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Fei Pu

Publications and source records attributed to Fei Pu.

At least 19 recordsLinked to original sources
arXiv2026

Two-time spatial decorrelation for the flat KPZ fixed point

We establish quantitative two-time spatial decorrelation for the Kardar--Parisi--Zhang fixed point with flat initial data. For every $s,t>0$,there exist constants $C,c>0$ such that \[ \big|{\rm Cov}(\mathfrak{h}(t,x),\mathfrak{h}(s,0))\big| \le C\exp\{-c|x|^3\},\qquad |x|\ge1. \] Unlike the fixed-time covariance, which is governed directly by the Airy$_1$ process, the two-time covariance involves the nonlinear variational evolution of the entire earlier height profile. Our proof combines cubic-exponential mixing of the Airy$_1$ process with a uniform localization estimate for intermediate optimizers in the directed landscape. As a consequence, the centered spatial averages, normalized by $N^{1/2}$, converge in finite-dimensional distributions to a centered Gaussian process whose covariance is the space-integrated two-time correlation of the flat KPZ fixed point.

math.PR↗
arXiv2025

Uniform dimension theorems for parabolic SPDEs

Consider the following $p$-dimensional system of It\^o type stochastic PDEs, \begin{align*}\left[\begin{aligned} &\partial_t u(t\,,x) = \partial^2_x u(t\,,x) + b(u(t\,,x)) + \sigma(u(t\,,x)) \xi(t\,,x)\\ &\text{for $(t\,,x)\in(0\,,\infty)\times\mathbb{T}$, subject to $u(0) \equiv u_0$ on $\mathbb{T}$}, \end{aligned}\right.\end{align*} where $\mathbb{T}$ denotes a given one-dimensional torus, the initial data $u_0:\mathbb{T}\to\mathbb{R}^p$ is assumed to be fixed and non-random and in $C^{1/2}(\mathbb{T}\,;\mathbb{R}^p)$, and $\xi$ denotes a $p$-dimensional space-time white noise. Under certain regularity conditions on $b$ and $\sigma$, it is proved that, if $p \ge 4$, then \begin{equation*} \mathrm{P}\{\operatorname{dim_{_H}} u(\{t\}\times F) = 2\operatorname{dim_{_H}} F \ \text{$\forall$compact $F\subset\mathbb{T}$, $t>0$}\}=1. \end{equation*} If in addition the matrix $\sigma(v)$ does not depend on $v\in\mathbb{R}^p$, and is nonsingular, then the above equality holds for all $p\ge2$.

math.PR↗
arXiv2025

Spatial fluctuation for stochastic heat equation with H\"older coefficients

In this paper, we establish associativity, spatial ergodicity and a central limit theorem for certain nonnegative solutions to the stochastic heat equation $\partial_t u=\frac12\partial_x^2 u+ u^\gamma \xi$ with $\gamma\in (0, 1)$. When $\gamma=\frac12$, we derive a limit for the moment generating function of the spatial integral and provide a lower bound on the spatial growth of the solution.

math.PR↗
arXiv2025

Sharp upper bounds on hitting probabilities for the solution to the stochastic heat equation on the line

For Gaussian random fields with values in $\mathbb{R}^d$, sharp upper and lower bounds on the probability of hitting a fixed set have been available for many years. These apply in particular to the solutions of systems of linear SPDEs. For non-Gaussian random fields, the available bounds are less sharp. For nonlinear systems of stochastic heat equations, a sharp lower bound was obtained in a previous paper by two of the authors. Here, we obtain the corresponding sharp upper bound. The proof requires a bound on the joint probability density function of a two-dimensional random vector whose components are the solution to the {\em nonlinear} stochastic heat equation and the supremum over a small rectangle of the solution to the {\em linear} stochastic heat equation, in terms of the size of the rectangle. This bound makes use of a formula that expresses the density of a {\em locally nondegenerate} random vector as an iterated Skorohod integral. The main effort is to estimate, using Malliavin calculus, each of the terms that arise from this formula.

math.PR↗
arXiv2025

Spatial decorrelation of KPZ from narrow wedge

We study the spatial decorrelation of the solution to the KPZ equation with narrow wedge initial data. For fixed $t>0$, we determine the decay rate of the spatial covariance function, showing that ${\rm Cov}[h(t,x),h(t,0)]\sim \frac{t}{x}$ as $x\to\infty$. In addition, we prove that the finite-dimensional distributions of the properly rescaled spatial average of the height function converge to those of a Brownian motion.

