Sourav Sarkar, Bálint Virág
We show that the law of the KPZ fixed point starting from arbitrary initial condition is absolutely continuous with respect to the law of Brownian motion $B$ on every compact interval. In particular, the Airy$_1$ process is absolutely continuous with respect to $B$ on any compact interval.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We consider the system of one-sided reflected Brownian motions which is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling invariant Markov process defined in [MQR17] and believed to govern the long time large scale fluctuations for all models in the KPZ universality class. Brownian last passage percolation was shown recently in [DOV18] to converge to the…
Let H(t,x) be the Hopf-Cole solution at time t of the Kardar-Parisi-Zhang (KPZ) equation starting with narrow wedge initial condition, i.e. the logarithm of the solution of the multiplicative stochastic heat equation starting from a Dirac delta. Also let H^{eq}(t,x) be the solution at time t of the KPZ equation with the same noise, but with initial condition given by a standard two-sided Brownian motion, so that H^{eq}(t,x)-H^{eq}(0,x) is itself distributed as a standard two-sided Brownian motion. We provide a simple proof of the following fact: for fixed t, H(t,x)-(H(t,x)-H^{eq}(t,0)) is loca…
The KPZ Upper-Tail Field as a KPZ Fixed Point with Brownian Initial Data
本文通过有向景观变分和局部布朗比较，将KPZ上尾场正时间扇区识别为从双边布朗V-型轮廓出发的KPZ不动点，负时间扇区用指数倾斜表示，并恢复其大尺度极限。
The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process that is conjectured to be at the core of the KPZ universality class. In this article we study two aspects the KPZ fixed point that share the same Brownian limiting behaviour: the local space regularity and the long time evolution. Most of the results that we will present here were obtained by either applying explicit formulas for the transition probabilities or applying the coupling method to discrete approximations. Instead we will use the variational description of the KPZ fixed point, allowing us the possibility of running the pr…
This paper seeks a quantitative comparison between the curves in the KPZ line ensemble [CH16] and a standard Brownian bridge under the $t^{1/3}$ vertical and $t^{2/3}$ horizontal scaling. The estimate we obtained is parallel to the one established in [Ham1], where the Airy line ensemble was studied. Our main tool is the soft Brownian Gibbs property enjoyed by the KPZ line ensemble. In view of the Gibbs property, the KPZ line ensemble differs from the Airy line ensemble mainly due to the intersecting nature of its curves, which results in the main technical difficulty in this paper. We develop …
We consider the asymptotic behavior of the KPZ fixed point $\{\mathsf H(x,t)\}_{x\in\mathbb R, t>0}$ conditioned on $\mathsf H(0,T)=L$ as $L$ goes to infinity. The main result is a conditional limit theorem for the fluctuations of $\mathsf H$ in the region near the line segment connecting the origin $(0,0)$ and $(0,T)$ for both step and flat initial conditions. The limit random field can be represented as a functional of two independent Brownian bridges, and in addition the limit random field depends also on the initial law of the KPZ fixed point. In particular for temporal fluctuations, the l…
Quantitative Brownian regularity of the KPZ fixed point with arbitrary initial data
研究KPZ固定点在任意初始数据下的空间增量与速率二布朗运动的强定量比较，证明其在紧集上的一致性，并应用得到空间增量的一侧大偏差不等式及中心KPZ固定点的韦纳密度熵有限性。
An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process (TASEP) with arbitrary right-finite initial condition. The method is by solving the biorthogonal ensemble/non-intersecting path representation found by [Sas05; BFPS07]. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data. In the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula, in terms of analogous kernels based on Browni…
In this paper we consider the limiting distribution of KPZ growth models with random but not stationary initial conditions introduced in [Chhita-Ferrari-Spohn 2018]. The one-point distribution of the limit is given in terms of a variational problem. By directly studying it, we deduce the right tail asymptotic of the distribution function. This gives a rigorous proof and extends the results obtained in [Meerson-Schmidt 2017].
