arXiv subjects

Zikai Chen

Publications and source records attributed to Zikai Chen.

6 recordsLinked to original sources

arXiv2026

Well-Posedness for SDEs with Logarithmical Critical Distributional Drifts

We study the stochastic differential equation $$d X_t=b(t,X_t)d t+\sqrt{2}d W_t$$ on $\mathbb R^d$, where $b$ is a time-dependent, divergence-free distributional drift of critical H\"older--Besov regularity $-1$, strengthened by an iterated-logarithmic correction. For every initial probability law, we construct a weak solution by smooth approximation and realize the singular drift as an additive functional. The main analytic ingredient is the Schauder estimate with a logarithmic smallness factor. Combined with uniform logarithmic Krylov estimates and a stochastic substitution formula for distributional test functions, this estimate allows us to apply a Zvonkin transformation and prove uniqueness in law among weak solutions satisfying the corresponding Krylov bounds. For solutions starting from deterministic points, we further show that their time-marginal distributions admit densities satisfying two-sided Aronson-type Gaussian estimates.

math.PR↗

arXiv2026

Kinetic Fokker--Planck Equations with Drifts in a Supercritical Range

We investigate kinetic Fokker--Planck equations with rough divergence-free drifts in anisotropic mixed Lebesgue spaces. The divergence-free structure yields an energy cancellation that remains effective under kinetic localization. Combining this cancellation with anisotropic regularization estimates and a De Giorgi iteration, we establish local boundedness for weak subsolutions and, consequently, global well-posedness for the Cauchy problem in a scaling-supercritical regime.

math.AP↗

arXiv2025

Kinetic SDEs with subcritical distributional drifts

In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in $\mathbb R^{2d}(d\geq2)$ driven by Brownian motion: $$\mathord{\rm d} X_t=V_t\mathord{\rm d} t,\ \mathord{\rm d} V_t=b(t,X_t,V_t)\mathord{\rm d} t+\sqrt{2}\mathord{\rm d} W_t,$$ where the subcritical distribution-valued drift $b$ belongs to the weighted anisotropic Hölder space $\mathbb L_T^{q_b}\mathbf C_{\boldsymbol{a}}^{α_b}(ρ_κ)$ with parameters $α_b\in(-1,0)$, $q_b\in(\frac{2}{1+α_b},\infty]$, $κ\in[0,1+α_b)$ and $÷_v b$ is bounded. We establish the well-posedness of weak solutions to the associated integral equation: $$X_t=X_0+\int_0^t V_s\mathord{\rm d} s,\ V_t=V_0+\lim_{n\to\infty}\int_0^t b_n(s,X_s,V_s)\mathord{\rm d}+\sqrt{2}W_t,$$ where $b_n:=b*Γ_n$ denotes the mollification of $b$ and the limit is taken in the $L^2$-sense. As an application, we discuss examples of $b$ involving Gaussian random fields.

math.PR↗

arXiv2025

On one typical Einstein-Weyl equation: inverse spectral transform for theCauchy problem, longtime behaviour of the solutions and implicit solutions

In this paper, we study one typical Einstein-Weyl equation. It arises from Ferapontov and Kruglikov's investigation on the integrability of several dispersionless partial differential equations and the geometry of their formal linearizations. First, by using Manakov-Santini IST (inverse spectral transform) method, we investigate its Cauchy problem including the direct problem from the initial data, the time evolution of the scattering and spectral data, and the inverse problem. Second, based on the nonlinear RH (Riemann-Hilbert) dressing, the longtime behaviour of the solutions is constructed. In addition, some implicit solutions are presented.

nlin.SI↗

arXiv2021

NAST: A Non-Autoregressive Generator with Word Alignment for Unsupervised Text Style Transfer

Autoregressive models have been widely used in unsupervised text style transfer. Despite their success, these models still suffer from the content preservation problem that they usually ignore part of the source sentence and generate some irrelevant words with strong styles. In this paper, we propose a Non-Autoregressive generator for unsupervised text Style Transfer (NAST), which alleviates the problem from two aspects. First, we observe that most words in the transferred sentence can be aligned with related words in the source sentence, so we explicitly model word alignments to suppress irrelevant words. Second, existing models trained with the cycle loss align sentences in two stylistic text spaces, which lacks fine-grained control at the word level. The proposed non-autoregressive generator focuses on the connections between aligned words, which learns the word-level transfer between styles. For experiments, we integrate the proposed generator into two base models and evaluate them on two style transfer tasks. The results show that NAST can significantly improve the overall performance and provide explainable word alignments. Moreover, the non-autoregressive generator achieves over 10x speedups at inference. Our codes are available at https://github.com/thu-coai/NAST.

cs.CL↗

arXiv2020

Stylized Dialogue Response Generation Using Stylized Unpaired Texts

Generating stylized responses is essential to build intelligent and engaging dialogue systems. However, this task is far from well-explored due to the difficulties of rendering a particular style in coherent responses, especially when the target style is embedded only in unpaired texts that cannot be directly used to train the dialogue model. This paper proposes a stylized dialogue generation method that can capture stylistic features embedded in unpaired texts. Specifically, our method can produce dialogue responses that are both coherent to the given context and conform to the target style. In this study, an inverse dialogue model is first introduced to predict possible posts for the input responses, and then this inverse model is used to generate stylized pseudo dialogue pairs based on these stylized unpaired texts. Further, these pseudo pairs are employed to train the stylized dialogue model with a joint training process, and a style routing approach is proposed to intensify stylistic features in the decoder. Automatic and manual evaluations on two datasets demonstrate that our method outperforms competitive baselines in producing coherent and style-intensive dialogue responses.

cs.CL↗