arXiv subjects

Zhongmin Qian

Publications and source records attributed to Zhongmin Qian.

At least 19 recordsLinked to original sources

arXiv2026

Image Inpainting via Stochastic Dynamics

Image inpainting aims to recover missing regions while preserving structural consistency. We propose a non-parametric method without network training based on data-guided stochastic dynamics. Starting from a masked image, the missing pixels are evolved through a reverse-time stochastic differential equation with a kernel-weighted correction estimated directly from a reference dataset. This empirical correction guides the reconstruction toward high-density regions of the data distribution without training a neural network or fitting a parametric density model. Experiments on MNIST, Fashion-MNIST, and MVTec show that the proposed method outperforms Mean Fill, Telea, and Navier-Stokes inpainting in PSNR, SSIM, and visual quality. On CelebA, it remains competitive and produces plausible completions for structure-sensitive occlusions. These results demonstrate the effectiveness of empirical reference statistics as a non-parametric prior for image inpainting.

cs.CV↗

arXiv2026

High Order Asymptotic Expansion at Infinity for Strong Solutions to Incompressible Navier-Stokes Equations

We discuss an interesting distinction between the incompressible Navier-Stokes equations (the velocity equations) and its vorticity form in whole space. We show that if the initial vorticity has a Gaussian bound then the bound is inherited up to the maximal lifespan of the strong solution. However, it turns out that the velocity equations don not share the same property. In fact, $L^p$-strong solutions to the velocity equations arising from ``well-localized" initial value generally behave at infinity like derivatives (of order $\geq3$) of the fundamental solution of Laplacian. To show this, a clean expansion up to maximal lifespan is derived : \begin{align} u(x,t)=-\nabla\sum_{|α|=0}^{d-1}\frac{(-1)^{|α|}}{α!}\partial^α\partial_{i,j}^2Γ(x)\int_0^t{\rm M}_α^{i,j}(s){\rm d}s+O\big(|x|^{-2d-1}\big)\nonumber \end{align} where ${\rm M}_α^{i,j}(t):=\int_{\mathbb R^d}y^αu^i(y,t)u^j(y,t){\rm d}y$ and $Γ$ is the fundamental solution of Laplacian. This improves the first order expansion given by L. Brandolese and F. Vigneron \cite{BV07}.

math.AP↗

arXiv2025

Tracking Brownian fluid particles in large eddy simulations

In this paper, we propose an approach for simulating wall-bounded incompressible turbulent flows by integrating the technology of random vortex method with the core principles of large-eddy simulations (LES). In particular, we employ the filtering function, interpreted as a spatial averaging operator, together with the integral representation theorem for parabolic equations, to construct a closed numerical scheme suitable for computing solutions to the Navier-Stokes equations. This framework numerically overcomes the difficulties associated with the non-locally integrable three-dimensional kernel inherent in the random vortex method, enabling efficient computation of flow fields via the Monte Carlo method. Several numerical experiments are presented for both laminar and turbulent flows in wall-bounded domains, to thereby reveal the underlying flow mechanisms near the wall boundary. The experimental results and systematic comparisons with alternative numerical approaches consistently demonstrate that the proposed method is numerically stable, possesses low theoretical complexity, and achieves acceptable computational efficiency.

physics.flu-dyn↗

arXiv2025

Signature Decomposition Method Applying to Pair Trading

High-frequency quantitative trading strategies have long been of significant interest in futures market. While advanced statistical arbitrage and deep learning enhance high-frequency data processing, they diminish opportunities for traditional methods and yield less interpretable, unstable strategies. Consequently, developing stable, interpretable quantitative strategies remains a priority in futures markets. In this study, we propose a novel pair trading strategy by leveraging the mathematical concept of path signature which serves as a feature representation of time series. Specifically, the path signature is decomposed into two new indicators: the path interactivity indicator segmented signature and the directional indicator covariation of increments, which serve as double filters in strategy design. Empirical experiments using minute-level futures data show our strategy significantly outperforms traditional pair trading, delivering higher returns, lower maximum drawdown, and higher Sharpe ratio. The proposed method enhances interpretability and robustness while maintaining strong returns, demonstrating the potential of path signatures in financial trading.

econ.GN↗

arXiv2025

Global-in-time convergence in infinity-ion-mass limit for bipolar Euler-Poisson equations

In this paper, the Cauchy problem for the multi-dimensional (M-D) bipolar Euler-Poisson equations with far field vacuum is considered. Based on physical observations and some elaborate analysis of this system's intrinsic symmetric hyperbolic-elliptic coupled structures, for a class of smooth initial data that are of small scaled density but possibly large mean velocity, we give one rigorous global-in-time convergence proof for regular solutions from M-D bipolar Euler-Poisson equations to M-D unipolar Euler-Poisson equations through the infinity-ion mass limit. Here the initial scaled density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial mean velocity are all positive. In order to deal with such kind of singular limits, the global-in-time uniform tame estimates of regular solutions to M-D bipolar Euler-Poisson equations with respect to the ratio of electron mass over ion mass are established, based on which the corresponding error estimates in smooth function spaces between the two systems considered are also given. To achieve these, our main strategy is to regard the original problem for M-D bipolar Euler-Poisson equations as the limit of a series of carefully designed approximate problems which have truncated convection operators and compactly supported initial data. For such artificial problems, we can derive careful a-priori estimates that are independent of the mass ratio, the size of the initial data' supports and the truncation parameters. Then the global uniform existence of regular solutions of the original problem are attained via careful compactness.