The directed landscape constructed in (Dauvergne-Ortmann-Virag '18) produces a directed, planar, random geometry, and is believed to be the universal scaling limit of two-dimensional first and last passage percolation models in the Kardar-Parisi-Zhang (KPZ) universality class. Geodesics in this random geometry form an important class of random continuous curves exhibiting fluctuation theory quite different from that of Brownian motion. In this vein, counterpart to Brownian local time, BLT (a self-similar measure supported on the set of zeros of Brownian motion), a local time for geodesics, GLT…
KPZ equation limit of sticky Brownian motion
本文研究了连续随机环境下的粒子运动，证明了在 moderate deviation regime 下其 quenched density 弱收敛至随机热方程，揭示了普适性，并推导了 $N$ 点粘性布朗运动最大值的涨落规律。
In this paper we show that an $H$-Brownian Gibbsian line ensemble is completely characterized by the finite-dimensional marginals of its lowest indexed curve for a large class of interaction Hamiltonians $H$. A particular consequence of our result is that the KPZ line ensemble is the unique line ensemble that satisfies the $H_1$-Brownian Gibbs property with $H_1(x) = e^x$ and whose lowest indexed curve is equal to the Hopf-Cole solution to the narrow wedge KPZ equation.
The KPZ fixed point and Brownian motion share the same null sets
研究显示KPZ固定点增量与扩散参数为2的布朗运动在紧集上互绝对连续，同时加性布朗运动与中心Airy sheet在紧集上绝对连续但非互绝对连续。应用包括证明KPZ固定点存在远离参考点的记录时间，刻画其图形击中概率的正性条件，以及计算这些随机交集的Hausdorff维度本质上界。
Brownian bridge limit of path measures in the upper tail of KPZ models
该研究证明了KPZ模型上尾中路径测度的布朗桥极限，通过几何和概率方法，揭示了聚合物测度的结构及淬火局部化指数。
The early time regime of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension, starting from a Brownian initial condition with a drift $w$, is studied using the exact Fredholm determinant representation. For large drift we recover the exact results for the droplet initial condition, whereas a vanishingly small drift describes the stationary KPZ case, recently studied by weak noise theory (WNT). We show that for short time $t$, the probability distribution $P(H,t)$ of the height $H$ at a given point takes the large deviation form $P(H,t) \sim \exp{\left(-Φ(H)/\sqrt{t} \right)}$. We obtain …
The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $γ$-Liouville quantum gravity (LQG) surface, $γ\in (0,2)$, decorated with a space-filling form of Schramm's SLE$_κ$, $κ= 16/γ^2 \in (4,\infty)$, $η$ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $η$ which can be defined as a function of $η$ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $η^{-1}(…
The half-space KPZ line ensemble and its scaling limit
本文构造半空间 KPZ 线系综，证明其在临界与超临界情形下经 1:2:3 KPZ 标度后的紧性，并识别相应的 Brown Gibbs 极限结构。
The geometry of coalescing random walks, the Brownian web distance and KPZ universality
研究共聚随机游走的几何结构，定义布朗网距离并探讨其与KPZ普遍性的关系，发现其尺度指数与KPZ不同但仍受Airy过程主导。
Bulk heights of the KPZ line ensemble
本文研究了KPZ线集合的bulk高度，通过积分分部公式推导出其渐进行为，揭示了H-Brownian Gibbs性质下的集中估计。
For each $t\geq 1$ we construct an $\mathbf{N}$-indexed ensemble of random continuous curves with three properties: 1. The lowest indexed curve is distributed as the time $t$ Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ) stochastic PDE with narrow wedge initial data; 2. The entire ensemble satisfies a resampling invariance which we call the $\mathbf{H}$-Brownian Gibbs property (with $\mathbf{H}(x)=e^{x}$); 3. Increments of the lowest indexed curve, when centered by $-t/24$ and scaled down vertically by $t^{1/3}$ and horizontally by $t^{2/3}$, remain uniformly absolutely continuous (i.e. …
Many models of one-dimensional local random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. For such a model, the interface profile at advanced time may be viewed in scaled coordinates specified via characteristic KPZ scaling exponents of one-third and two-thirds. When the long time limit of this scaled interface is taken, it is expected -- and proved for a few integrable models -- that, up to a parabolic shift, the Airy$_2$ process $\mathcal{A}:\mathbb{R} \to \mathbb{R}$ is obtained. This process may be embedded via the Robinson-Schensted-Knuth correspondence a…
We consider large-scale point fields which naturally appear in the context of the Kardar-Parisi-Zhang (KPZ) phenomenon. Such point fields are geometrical objects formed by points of mass concentration, and by shocks separating the sources of these points. We introduce similarly defined point fields for processes of coalescing fractional Brownian motions (cfBm). The case of the Hurst index 2/3 is of particular interest for us since, in this case, the power law of the density decay is the same as that in the KPZ phenomenon. In this paper, we present strong numerical evidence that statistical pro…
