arXiv subjects

Cheuk Yin Lee

Publications and source records attributed to Cheuk Yin Lee.

At least 19 recordsLinked to original sources

arXiv2026

Sharp regularity and small ball probabilities for the stochastic heat equation on bounded domains

We consider the stochastic heat equation $\partial_t u(t,x) = Δu(t,x) + \dot{W}_α(t,x)$ on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where $\dot{W}_α$ is a Gaussian noise that is white in time and whose spatial covariance is the kernel of $(-Δ)^{-α}$ with $α>0$. We prove that a unique pointwise defined mild solution exists if and only if $α>d/2-1$. In this case, if in addition the domain is $C^2$, we also establish spatial and temporal Holder regularity of the solution. When $d/2-1<α<d/2$, we show that the Holder exponents are optimal and obtain exact local and uniform moduli of continuity, a Chung-type law of the iterated logarithm, and sharp small ball probability estimates for the solution.

math.PR↗

arXiv2026

Fourier dimension of the graph of fractional Brownian motion with $H \ge 1/2$

We prove that the Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than $1/2$ is almost surely 1. This extends the result of Fraser and Sahlsten (2018) for the Brownian motion and confirms part of the conjecture of Fraser, Orponen and Sahlsten (2014). We introduce a combinatorial integration by parts formula to compute the moments of the Fourier transform of the graph measure. The proof of our main result is based on this integration by parts formula together with Faà di Bruno's formula and strong local nondeterminism of fractional Brownian motion. We also show that the graph of a symmetric $α$-stable process has Fourier dimension 1 almost surely when $α\in [1,2]$ and is a Salem set when $α= 1$.

math.PR↗

arXiv2026

Points of slow growth for parabolic SPDEs

Consider the stochastic PDE, $\partial_tu = \partial^2_x u + σ(u) \dot{W}$ on $\mathbb{R}_+\times\mathbb{R}$, subject to $u(0)\equiv1$, where $\dot{W}$ denotes space-time white noise on $\mathbb{R}_+\times\mathbb{R}$ and $σ:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous. It is known that $u(t\,,x)-1$ has approximately a Gaussian distribution for every $x$ when $t\approx0$. Here we prove that there exist random points $x\in\mathbb{R}$ where the fluctuations of the solution near times zero are almost surely of sharp order $t^{1/4}$. Our work bears some loose resemblance to the study of the slow points of Brownian motion increments, though significant challenges arise due to the infinite-dimensional nature of the present problem.

math.PR↗

arXiv2026

On the slow points of fractional Brownian motion

Esser and Loosveldt have recently resolved a long-standing open problem in the folklore by proving that fractional Brownian motion (fBm) has slow points in the sense of Kahane, following a rich theory of slow points developed for Brownian motion and other, related, self-similar Markov processes. We presently introduce another method for the study of slow points in order to compute the Hausdorff dimension of fBm slow points. Our method follows recent ideas on the points of slow growth for SPDEs but also requires a number of new localization ideas that are likely to have other applications.

math.PR↗

arXiv2026

Sharp moduli of continuity for Gaussian fields and stochastic PDEs via correlation bounds

Exact uniform and local moduli of continuity for anisotropic Gaussian random fields are established under a general framework based on correlation bounds for pairwise increments. This framework does not necessarily require strong local nondeterminism (SLND), stationarity of increments, or spectral-type representations. As an application, we solve an open problem about sharp moduli of continuity for a class of linear stochastic PDEs in a particular case where the solution is $C^{1-}$ in space and SLND is not available. In this case, we establish decorrelation of the spatial increments via new localization estimates, which may be of independent interest. We also briefly discuss an example about Gaussian Volterra processes.

math.PR↗

arXiv2026

On the spatio-temporal increments of nonlinear parabolic SPDEs and the open KPZ equation

We study spatio-temporal increments of the solutions to nonlinear parabolic SPDEs on a bounded interval with Dirichlet, Neumann, or Robin boundary conditions. We identify the exact local and uniform spatio-temporal moduli of continuity for the sample functions of the solutions. These moduli of continuity results imply the existence of random points in space-time at which spatio-temporal oscillations are exceptionally large. We also establish small-ball probability estimates and Chung-type laws of the iterated logarithm for spatio-temporal increments. Our method yields extension of some of these results to the open KPZ equation on the unit interval with inhomogeneous Neumann boundary conditions. Our key ingredients include new strong local non-determinism results for linear stochastic heat equation under various types of boundary conditions, and detailed estimates for the errors in linearization of spatio-temporal increments of the solution to the nonlinear equation.

math.PR↗

arXiv2026

Propagation of Singularities for the Damped Stochastic Klein-Gordon Equation

For the $1+1$ dimensional damped stochastic Klein-Gordon equation, we show that random singularities associated with the law of the iterated logarithm exist and propogate in the same way as the stochastic wave equation. This provides evidence for possible connections to microlocal analysis, ie. the exact regularity and singularities described in this paper should admit wavefront set type descriptions whose propagation is determined by the highest order terms of the linear operator. Despite the results being exactly the same as those of the wave equation, our proofs are significantly different than the proofs for the wave equation. Miraculously, proving our results for the critically damped equation implies them for the general equation, which significantly simplifies the problem. Even after this simplification, many important parts of the proof are significantly different than (and we think are more intuitive from the PDE viewpoint compared to) existing proofs for the wave equation.