math.PR↗
arXiv2025

Macroscopic Hausdorff dimension of the level sets of the Airy processes

We study the Macroscopic Hausdorff dimension of the upper and lower level sets of the Airy processes, following the general method developed in Khoshnevisan et al. \cite{KKX17}. For the Airy$_1$ process, the approach to macroscopic Hausdorff dimension of level sets hinges on some inequalities for its joint probabilities, while for the Airy$_2$ process, we make use of some quantitative estimates on the tail probabilities of its maximum and minimum over an interval.

math.PR↗
arXiv2023

Lower bound on spatial asymptotic of parabolic Anderson model with narrow wedge initial condition

Let $\{u(t\,,x): (t,x)\in (0, \infty)\times \mathbb{R}\}$ be the solution to parabolic Anderson model with narrow wedge initial condition. Using the association property of parabolic Anderson model, we establish a lower bound on spatial asymptotic of the solution: \begin{align*} \liminf_{R\to\infty}\frac{ \max_{|x|\leq R}\left(\log u(t\,,x) + \frac{x^2}{2t}\right)}{(\log R)^{2/3}} \geq \frac14\left(\frac{t}2\right)^{1/3}, \quad \text{a.s.} \end{align*}

math.PR↗
arXiv2023

Ergodicity, CLT and asymptotic maximum of the Airy$_1$ process

We first show that the Airy$_1$ process is associated using the association property of the solution to the stochastic heat equation and convergence of the KPZ equation to the KPZ fixed point. Then we apply Newman's inequality to establish the ergodicity and central limit theorem for the Airy$_1$ process. Combined with the asymptotic behavior of the tail probability, we derive a Poisson limit theorem for the Airy$_1$ process and give a precise estimate on the asymptotic behavior of the maximum of the Airy$_1$ process over an interval. Analogous results for the Airy$_2$ process are also presented.

math.PR↗
arXiv2021

Gaussian fluctuation for spatial average of super-Brownian motion

Let $\{u(t\,, x)\}_{(t, x)\in \mathbb{R}_+\times \mathbb{R}}$ be the density of one-dimensional super-Brownian motion starting from Lebesgue measure. Using the Laplace functional of super-Brownian motion, we prove that as $N\to \infty$, the normalized spatial integral $N^{-1/2}\int_0^{xN}[u(t\,, z)-1 ]\rm{d} z$ converges jointly in $(t, x)$ to Brownian sheet in distribution.

math.PR↗
arXiv2021

Gaussian fluctuation for Gaussian Wishart matrices of overall correlation

In this note, we study the Gaussian fluctuations for the Wishart matrices $d^{-1}\mathcal{X}_{n, d}\mathcal{X}^{T}_{n, d}$, where $\mathcal{X}_{n, d}$ is a $n\times d$ random matrix whose entries are jointly Gaussian and correlated with row and column covariance functions given by $r$ and $s$ respectively such that $r(0)=s(0)=1$. Under the assumptions $s\in \ell^{4/3}(\mathbb{Z})$ and $\|r\|_{\ell^1(\mathbb{Z})}< \sqrt{6}/2$, we establish the $\sqrt{n^3/d}$ convergence rate for the Wasserstein distance between a normalization of $d^{-1}\mathcal{X}_{n, d}\mathcal{X}^{T}_{n, d}$ and the corresponding Gaussian ensemble. This rate is the same as the optimal one computed in \cite{JL15,BG16,BDER16} for the total variation distance, in the particular case where the Gaussian entries of $\mathcal{X}_{n, d}$ are independent. Similarly, we obtain the $\sqrt{n^{2p-1}/d}$ convergence rate for the Wasserstein distance in the setting of random $p$-tensors of overall correlation. Our analysis is based on the Malliavin-Stein approach.

math.PR↗
arXiv2020

Gaussian fluctuation for spatial average of parabolic Anderson model with Neumann/Dirichlet/periodic boundary conditions

Consider the parabolic Anderson model $\partial_tu=\frac{1}{2}\partial_x^2u+u\, \eta$ on the interval $[0, L]$ with Neumann, Dirichlet or periodic boundary conditions, driven by space-time white noise $\eta$. Using Malliavin-Stein method, we establish the central limit theorem for the fluctuation of the spatial integral $\int_0^Lu(t\,, x)\, \mathrm{d} x$ as $L$ tends to infinity, where the limiting Gaussian distribution is independent of the choice of the boundary conditions and coincides with the Gaussian fluctuation for the spatial average of parabolic Anderson model on the whole space $\mathbb{R}$.