Bi-infinite systems of singularly interacting Brownian particles and the KPZ equation
研究实线上具奇异非对称相互作用的布朗粒子双无穷系统，通过碰撞链局部化等方法，建立适定性理论与确定平稳分布，证明在弱非对称下粒子数函数平衡涨落收敛到乘性随机热方程，将其置于KPZ普适类。
Invariant measures for the open KPZ equation: the Gaussian case
本文基于Quastel和Gu的方法，研究开放KPZ方程的不变测度，证明恒定漂移的布朗运动在考虑高度移位时的不变性。
Multipoint distributions of the KPZ fixed point with compactly supported initial conditions
本文通过对 TASEP 多点分布取 1:2:3 KPZ 标度极限，并利用初始条件核的概率表示，首次给出紧支撑一般初始条件下 KPZ 不动点的时空联合分布公式。
We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data H(0,X)=B(X), for B(X) a two-sided standard Brownian motion) and show that as time T goes to infinity, the fluctuations of the height function H(T,X) grow like T^{1/3} and converge to those previously encountered in the study of the stationary totally asymmetric simple exclusion process, polynuclear growth model and last passage percolation. The starting point for this work is our derivation of a Fredholm determinant formula for Macdonald processes which degenerates to a corresponding formula fo…
Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
本文研究带 m 个漫游者的 Airy 过程上确界分布，证明其为 Tracy–Widom GOE 分布的 2m 参数形变，并给出 Fredholm 行列式、Painlevé II、KPZ 不动点、PDE 与 KdV 等多种刻画。
An invariance principle for the 1D KPZ equation
本文证明在适当缩放下，一类满足等变、对称且光滑的离散随机表面模型收敛于1D KPZ方程的Cole-Hopf解，建立类似Donsker定理的不变原理。
We consider the KPZ fixed point starting from a general class of initial data. In this article, we study the growth of the large peaks of the KPZ fixed point at a spatial point $0$ when time $t$ goes to $\infty$ and when $t$ approaches $1$. We prove that for a very broad class of initial data, as $t\to \infty$, the limsup of the KPZ fixed point height function when scaled by $t^{1/3}(\log\log t)^{2/3}$ almost surely equals a constant. The value of the constant is $(3/4)^{2/3}$ or $(3/2)^{2/3}$ depending on the initial data being non-random or Brownian respectively. Furthermore, we show that th…
In this article we consider the KPZ fixed point starting from a two-sided Brownian motion with an arbitrary diffusion coefficient. We apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point (covariance) function of the spatial derivative process and the location of the maximum of an Airy process plus Brownian motion minus a parabola. Integration by parts also allows us to deduce the density of this location in terms of the second derivative of the variance of the KPZ fixed point. In the stationary regime, we find the same density related…
We consider the Cole-Hopf solution of the (1+1)-dimensional KPZ equation started from the narrow wedge initial condition. In this article, we ask how the peaks and valleys of the KPZ height function (centered by time/24) at any spatial point grow as time increases. Our first main result is about the law of iterated logarithms for the KPZ equation. As time variable $t$ goes to $\infty$, we show that the limsup of the KPZ height function with the scaling by $t^{1/3}(\log\log t)^{2/3}$ is almost surely equal to $(\frac{3}{4\sqrt{2}})^{2/3}$ whereas the liminf of the height function with the scali…
The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process, recently introduced by Matetski, Quastel, Remenik (arXiv:1701.00018), that describes the limit fluctuations of the height function associated to the totally asymmetric simple exclusion process (TASEP), and it is conjectured to be at the centre of the KPZ universality class. Our main result is that the KPZ incremental process converges weakly to its invariant measure, given by a two-sided Brownian motion with zero drift and diffusion coefficient 2. The heart of the proof is the coupling method that allows us to compare the TASEP hei…
The stationary measures of the Kardar-Parisi-Zhang equation on an interval have been computed recently. We present a rather direct derivation of this result by taking the weak asymmetry limit of the matrix product ansatz for the asymmetric simple exclusion process. We rely on the matrix product ansatz representation of Enaud and Derrida, which allows to express the steady-state in terms of re-weighted simple random walks. In the continuum limit, its measure becomes a path integral (or re-weighted Brownian motion) of the form encountered in Liouville quantum mechanics, recovering the recent for…