math.PR↗

arXiv2026

Growth rate of liquidity provider's wealth in G3Ms

We study how trading fees and continuous-time arbitrage affect the profitability of liquidity providers (LPs) in Geometric Mean Market Makers (G3Ms). We use stochastic reflected diffusion processes to analyze the dynamics of a G3M model under the arbitrage-driven market. Our research focuses on calculating LP wealth and extends the findings of Tassy and White related to the constant product market maker (Uniswap v2) to a wider range of G3Ms, including Balancer. This allows us to calculate the long-term expected logarithmic growth of LP wealth, offering new insights into the complex dynamics of AMMs and their implications for LPs in decentralized finance.

q-fin.MF↗

arXiv2026

Polarity of points for Gaussian random fields in critical dimension

We study the property of hitting points for a class of $\mathbb{R}^d$-valued continuous Gaussian random fields on $\mathbb{R}^N$ with stationary increments, i.i.d. coordinates, and a regularly varying variance function $σ$ of index $0<H<1$. We first prove that if \[ \lim_{r\to 0^+} \frac{r^N}{σ^d\left(r\left( \log\log\frac{1}{r}\right)^{-1/N}\right)} = \infty, \] then every fixed point is polar (i.e., not hit almost surely). In general, this criterion may not be optimal in the critical dimension $d=N/H$. To aim for an optimal condition, we consider the specific case $σ(r) = r^H (\log(1/r))^γ$ and prove that, in the critical dimension $d=N/H$, points are polar if and only if $γ\le 1/d$, or equivalently in this specific case, \[ \int_{0^+} \frac{r^{N-1}}{σ^d(r)} dr = \infty. \] This integral condition is also necessary for points to be polar under general assumptions. Our main contribution lies in the proof of sufficiency of this condition in the specific case, where we extend a covering argument of Talagrand (1998) based on sojourn time estimates to obtain Hausdorff measure bounds and solve polarity of points in the critical dimension.

math.PR↗

arXiv2026

Strong local nondeterminism for stochastic time-fractional slow and fast diffusion equations

We study a class of stochastic time-fractional equations on $\mathbb{R}^d$ driven by a centered Gaussian noise, involving a Caputo time derivative of order $β>0$, a fractional (power) Laplacian of order $α>0$, and a Riemann-Liouville time integral of order $γ\ge0$ acting on the noise. The noise is fractional in time (index $H$) and Riesz-type in space (index $\ell$). We derive sharp Dalang-type necessary and sufficient conditions for the existence of a random field solution across almost full parameter range $(α,β,γ;H,\ell)$. Under the Dalang-type conditions, we prove sharp variance bounds for temporal and spatial increments, as well as strong local nondeterminism in time in several regimes (two-sided version for $β=1$ and for parts of the case $β=2$; one-sided version for $0<β<2$) and strong local nondeterminism in space for the whole range of parameters. As applications, we derive exact uniform and local moduli of continuity, Chung-type laws of the iterated logarithm, and quantitative bounds on small ball probabilities. Along the way, we obtain sharp asymptotics for the fundamental solution kernels at $0$ and $\infty$, which may be of independent interest.

math.PR↗

arXiv2025

Uniform dimension theorems for parabolic SPDEs

Consider the following $p$-dimensional system of Itô type stochastic PDEs, \begin{align*}\left[\begin{aligned} &\partial_t u(t\,,x) = \partial^2_x u(t\,,x) + b(u(t\,,x)) + σ(u(t\,,x)) ξ(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 $ξ$ denotes a $p$-dimensional space-time white noise. Under certain regularity conditions on $b$ and $σ$, 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 $σ(v)$ does not depend on $v\in\mathbb{R}^p$, and is nonsingular, then the above equality holds for all $p\ge2$.

math.PR↗

arXiv2025

On the passage times of self-similar Gaussian processes on curved boundaries

Let $T_{c,β}$ denote the smallest $t\ge1$ that a continuous, self-similar Gaussian process with self-similarity index $α>0$ moves at least $\pm c t^β$ units. We prove that: (i) If $β>α$, then $T_{c,β}=\infty$ with positive probability; (ii) If $β<α$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then $T_{c,β}$ has moments of all order; and (iii) If $β=α$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then there exists a continuous, strictly decreasing function $λ:(0\,,\infty)\to(0\,,\infty)$ such that $\mathrm{E}(T_{c,β}^μ)$ is finite when $0<μ<λ(c)$ and infinite when $μ>λ(c)$. Together these results extend a celebrated theorem of Breiman (1967) and Shepp (1967) for passage times of a Brownian motion on the critical square-root boundary. We briefly discuss two examples: One about fractional Brownian motion, and another about a family of linear stochastic partial differential equations.

