arXiv subjects

Liying Tao

Publications and source records attributed to Liying Tao.

5 recordsLinked to original sources

arXiv2025

Global well-posedness of the energy-critical stochastic Hartree nonlinear wave equation

We consider the Cauchy problem for the stochastic Hartree nonlinear wave equations (SHNLW) with a cubic convolution nonlinearity and an additive stochastic forcing on the Euclidean space. Our goal in this paper is two-fold. (i) We study the defocusing energy-critical SHNLW on $\mathbb{R}^d$, for $d \geq 5$, and prove that they are globally well-posed with deterministic initial data in the energy space. (ii) Next, we consider the well-posedness of the defocusing energy-critical SHNLW with randomized initial data below the energy space. In particular, when $d=5$, we prove it is almost surely globally well-posed. As a byproduct, by removing the stochastic forcing our result covers the study of the (deterministic) Hartree nonlinear wave equation (HNLW) with randomized initial data below the energy space. The main ingredients in the globalization argument involve the probabilistic perturbation approach by Bényi-Oh-Pocovnicu (2015) and Pocovnicu (2017), time integration by parts trick of Oh-Pocovnicu (2016), and an estimate of the Hartree potential energy.

math.AP↗

arXiv2025

Optimization of Low-Latency Spiking Neural Networks Utilizing Historical Dynamics of Refractory Periods

The refractory period controls neuron spike firing rate, crucial for network stability and noise resistance. With advancements in spiking neural network (SNN) training methods, low-latency SNN applications have expanded. In low-latency SNNs, shorter simulation steps render traditional refractory mechanisms, which rely on empirical distributions or spike firing rates, less effective. However, omitting the refractory period amplifies the risk of neuron over-activation and reduces the system's robustness to noise. To address this challenge, we propose a historical dynamic refractory period (HDRP) model that leverages membrane potential derivative with historical refractory periods to estimate an initial refractory period and dynamically adjust its duration. Additionally, we propose a threshold-dependent refractory kernel to mitigate excessive neuron state accumulation. Our approach retains the binary characteristics of SNNs while enhancing both noise resistance and overall performance. Experimental results show that HDRP-SNN significantly reduces redundant spikes compared to traditional SNNs, and achieves state-of-the-art (SOTA) accuracy both on static datasets and neuromorphic datasets. Moreover, HDRP-SNN outperforms artificial neural networks (ANNs) and traditional SNNs in noise resistance, highlighting the crucial role of the HDRP mechanism in enhancing the performance of low-latency SNNs.

cs.NE↗

arXiv2025

Critical threshold for weakly interacting log-correlated focusing Gibbs measures

We study log-correlated Gibbs measures on the $d$-dimensional torus with weakly interacting focusing quartic potentials whose coupling constants tend to $0$ as we remove regularization. In particular, we exhibit a phase transition for this model by identifying a critical threshold, separating the weakly and strongly coupling regimes; in the weakly coupling regime, we show that the frequency-truncated measures converge to the base Gaussian measure (possibly with a renormalized $L^2$-cutoff), whereas, in the strongly coupling regime, we prove non-convergence of the frequency-truncated measures, even up to a subsequence. Our result answers an open question posed by Brydges and Slade (1996).

math.PR↗

arXiv2025

Global well-posedness of the energy-critical stochastic nonlinear Schrödinger equation on the three-dimensional torus

We study the Cauchy problem of the defocusing energy-critical stochastic nonlinear Schrödinger equation (SNLS) on the three dimensional torus, forced by an additive noise. We adapt the atomic spaces framework in the context of the energy-critical nonlinear Schrödinger equation, and employ probabilistic perturbation arguments in the context of stochastic PDEs, establishing the global well-posedness of the defocusing energy-critical quintic SNLS in the energy space. It is the first global well-posedness result for the periodic SNLS in a critical space.

math.AP↗

arXiv2023

Almost sure scattering for defocusing energy critical Hartree equation on $\R^5$

We consider the defocusing energy-critical Hartree equation $i\pa_tu+Δu=(|\cdot|^{-4}\ast|u|^2)u$ in spatial dimension $d=5$ and prove almost sure scattering with initial data $u_0\in H^s_x(\R^5)$ for any $s\in\R$. The proof relies on the modified interaction Morawetz estimate, the stability theories, the ``Narrowed'' Wiener randomization. We are inspired to consider this problem by the work of Shen-Soffer-Wu \cite{Shen-Soffer-Wu 1}, which treated the analogous problem for the energy-critical Schrödinger equation. The new ingredient in this paper are that we take an alternative proof to give the interaction Morawetz estimate. And the nonlocal nonlinearity term will bring some difficulties.

math.AP↗