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Alexander Dunlap

Publications and source records attributed to Alexander Dunlap.

At least 19 recordsLinked to original sources
arXiv2025

Invariant measures for the open KPZ equation: an analytic perspective

The ergodic theory of the open KPZ equation has seen significant progress in recent years, with explicit invariant measures described in a series of works by Corwin--Knizel, Barraquand--Le Doussal, and Bryc--Kuznetsov--Wang--Weso{\l}owski. In this paper, we provide a stochastic analytic proof of the formula for the invariant measures. Our approach starts from the Gaussian invariant measure for the case of homogeneous boundary conditions. We approximate the inhomogeneous problem by a homogeneous one with a singular boundary potential. Using tools including change of measure, time reversal for Markov processes, and It\^o's formula, we then reduce the problem to analyzing the KPZ nonlinearity in a thin boundary layer. Finally, using the theory of regularity structures, we establish a central limit theorem for the time-integrated nonlinearity near the boundary, which completes the proof of the invariance. Although it is known that different boundary parameters give rise to distinct physical regimes for the invariant measures, our method is robust and does not rely on any particular choice of boundary parameters.

math.PR↗
arXiv2025

A critical stochastic heat equation with long-range noise

We consider a semilinear stochastic heat equation in spatial dimension at least $3$, forced by a noise that is white in time with a covariance kernel that decays like $\lvert x\rvert^{-2}$ as $\lvert x\rvert\to\infty$. We show that in an appropriate diffusive scaling limit with a logarithmic attenuation of the noise, the pointwise statistics of the solution can be approximated by the solution to a forward-backward stochastic differential equation (FBSDE). The scaling and structure of the problem is similar to that of the two-dimensional stochastic heat equation forced by an approximation of space-time white noise considered by the first author and Gu (Ann. Probab., 2022). However the resulting FBSDE is different due to the long-range correlations of the noise.

math.PR↗
arXiv2025

McKean-Vlasov limits of scaling-critical reaction-diffusion equations with random initial data

We study a large class of scaling-critical reaction-diffusion equations in two spatial dimensions, where the initial data is white noise mollified at scale $\varepsilon^2$ and the reaction term is attenuated by a factor of $(\log\varepsilon^{-1})^{-1}$. We show that as $\varepsilon\to 0$, the solution converges to the solution of a McKean-Vlasov equation, which is Gaussian with standard deviation given by the solution to an ODE. Our result covers the case of the reaction term $f(u)=u^3$, and thus gives a new proof of the limiting behavior for the Allen-Cahn equation discovered in the recent work of Gabriel, Rosati, and Zygouras (Probab. Theory Related Fields 192: 1373-1446, 2025).

math.PR↗
arXiv2025

Invariant measures and shocks in the KPZ fixed point

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 $\theta > 0$, and are supported on functions $f$ satisfying $\lim_{|x| \to \infty} \frac{f(x)}{|x|} = 2\theta$. Each can be described as the sum of a Brownian motion and an independent Bessel-$3$ process with drift. We show that these measures appear as the $L \to \infty$ limit of the (conjectural) stationary measures for the conjectural open KPZ fixed point on $[0,L]$, after recentering by an appropriately defined shock location. Furthermore, we show that, with respect to the standard, deterministic recentering at $x = 0$, all extremal invariant measures for the KPZ fixed point are Brownian motions with drift. To do this, we first show that any extremal invariant measures must be supported on functions having fixed asymptotic slopes at $\pm \infty$. Using a one-force-one-solution principle from the work of Busani, Sepp\"al\"ainen, and the second author, this rules out all other invariant measures except those having left slope $-2\theta$ and right slope $+2\theta$ for some $\theta > 0$. To handle this case, we derive the limiting fluctuations of the shock for a special choice of initial condition. Additionally, we derive the limiting fluctuations of the shock for the case of the invariant measure from the perspective of a shock, and for the case of initial data $f(x) = 2\theta|x|$.

math.PR↗
arXiv2024

Simultaneous global inviscid Burgers flows with periodic Poisson forcing

We study the inviscid Burgers equation on the circle $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ forced by the derivative of a Poisson point process on $\mathbb{R}\times\mathbb{T}$. We construct global solutions with mean $\theta$ simultaneously for all $\theta\in\mathbb{R}$, and in addition construct their associated global shocks (which are unique except on a countable set of $\theta$). We then show that as $\theta$ changes, the solution only changes through the movement of the global shock, and give precise formulas for this movement. This is an analogue of previous results by the author and Yu Gu in the viscous case with white-in-time forcing, which related the derivative of the solution in $\theta$ to the density of a particle diffusing in the Burgers flow.

math.PR↗
arXiv2024

Viscous shock fluctuations in KPZ

We study ``V-shaped'' solutions to the KPZ equation, those having opposite asymptotic slopes $\theta$ and $-\theta$, with $\theta>0$, at positive and negative infinity, respectively. Answering a question of Janjigian, Rassoul-Agha, and Sepp\"al\"ainen, we show that the spatial increments of V-shaped solutions cannot be statistically stationary in time. This completes the classification of statistically time-stationary spatial increments for the KPZ equation by ruling out the last case left by those authors. To show that these V-shaped time-stationary measures do not exist, we study the location of the corresponding ``viscous shock,'' which, roughly speaking, is the location of the bottom of the V. We describe the limiting rescaled fluctuations, and in particular show that the fluctuations of the shock location are not tight, for both stationary and flat initial data. We also show that if the KPZ equation is started with V-shaped initial data, then the long-time limits of the time-averaged laws of the spatial increments of the solution are mixtures of the laws of the spatial increments of $x\mapsto B(x)+\theta x$ and $x\mapsto B(x)-\theta x$, where $B$ is a standard two-sided Brownian motion.