Characterization of the directed landscape from the KPZ fixed point
本文证明了 directed landscape 是满足独立增量、单调性和平移交换性的 KPZ fixed point 的唯一耦合，统一了 KPZ universality class 的两个核心对象，并提供通用框架证明了多种模型向 directed landscape 的收敛。
On the two-point function of the one-dimensional KPZ equation
本文应用Malliavin微积分推导出一维KPZ方程斜率的两点函数，用随机热方程的聚合物端点退火分布表示，并证明该分布由KPZ方程解的方差导数给出。
In 2002, Johansson conjectured that the maximum of the Airy$_2$ process minus the parabola $x^2$ is almost surely achieved at a unique location. This result was proved a decade later by Corwin and Hammond; Moreno Flores, Quastel and Remenik; and Pimentel. Up to scaling, the Airy$_2$ process minus the parabola $x^2$ arises as the fixed time spatial marginal of the KPZ fixed point when started from narrow wedge initial data. We extend this maximizer uniqueness result to the fixed time spatial marginal of the KPZ fixed point when begun from any element of a very broad class of initial data. None …
On the spatio-temporal increments of nonlinear parabolic SPDEs and the open KPZ equation
研究有界区间上非线性抛物型SPDE解的时空增量，确定了精确的局部和一致模连续性，并推广到开放KPZ方程。
We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process space-time stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finite-dimensional distributions of this process. Our method differs from the one …
We consider the stochastic continuity equation perturbed by a fractional Brownian motion and the drift is allowed to be discontinuous. We show that for almost all paths of the fractional Brownian motion there exists a solution to the equation.
Spatial decorrelation of KPZ from narrow wedge
本文研究KPZ方程从窄楔初值出发的空间去相关，确定了协方差衰减速率并证明重标度空间平均收敛于布朗运动。
This article is concerned with modulus of continuity of Brownian local times. Specifically, we focus on 3 closely related problems: (a) Limit theorem for a Brownian modulus of continuity involving Riesz potentials, where the limit law is an intricate Gaussian mixture. (b) Central limit theorems for the projections of L2 modulus of continuity for a 1-dimensional Brownian motion. (c) Extension of the second result to a 2-dimensional Brownian motion. Our proofs rely on a combination of stochastic calculus and Malliavin calculus tools, plus a thorough analysis of singular integrals.
We show that the local time of one-dimensional super-Brownian motion is locally $γ$-Hölder continuous near the boundary if $0<γ<3$ and fails to be locally $γ$-Hölder continuous if $γ>3$.
The purpose of this note is to prove a central limit theorem for the $L^3$-modulus of continuity of the Brownian local time using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^3$-modulus of the Brownian local time.
In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, one endpoint of such polymers is fixed, say at $(0,0) \in \mathbb{R}^2$, and the other is varied horizontall…
Markov limits of steady states of the KPZ equation on an interval
本文证明了 KPZ 方程平稳测度两种概率描述的等价性，利用 Markov 描述严格验证了相关结果，并分析了有限区间上 KPZ 方程稳态在大尺度下的 Markov 极限行为。
We present an exact solution for the height distribution of the KPZ equation at any time $t$ in a half space with flat initial condition. This is equivalent to obtaining the free energy distribution of a polymer of length $t$ pinned at a wall at a single point. In the large $t$ limit a binding transition takes place upon increasing the attractiveness of the wall. Around the critical point we find the same statistics as in the Baik-Ben--Arous-Péché transition for outlier eigenvalues in random matrix theory. In the bound phase, we obtain the exact measure for the endpoint and the midpoint of the…
A note on the continuity in the Hurst index of the solution of rough differential equations driven by a fractional Brownian motion
本文在粗糙路径框架下，证明了当Hurst参数H在(1/3, 1/2]区间内时，由分数布朗运动驱动的随机微分方程的解关于该参数具有连续性。
An upper tail field of the KPZ fixed point
本文研究了具有narrow-wedge初始条件的KPZ不动点，通过在高点邻域适当放大，构造出定义于全2维空间的上尾场，并证明其在正负时间区间分别表现为KPZ不动点与Brown型场。
We give a result of stability in law of the local time of the fractional Brownian motion with respect to small perturbations of the Hurst parameter. Concretely, we prove that the law (in the space of continuous functions) of the local time of the fractional Brownian motion with Hurst parameter $H$ converges weakly to that of the local time of $B^{H_0}$, when $H$ tends to $H_0$.