math.AP↗

arXiv2024

Random large eddy simulation for 3-dimensional incompressible viscous flows

We develop a numerical method for simulation of incompressible viscous flows by integrating the technology of random vortex method with the core idea of Large Eddy Simulation (LES). Specifically, we utilize the filtering method in LES, interpreted as spatial averaging, along with the integral representation theorem for parabolic equations, to achieve a closure scheme which may be used for calculating solutions of Navier-Stokes equations. This approach circumvents the challenge associated with handling the non-locally integrable 3-dimensional integral kernel in the random vortex method and facilitates the computation of numerical solutions for flow systems via Monte-Carlo method. Numerical simulations are carried out for both laminar and turbulent flows, demonstrating the validity and effectiveness of the method.

physics.flu-dyn↗

arXiv2024

Random vortex and expansion-rate model for Oberbeck-Boussinesq fluid flows

By using a formulation of a class of compressible viscous flows with a heat source via vorticity and expansion-rate, we study the Oberbeck-Boussinesq flows. To this end we establish a new integral representation for solutions of parabolic equations subject to certain boundary condition, which allows us to develop a random vortex method for certain compressible flows and to compute numerically solutions of their dynamical models. Numerical experiments are carried out, which not only capture detailed Bénard convection but also are capable of providing additional information on the fluid density and the dynamics of expansion-rate of the flow.

math.AP↗

arXiv2024

Rough Path Renormalization from Stratonovich to Itô for Fractional Brownian Motion

This paper develops an Itô-type fractional pathwise integration theory for fractional Brownian motion with Hurst parameters $ H \in (\frac{1}{3}, \frac{1}{2}] $, using the Lyons' rough path framework. This approach is designed to fill gaps in conventional stochastic calculus models that fail to account for temporal persistence prevalent in dynamic systems such as those found in economics, finance, and engineering. The pathwise-defined method not only meets the zero expectation criterion but also addresses the challenges of integrating non-semimartingale processes, which traditional Itô calculus cannot handle. We apply this theory to fractional Black-Scholes models and high-dimensional fractional Ornstein-Uhlenbeck processes, illustrating the advantages of this approach. Additionally, the paper discusses the generalization of Itô integrals to rough differential equations (RDE) driven by fBM, emphasizing the necessity of integrand-specific adaptations in the Itô rough path lift for stochastic modeling.

math.PR↗

arXiv2024

Lévy Area Analysis and Parameter Estimation for fOU Processes via Non-Geometric Rough Path Theory

This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path. Our approach is particularly suitable for high-frequency data. To formulate the parameter estimators, we introduce a theory of pathwise Itô integrals with respect to fractional Brownian motion. By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated Lévy area processes, we demonstrate that our estimators are strongly consistent and pathwise stable. Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings, and may have practical implications for fields including finance, economics, and engineering.

math.PR↗

arXiv2024

Entropy Estimate for Degenerate SDEs with Applications to Nonlinear Kinetic Fokker-Planck Equations

The relative entropy for two different degenerate diffusion processes is estimated by using the Wasserstein distance of initial distributions and the difference between coefficients. As applications, the entropy cost inequality and exponential ergodicity in entropy are derived for distribution dependent stochastic Hamiltonian systems associated with nonlinear kinetic Fokker Planck equations.

math.PR↗

arXiv2024

Mean field equations arising from random vortex dynamics

We consider Mckean-Vlasov type stochastic differential equations with multiplicative noise arising from the random vortex method. Such an equation can be viewed as the mean-field limit of interacting particle systems with singular interacting kernels such as the Biot-Savart kernel. A new estimate for the transition probability density of diffusion processes will be formulated to handle the singularity of the interacting kernel. The existence and uniqueness of the weak solution of such SDEs will be established as the main result.

math.PR↗

arXiv2024

Random vortex dynamics and Monte-Carlo simulations for wall-bounded viscous flows

Functional integral representations for solutions of the motion equations for wall-bounded incompressible viscous flows, expressed (implicitly) in terms of distributions of solutions to stochastic differential equations of McKean-Vlasov type, are established by using a perturbation technique. These representations are used to obtain exact random vortex dynamics for wall-bounded viscous flows. Numerical schemes therefore are proposed and the convergence of the numerical schemes for random vortex dynamics with an additional force term is established. Several numerical experiments are carried out for demonstrating the motion of a viscous flow within a thin layer next to the fluid boundary.