math.PR↗
arXiv2020

Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein's method

Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a $d$-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang's condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form $N^{-d} \int_{[0,N]^d} g(u(t\,,x))\, \mathrm{d} x$, as $N\rightarrow\infty$, where $g$ is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein's method for normal approximations. Our results include a central limit theorem for the {\it Hopf-Cole} solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.

math.PR↗
arXiv2020

Spatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension $d\geq 1$

Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$ which satisfies Dalang's condition. Let $\boldsymbol{p}_t(x):=(2\pi t)^{-d/2}\exp\{-\|x\|^2/(2t)\}$ denote the standard Gaussian heat kernel on $\mathbb{R}^d$. We prove that for all $t>0$, the process $U(t):=\{u(t\,, x)/\boldsymbol{p}_t(x): x\in \mathbb{R}^d\}$ is stationary using Feynman-Kac's formula, and is ergodic under the additional condition $\hat{f}\{0\}=0$, where $\hat{f}$ is the Fourier transform of $f$. Moreover, using Malliavin-Stein method, we investigate various central limit theorems for $U(t)$ based on the quantitative analysis of $f$. In particular, when $f$ is given by Riesz kernel, i.e., $f(\mathrm{d} x) = \|x\|^{-\beta}\mathrm{d} x$, we obtain a multiple phase transition for the CLT for $U(t)$ from $\beta\in(0\,,1)$ to $\beta=1$ to $\beta\in(1\,,d\wedge 2)$.

math.PR↗
arXiv2020

Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition

Let $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}}$ denote the solution to the parabolic Anderson model with initial condition $\delta_0$ and driven by space-time white noise on $\mathbb{R}_+\times\mathbb{R}$, and let $p_t(x):= (2\pi t)^{-1/2}\exp\{-x^2/(2t)\}$ denote the standard Gaussian heat kernel on the line. We use a non-trivial adaptation of the methods in our companion papers \cite{CKNP,CKNP_b} in order to prove that the random field $x\mapsto u(t\,,x)/p_t(x)$ is ergodic for every $t >0$. And we establish an associated quantitative central limit theorem following the approach based on the Malliavin-Stein method introduced in Huang, Nualart, and Viitasaari \cite{HNV2018}.

math.PR↗
arXiv2019

Central limit theorems for parabolic stochastic partial differential equations

Let $\{u(t\,,x)\}_{t\ge 0, x\in \mathbb{R}^d}$ denote the solution of a $d$-dimensional nonlinear stochastic heat equation that is driven by a Gaussian noise, white in time with a homogeneous spatial covariance that is a finite Borel measure $f$ and satisfies Dalang's condition. We prove two general functional central limit theorems for occupation fields of the form $N^{-d} \int_{\mathbb{R}^d} g(u(t\,,x)) \psi(x/N)\, \mathrm{d} x$ as $N\rightarrow \infty$, where $g$ runs over the class of Lipschitz functions on $\mathbb{R}^d$ and $\psi\in L^2(\mathbb{R}^d)$. The proof uses Poincar\'e-type inequalities, Malliavin calculus, compactness arguments, and Paul L\'evy's classical characterization of Brownian motion as the only mean zero, continuous L\'evy process. Our result generalizes central limit theorems of Huang et al \cite{HuangNualartViitasaari2018,HuangNualartViitasaariZheng2019} valid when $g(u)=u$ and $\psi = \mathbf{1}_{[0,1]^d}$.

math.PR↗
arXiv2019

Spatial ergodicity for SPDEs via Poincaré-type inequalities

Consider a parabolic stochastic PDE of the form $\partial_t u=\frac{1}{2}Δu + σ(u)η$, where $u=u(t\,,x)$ for $t\ge0$ and $x\in\mathbb{R}^d$, $σ:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous and non random, and $η$ is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation $f$. If, in addition, $u(0)\equiv1$, then we prove that, under a mild decay condition on $f$, the process $x\mapsto u(t\,,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergodicity of $u$ teaches us about the intermittent nature of the solution to such SPDEs \cite{BertiniCancrini1995,KhCBMS}. Our results provide rigorous justification of such discussions. Our methods hinge on novel facts from harmonic analysis and functions of positive type, as well as from Malliavin calculus and Poincaré inequalities. We further showcase the utility of these Poincaré inequalities by: (a) describing conditions that ensure that the random field $u(t)$ is mixing for every $t>0$; and by (b) giving a quick proof of a conjecture of Conus et al \cite{CJK12} about the "size" of the intermittency islands of $u$. The ergodicity and the mixing results of this paper are sharp, as they include the classical theory of Maruyama \cite{Maruyama} (see also Dym and McKean \cite{DymMcKean}) in the simple setting where the nonlinear term $σ$ is a constant function.

math.PR↗