arXiv subjects

Dongsheng Wu

Publications and source records attributed to Dongsheng Wu.

10 recordsLinked to original sources

arXiv2026

A trace formula derivation of the functional equation for symmetric square L-functions

This paper revisits the functional equation of the symmetric-square $L$-function from the perspective of trace formulas. We show that, for level-one holomorphic cusp forms and even Hecke--Maass cusp forms, this symmetry can be recovered directly from the Petersson and Kuznetsov trace formulas. Although the functional equation itself is classical, our aim is to provide a concrete and transparent example of how the analytic structure of an $L$-function can emerge from a comparison of the spectral and geometric sides of a trace formula. The argument leads to an averaged reciprocity identity and treats the holomorphic and Maass settings within the same general framework. In this way, the paper offers a simple model for extending trace-formula methods beyond standard $L$-functions.

math.NT↗

arXiv2022

GCN-MIF: Graph Convolutional Network with Multi-Information Fusion for Low-dose CT Denoising

Being low-level radiation exposure and less harmful to health, low-dose computed tomography (LDCT) has been widely adopted in the early screening of lung cancer and COVID-19. LDCT images inevitably suffer from the degradation problem caused by complex noises. It was reported that deep learning (DL)-based LDCT denoising methods using convolutional neural network (CNN) achieved impressive denoising performance. Although most existing DL-based methods (e.g., encoder-decoder framework) can implicitly utilize non-local and contextual information via downsampling operator and 3D CNN, the explicit multi-information (i.e., local, non-local, and contextual) integration may not be explored enough. To address this issue, we propose a novel graph convolutional network-based LDCT denoising model, namely GCN-MIF, to explicitly perform multi-information fusion for denoising purpose. Concretely, by constructing intra- and inter-slice graph, the graph convolutional network is introduced to leverage the non-local and contextual relationships among pixels. The traditional CNN is adopted for the extraction of local information. Finally, the proposed GCN-MIF model fuses all the extracted local, non-local, and contextual information. Extensive experiments show the effectiveness of our proposed GCN-MIF model by quantitative and visualized results. Furthermore, a double-blind reader study on a public clinical dataset is also performed to validate the usability of denoising results in terms of the structural fidelity, the noise suppression, and the overall score. Models and code are available at https://github.com/tonyckc/GCN-MIF_demo.

eess.IV↗

arXiv2018

Sharp Space-Time Regularity of the Solution to Stochastic Heat Equation Driven by Fractional-Colored Noise

In this paper, we study the following stochastic heat equation \[ \partial_tu=\mathcal{L} u(t,x)+\dot{B},\quad u(0,x)=0,\quad 0\le t\le T,\quad x\in\mathbb{R}d, \] where $\mathcal{L}$ is the generator of a Lévy process $X$ taking value in $\mathbb{R}^d$, $B$ is a fractional-colored Gaussian noise with Hurst index $H\in\left(\frac12,\,1\right)$ for the time variable and spatial covariance function $f$ which is the Fourier transform of a tempered measure $μ.$ After establishing the existence of solution for the stochastic heat equation, we study the regularity of the solution $\{u(t,x),\, t\ge 0,\, x\in\mathbb{R}^d\}$ in both time and space variables. Under mild conditions, we give the exact uniform modulus of continuity and a Chung-type law of iterated logarithm for the sample function $(t,x)\mapsto u(t,x)$. Our results generalize and strengthen the corresponding results of Balan and Tudor (2008) and Tudor and Xiao (2017).

math.PR↗

arXiv2015

Smoothness of Local Times and Self-Intersection Local Times of Gaussian Random Fields

This paper is concerned with the smoothness (in the sense of Meyer-Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.

math.PR↗

arXiv2012

Critical Brownian sheet does not have double points

We derive a decoupling formula for the Brownian sheet which has the following ready consequence: An $N$-parameter Brownian sheet in $\mathbf{R}^d$ has double points if and only if $d<4N$. In particular, in the critical case where $d=4N$, the Brownian sheet does not have double points. This answers an old problem in the folklore of the subject. We also discuss some of the geometric consequences of the mentioned decoupling, and establish a partial result concerning $k$-multiple points in the critical case $k(d-2N)=d$.