math.NA↗

arXiv2024

Dirac operators and field equations of the gravitational field and matter fields

Dirac operators on curved space-times are introduced with the help of a new point-view that observers have to be included in the formulation of natural laws. The class of Dirac operators are Lorentz invariant in the sense that the transformation rule is specified under diffeomorphisms of the space-time which preserve the time orientation and the gravitational field. Moreover these Dirac operators, like the original Dirac's operator with the special relativity, satisfy the Hamiltonian relation required by the general theory of relativity up to a correction due to the setup of a reference frame. In order to generalise (or discover) Dirac operators, which have been an important building block in the quantum field theories (QFTs), to curved space-times, we bring observers (which are elements of the principal orthonormal bundle over the space-time with its structure group being the proper Lorentz group) into the description of Fermion fields in the presence of a gravitational field. As a consequence field equations combining the gravitational field and matter fields may be formulated. This work suggests that observational effects (due to the setup of references) in the presence of a strong gravitational field, i.e. matter, may be unavoidable.

gr-qc↗

arXiv2023

On vorticity and expansion-rate of fluid flows, conditional law duality and their representations

By using a formulation of motion equations for a viscous (compressible) fluid flow in terms of the vorticity and the rate of expansion as the main fluid dynamical variables, an approximation model is established for compressible flows with slowly varied (over the space) fluid density. The probabilistic tools and the main ingredient such as the duality of conditional laws and the forward type Feynman-Kac formula are established for elliptic operators of second order, in order to formulate the corresponding random vortex method for a class of viscous compressible fluid flows, based on their approximation motion equations.

math.AP↗

arXiv2023

Monte-Carlo method for incompressible fluid flows past obstacles

We establish stochastic functional integral representations for incompressible fluid flows occupying wall-bounded domains using the conditional law duality for a class of diffusion processes. These representations are used to derive a Monte-Carlo scheme based on the corresponding exact random vortex formulation. We implement several numerical experiments based on the Monte-Carlo method without appealing to the boundary layer flow computations, to demonstrate the methodology.

physics.flu-dyn↗

arXiv2023

Twin Brownian particle method for the study of Oberbeck-Boussinesq fluid flows

We establish stochastic functional integral representations for solutions of Oberbeck-Boussinesq equations in the form of McKean-Vlasov-type mean field equations, which can be used to design numerical schemes for calculating solutions and for implementing Monte-Carlo simulations of Oberbeck-Boussinesq flows. Our approach is based on the duality of conditional laws for a class of diffusion processes associated with solenoidal vector fields, which allows us to obtain a novel integral representation theorem for solutions of some linear parabolic equations in terms of the Green function and the pinned measure of the associated diffusion. We demonstrate via numerical experiments the efficiency of the numerical schemes, which are capable of revealing numerically the details of Oberbeck-Boussinesq flows within their thin boundary layer, including B{é}nard's convection feature.

physics.flu-dyn↗

arXiv2023

Volatility forecasting with machine learning and intraday commonality

We apply machine learning models to forecast intraday realized volatility (RV), by exploiting commonality in intraday volatility via pooling stock data together, and by incorporating a proxy for the market volatility. Neural networks dominate linear regressions and tree-based models in terms of performance, due to their ability to uncover and model complex latent interactions among variables. Our findings remain robust when we apply trained models to new stocks that have not been included in the training set, thus providing new empirical evidence for a universal volatility mechanism among stocks. Finally, we propose a new approach to forecasting one-day-ahead RVs using past intraday RVs as predictors, and highlight interesting time-of-day effects that aid the forecasting mechanism. The results demonstrate that the proposed methodology yields superior out-of-sample forecasts over a strong set of traditional baselines that only rely on past daily RVs.

q-fin.ST↗

arXiv2022

New Approach for Vorticity Estimates of Solutions of the Navier-Stokes Equations

We develop a new approach for regularity estimates, especially vorticity estimates, of solutions of the three-dimensional Navier-Stokes equations with periodic initial data, by exploiting carefully formulated linearized vorticity equations. An appealing feature of the linearized vorticity equations is the inheritance of the divergence-free property of solutions, so that it can intrinsically be employed to construct and estimate solutions of the Navier-Stokes equations. New regularity estimates of strong solutions of the three-dimensional Navier-Stokes equations are obtained by deriving new explicit a priori estimates for the heat kernel (i.e., the fundamental solution) of the corresponding heterogeneous drift-diffusion operator. These new a priori estimates are derived by using various functional integral representations of the heat kernel in terms of the associated diffusion processes and their conditional laws, including a Bismut-type formula for the gradient of the heat kernel. Then the a priori estimates of solutions of the linearized vorticity equations are established by employing a Feynman-Kac-type formula. The existence of strong solutions and their regularity estimates up to a time proportional to the reciprocal of the square of the maximum initial vorticity are established. All the estimates established in this paper contain known constants that can be explicitly computed.

math.AP↗