematical Analysis · January 1, 2025
We consider nonnegative solutions of the quasilinear heat equation in one dimension. Our solutions may vanish and may be unbounded. The equation is then degenerate, and weak solutions are generally nonunique. We introduce a notion of strong solution that e ...
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Journal articleJournal of Functional Analysis · December 15, 2024
For the one dimensional Burgers equation with a random and periodic forcing, it is well-known that there exists a family of invariant measures, each corresponding to a different average velocity. In this paper, we consider the coupled invariant measures an ...
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Journal articleJournal of Machine Learning Research · April 2024
Sum-of-norms clustering is a popular convexification of $K$-means clustering. We show that, if the dataset is made of a large number of independent random variables distributed according to the uniform measure on the union of two disjoint balls of unit rad ...
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Journal articleCommunications 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. ...
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Journal articleNonlinearity · 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 ...
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Journal articleAnnales 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 ...
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Journal articleSIAM 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 ...
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Journal articleAnnals 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 ...
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Journal articleJournal 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 ...
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Journal articleArchive 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 ...
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Journal articleCommunications 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 ...
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Journal articlePublications 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 ...
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Journal articleNonlinearity · 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 ...
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Journal articleElectronic 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 = −(−∆ ...
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Journal articleCommunications 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 ...
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Journal articleProbability 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 ...
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Journal articleAnnals 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 ...
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Journal articleAnnals 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 ...
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https://orcid.org/0000-0002-0754-5826