math.PR↗
arXiv2024

Edwards-Wilkinson fluctuations in subcritical 2D stochastic heat equations

We study 2D nonlinear stochastic heat equations under a logarithmically attenuated white-noise limit with subcritical coupling. We show that solutions asymptotically exhibit Edwards-Wilkinson fluctuations. This extends work of Ran Tao, which required a stricter condition on the coupling. Part of the limiting fluctuation is measurable with respect to the original noise and the remainder is independent. We also show that these statistics are universal in the sense that they are independent of the fine details of the model.

math.PR↗
arXiv2024

Additive-multiplicative stochastic heat equations, stationary solutions, and Cauchy statistics

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 measures exist for all values of the parameters. We also show that if the multiplicative noise is sufficiently strong, the invariant measure has Cauchy-distributed marginals. Using the same techniques, we prove a similar result on Cauchy-distributed marginals for a logarithmically attenuated version of the problem in two spatial dimensions. The proofs rely on stochastic analysis and elementary potential theory.

math.PR↗
arXiv2023

Branching Brownian motion with generation-dependent diffusivity and nonlocal partial differential equations

We study a voting model on a branching Brownian motion process on $\mathbb{R}$ in which the diffusivity of each child particle is increased from that of the parent by a factor of $\gamma>1$. The probability distribution of the overall vote is given in terms of the solution to a nonlocal nonlinear PDE. We exhibit conditions on the nonlinearity such that the long-time behavior of the distribution undergoes a phase transition in $\gamma$. If $\gamma$ is sufficiently large, then the long-time distribution converges to uniform. If $\gamma$ is close enough to $1$, then the long-time distribution depends in a nontrivial way on the location of the initial particle. The limiting dependence is given by a steady-state solution to the nonlocal PDE. Our study gives a probabilistic interpretation of a class of semilinear nonlocal PDEs. Interestingly, while the PDE are nonlocal, the underlying random process does not require any non-local interactions.

math.AP↗
arXiv2023

Jointly stationary solutions of periodic Burgers flow

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 and study how they change as the velocity parameter varies. We show that the derivative of the invariant measure with respect to the velocity parameter exists, and it can be interpreted as the steady state of a diffusion advected by the Burgers flow

math.PR↗
arXiv2023

Renormalization flow for the 2D nonlinear stochastic heat equation: pointwise statistics and universality

We consider a two-dimensional stochastic heat equation with noise correlated at scale $\rho \ll 1$ and of strength $|\log\rho|^{-1/2}\sigma(v)$ depending nonlinearly on the solution $v$. Under certain conditions, the first author and Gu have shown that the one-point statistics of $v$ converge in law as $\rho\to 0$ to the terminal value of an associated forward-backward SDE. Here, we show that the 2D stochastic heat equation is stable under renormalization with a new effective nonlinearity tied to the decoupling function of the forward-backward SDE. This allows us to extend the pointwise results to a much broader class of nonlinearities. We also show that these limiting pointwise statistics are insensitive to the fine details of the noise, and thus universal.

math.PR↗
arXiv2023

Uniqueness and root-Lipschitz regularity for a degenerate heat equation

We consider nonnegative solutions of the quasilinear heat equation $\partial_t u = \tfrac{1}{2} u \partial_x^2 u$ 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 ensures uniqueness. For suitable initial data, we prove a lower bound on the time for which a strong solution $u$ exists and $\sqrt{u}$ remains globally Lipschitz in space. In a companion paper, we show that this condition is important in the study of two-dimensional nonlinear stochastic heat equations.

math.AP↗
arXiv2022

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

In this paper, we study the localization length of the $1+1$ 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 distribution in the thermodynamic limit, and derive an explicit density formula of the limiting distribution. As a consequence, we prove the $\tfrac32$-power law decay of the density, confirming the physics prediction of Hwa-Fisher \cite{fisher}. Our proof uses the recent result of Das-Zhu \cite{daszhu}.

math.PR↗
arXiv2022

Invariant measures for stochastic conservation laws on the line

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 spatially-homogeneous invariant measure for each mean in a non-explicit unbounded set. This generalizes previous work on the stochastic Burgers equation.

math.PR↗
arXiv2021

Local versions of sum-of-norms clustering

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 stochastic ball model. More precisely, we prove a quantitative bound on the error incurred in the clustering of disjoint connected sets. Our bound is expressed in terms of the number of datapoints and the localization length of the functional.

cs.LG↗
arXiv2021

A quenched local limit theorem for stochastic flows

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 particle after a long time can be approximated pointwise by the product of a deterministic Gaussian density and a spacetime-stationary random field $U$. If the velocity field is additionally assumed to be incompressible, then $U\equiv 1$ almost surely and we obtain a local central limit theorem.

math.PR↗
arXiv2021

Sum-of-norms clustering does not separate nearby balls

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 radius, and if the balls are sufficiently close to one another, then sum-of-norms clustering will typically fail to recover the decomposition of the dataset into two clusters. As the dimension tends to infinity, this happens even when the distance between the centers of the two balls is taken to be as large as $2\sqrt{2}$. In order to show this, we introduce and analyze a continuous version of sum-of-norms clustering, where the dataset is replaced by a general measure. In particular, we state and prove a local-global characterization of the clustering that seems to be new even in the case of discrete datapoints.

cs.LG↗