math.PR↗

arXiv2025

On the Fourier dimension of fractional Brownian graphs

In this note we prove that the Fourier dimension of the graph $G(B)$ of a fractional Brownian motion $B$ with Hurst parameter $H\in(0,1/2)$ is equal to 1. This finishes to solve a conjecture by Fraser and Sahlsten. It also yields an exact formula for the gap $\dim_{\rm H}(G(B)) - \dim_{\rm F}(G(B))$ between the Hausdorff dimension and the Fourier dimension of $G(B)$. The proof is based on an intricate combinatorics procedure for multiple integrals related to the covariance function of the fractional Brownian motion.

math.PR↗

arXiv2025

Hitting probabilities, thermal capacity, and Hausdorff dimension results for the Brownian sheet

Let $W= \{W(t): t \in \mathbb{R}_+^N \}$ be an $(N, d)$-Brownian sheet and let $E \subset (0, \infty)^N$ and $F \subset \mathbb{R}^d$ be compact sets. We prove a necessary and sufficient condition for $W(E)$ to intersect $F$ with positive probability and determine the essential supremum of the Hausdorff dimension of the intersection set $W(E)\cap F$ in terms of the thermal capacity of $E \times F$. This extends the previous results of Khoshnevisan and Xiao (2015) for the Brownian motion and Khoshnevisan and Shi (1999) for the Brownian sheet in the special case when $E \subset (0, \infty)^N$ is an interval.

math.PR↗

arXiv2025

Local times of anisotropic Gaussian random fields and stochastic heat equation

We study the local times of a large class of Gaussian random fields satisfying strong local nondeterminism with respect to an anisotropic metric. We establish moment estimates and Hölder conditions for the local times of the Gaussian random fields. Our key estimates rely on geometric properties of Voronoi partitions with respect to an anisotropic metric and the use of Besicovitch's covering theorem. As a consequence, we deduce sample path properties of the Gaussian random fields that are related to Chung's law of the iterated logarithm and modulus of non-differentiability. Moreover, we apply our results to systems of stochastic heat equations with additive Gaussian noise and determine the exact Hausdorff measure function with respect to the parabolic metric for the level sets of the solutions.

math.PR↗

arXiv2023

Polarity of points for systems of nonlinear stochastic heat equations in the critical dimension

Let $u(t, x) = (u_1(t, x), \dots, u_d(t, x))$ be the solution to the systems of nonlinear stochastic heat equations \[ \begin{split} \frac{\partial}{\partial t} u(t, x) &= \frac{\partial^2}{\partial x^2} u(t, x) + σ(u(t, x)) \dot{W}(t, x),\\ u(0, x) &= u_0(x), \end{split} \] where $t \ge 0$, $x \in \mathbb{R}$, $\dot{W}(t, x) = (\dot{W}_1(t, x), \dots, \dot{W}_d(t, x))$ is a vector of $d$ independent space-time white noises, and $σ: \mathbb{R}^d \to \mathbb{R}^{d\times d}$ is a matrix-valued function. We say that a subset $S$ of $\mathbb{R}^d$ is polar for $\{u(t, x), t \ge 0, x \in \mathbb{R}\}$ if \[ \mathbb{P}\{u(t,x) \in S \text{ for some } t>0 \text{ and } x\in\mathbb{R} \}=0. \] The main result of this paper shows that, in the critical dimension $d=6$, all points in $\mathbb{R}^d$ are polar for $\{u(t, x), t \ge 0, x \in \mathbb{R}\}$. This solves an open problem of Dalang, Khoshnevisan and Nualart (2009, 2013) and Dalang, Mueller and Xiao (2021). We also provide a sufficient condition for a subset $S$ of $\mathbb{R}^d$ to be polar.

math.PR↗

arXiv2023

Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds

We consider nonlinear parabolic stochastic PDEs on a bounded Lipschitz domain driven by a Gaussian noise that is white in time and colored in space, with Dirichlet or Neumann boundary condition. We establish existence, uniqueness and moment bounds of the random field solution under measure-valued initial data $ν$. We also study the two-point correlation function of the solution and obtain explicit upper and lower bounds. For $C^{1, α}$-domains with Dirichlet condition, the initial data $ν$ is not required to be a finite measure and the moment bounds can be improved under the weaker condition that the leading eigenfunction of the differential operator is integrable with respect to $|ν|$. As an application, we show that the solution is fully intermittent for sufficiently high level $λ$ of noise under the Dirichlet condition, and for all $λ> 0$ under the Neumann condition.

math.PR↗

arXiv2022

Hitting probabilities of Gaussian random fields and collision of eigenvalues of random matrices

Let $X= \{X(t), t \in \mathbb R^N\}$ be a centered Gaussian random field with values in $\mathbb R^d$ satisfying certain conditions and let $F \subset \mathbb R^d$ be a Borel set. In our main theorem, we provide a sufficient condition for $F$ to be polar for $X$, i.e. $\mathbb P \big( X(t) \in F \hbox{ for some } t \in \mathbb R^N \big) = 0$, which improves significantly the main result in Dalang et al [7], where the case of $F$ being a singleton was considered. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, by using our main theorem, we solve a problem on the existence of collisions of the eigenvalues of random matrices with Gaussian random field entries that was left open in Jaramillo and Nualart [14] and Song et al [21].

math.PR↗