math.PR↗

arXiv2011

$α$-Time Fractional Brownian Motion: PDE Connections and Local Times

For $0<α\leq 2$ and $0<H<1$, an $α$-time fractional Brownian motion is an iterated process $Z = \{Z(t)=W(Y(t)), t \ge 0\}$ obtained by taking a fractional Brownian motion $\{W(t), t\in \RR{R} \}$ with Hurst index $0<H<1$ and replacing the time parameter with a strictly $α$-stable Lévy process $\{Y(t), t\geq 0 \}$ in $\RR{R}$ independent of $\{W(t), t \in \R\}$. It is shown that such processes have natural connections to partial differential equations and, when $Y$ is a stable subordinator, can arise as scaling limit of randomly indexed random walks. The existence, joint continuity and sharp Hölder conditions in the set variable of the local times of a $d$-dimensional $α$-time fractional Brownian motion $X = \{X(t), t \in \R_+$\} defined by $ X(t)=\big(X_{1}(t),..., X_{d}(t) \big), $ where $t\geq 0$ and $X_{1},..., X_{d}$ are independent copies of $Z$, are investigated. Our methods rely on the strong local ' nondeterminism of fractional Brownian motion.

math.PR↗

arXiv2009

Regularity of Intersection Local Times of Fractional Brownian Motions

Let $B^{α_i}$ be an $(N_i,d)$-fractional Brownian motion with Hurst index ${α_i}$ ($i=1,2$), and let $B^{α_1}$ and $B^{α_2}$ be independent. We prove that, if $\frac{N_1}{α_1}+\frac{N_2}{α_2}>d$, then the intersection local times of $B^{α_1}$ and $B^{α_2}$ exist, and have a continuous version. We also establish Hölder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and intersection points. One of the main motivations of this paper is from the results of Nualart and Ortiz-Latorre ({\it J. Theor. Probab.} {\bf 20} (2007)), where the existence of the intersection local times of two independent $(1,d)$-fractional Brownian motions with the same Hurst index was studied by using a different method. Our results show that anisotropy brings subtle differences into the analytic properties of the intersection local times as well as rich geometric structures into the sets of intersection times and intersection points.

math.PR↗

arXiv2008

Local times of multifractional Brownian sheets

Denote by $H(t)=(H_1(t),...,H_N(t))$ a function in $t\in{\mathbb{R}}_+^N$ with values in $(0,1)^N$. Let $\{B^{H(t)}(t)\}=\{B^{H(t)}(t),t\in{\mathbb{R}}^N_+\}$ be an $(N,d)$-multifractional Brownian sheet (mfBs) with Hurst functional $H(t)$. Under some regularity conditions on the function $H(t)$, we prove the existence, joint continuity and the Hölder regularity of the local times of $\{B^{H(t)}(t)\}$. We also determine the Hausdorff dimensions of the level sets of $\{B^{H(t)}(t)\}$. Our results extend the corresponding results for fractional Brownian sheets and multifractional Brownian motion to multifractional Brownian sheets.

math.PR↗

arXiv2008

Joint continuity of the local times of fractional Brownian sheets

Let $B^H=\{B^H(t),t\in{\mathbb{R}_+^N}\}$ be an $(N,d)$-fractional Brownian sheet with index $H=(H_1,...,H_N)\in(0,1)^N$ defined by $B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ where $B^H_1,...,B^H_d$ are independent copies of a real-valued fractional Brownian sheet $B_0^H$. We prove that if $d<\sum_{\ell=1}^NH_{\ell}^{-1}$, then the local times of $B^H$ are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global Hölder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$.

math.PR↗

arXiv2006

Fractal properties of the random string processes

Let $\{u_t(x),t\ge 0, x\in {\mathbb{R}}\}$ be a random string taking values in ${\mathbb{R}}^d$, specified by the following stochastic partial differential equation [Funaki (1983)]: \[\frac{\partial u_t(x)}{\partial t}=\frac{{\partial}^2u_t(x)}{\partial x^2}+\dot{W},\] where $\dot{W}(x,t)$ is an ${\mathbb{R}}^d$-valued space-time white noise. Mueller and Tribe (2002) have proved necessary and sufficient conditions for the ${\mathbb{R}}^d$-valued process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$ to hit points and to have double points. In this paper, we continue their research by determining the Hausdorff and packing dimensions of the level sets and the sets of double times of the random string process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$. We also consider the Hausdorff and packing dimensions of the range and graph of the string.

math.PR↗