We endow the $\sqrt{8/3}$-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere ${\mathbf S}^2$. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric $d_{\mathcal Q}$ on a countable dense subset of ${\mathbf S}^2$. Here we show that $d_{\mathcal Q}$ a.s. extends uniquely and continuously to a metric $\bar{d}_{\mathcal Q}$ on all of ${\mathbf S}^2$. Letting $d$ …
Invariant measures and shocks in the KPZ fixed point
本文从KPZ固定点激波视角构造了一族不变测度，证明其在激波重新定位后作为L趋于无穷时的极限，并展示所有极值不变测度均为具有固定渐近斜率的布朗运动。
We collect some applications of the variational formula established by Schröder (1988) and Rueß(2013) for the quenched Lyapunov exponent of Brownian motion in stationary and ergodic nonnegative potential. We show for example that the Lyapunov exponent for nondeterministic potential is strictly lower than the Lyapunov exponent for the averaged potential. The behaviour of the Lyapunov exponent under independent perturbations of the underlying potential is examined. And with the help of counterexamples we are able to give a detailed picture of the continuity properties of the Lyapunov exponent.
Edgeworth-type expansion for the one-point distribution of the KPZ fixed point with a large height at a prior location
本文研究KPZ不动点在先前位置大高度条件下的单点分布，证明其收敛于GUE Tracy-Widom分布并给出Edgeworth型低阶展开，与后期大高度情形的高斯型极限共同构成完整相图。
We prove absolute continuity of Gaussian measures associated to complex Brownian bridges under certain gauge transformations. As an application we prove that the invariant measure for the periodic derivative nonlinear Schrödinger equation obtained by Nahmod, Oh, Rey-Bellet and Staffilani in [20], and with respect to which they proved almost surely global well-posedness, coincides with the weighted Wiener measure constructed by Thomann and Tzvetkov [24]. Thus, in particular we prove the invariance of the measure constructed in [24].
Uniform in time modulus of continuity of Brownian motion
本文研究标准布朗运动连续模的时间一致控制，通过控制特定随机变量的指数矩，推导出扩散过程的稳定性不等式，并应用于两个简单框架。
Brownian motions in the infinite-dimensional group of all unitary operators are studied under strong continuity assumption rather than norm continuity. Every such motion can be described in terms of a countable collection of independent one-dimensional Brownian motions. The proof involves continuous tensor products and continuous quantum measurements. A by-product: a Brownian motion in a separable F-space (not locally convex) is a Gaussian process.
Temporal increments of the KPZ equation with general initial data
本文研究具有一般初始数据的(1+1)维KPZ方程的时间过程，证明其时间增量可由Hurst参数为1/4的分数布朗运动近似，并由此推导出变差、重对数律和Hausdorff维数等样本路径性质。
Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion and \[ α_{t}:=\int_{-\infty}^{\infty} (L^{x}_{t})^{2} dx . \] Let $η=N(0,1)$ be independent of $α_{t}$. For each fixed $t$ \[ {\int_{-\infty}^{\infty} (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}} \stackrel{\mathcal{L}}{\to}({64 \over 3})^{1/2}\sqrt{α_{t}} η, \] as $h\rar 0$. Equivalently \[ {\int_{-\infty}^{\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- 4t\over t^{3/4}} \stackrel{\mathcal{L}}{\to}({64 \over 3} )^{1/2}\sqrt{α_{1}} η, \] as $t\rar\infty$.
Fractional Brownian Motion: Local Modulus of Continuity with Refined Almost Sure Upper Bound and First Exit Time from One-sided Barrier
本文基于最优速率小波级数表示，推导出分数布朗运动局部连续模的改进几乎必然上界及其 $p$ 阶矩上界，填补了迭代对数律的空白，并据此给出了从正值边界首次退出时间的改进上尾概率渐近估计。
We study the long-time regime of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimensions for the Brownian and droplet initial conditions and present a simple derivation of the tail of the large deviations of the height on the negative side $λH<0$. We show that for both initial conditions, the cumulative distribution functions take a large deviations form, with a tail for $- \tilde s \gg 1$ given by $-\log \mathbb{P}\left(\frac{H}{t}<\tilde{s}\right)=t^2 \frac{4 }{15 π} (-\tilde{s})^{5/2} $. This exact expression was already observed at small time for both initial conditions suggesting that …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。