arXiv subjects

Zhenxin Liu

Publications and source records attributed to Zhenxin Liu.

At least 19 recordsLinked to original sources

arXiv2026

A stochastic convex integration scheme for the intermittent Onsager theorem in two-dimensional domains

For any $γ\in \left[ {0,\frac{1}{3}} \right)$, we construct $γ$-Hölder-continuous martingale solutions to the two-dimensional stochastic incompressible Euler equations. These martingale solutions exhibit dissipative behavior and intermittency, thereby establishing the flexible part of the intermittent Onsager theorem. The proof rests on a stochastic and intermittent variant of the Newton--Nash iteration combined with the Littlewood--Paley decomposition scheme. This composite iteration scheme incorporates new stochastic pressure, Reynolds stress and intermittent perturbations to formalize the stochastic and intermittent fluctuations.

math.AP↗

arXiv2026

Averaging Principle and Pullback Attractor Convergence for McKean--Vlasov Stochastic Reaction--Diffusion Equations

We establish three averaging principles for distribution-dependent stochastic reaction--diffusion equations with rapidly oscillating coefficients on the torus $\mathbb T^d$, $d\le3$. First, solutions converge in mean square, uniformly on finite time intervals, to solutions of the averaged equation. Under a contraction condition, both the original and averaged equations admit unique bounded entire solutions whose mean-square distance vanishes uniformly for all $t\in\mathbb R$. At the level of probability laws, the original nonautonomous equation possesses a family of pullback attractors, whereas the averaged equation has a global attractor; the former converge upper-semicontinuously to the latter, uniformly over the coefficient hull. As an application, we present a class of stochastic reaction--diffusion models motivated by large-scale interacting systems.

math.DS↗

arXiv2026

Continuity of measure-theoretic entropy for stochastic differential equations

For stochastic differential equations, we establish a relationship between the measure-theoretic entropy of the stochastic flow and the rate of volume growth of stable submanifolds under iteration. By combining the result of Kifer and Yomdin (1988), we show that under a suitable integrability condition, for systems with $C^{\infty}$ coefficients, the measure-theoretic entropy is upper semicontinuous with respect to the coefficients of SDEs.

math.DS↗

arXiv2026

A criterion for the well-posedness of McKean-Vlasov stochastic differential equations

We establish strong existence and pathwise uniqueness for McKean-Vlasov stochastic differential equations with coefficients satisfying a distribution-dependent Lyapunov condition. Under a hybrid Perron-Nagumo condition that permits a non-integrable singularity at the initial time, pathwise uniqueness holds within the class of strong solutions satisfying the corresponding Lyapunov estimate. For existence, we truncate the coefficients on nested bounded domains, construct absorbed local weak solutions, pass to a weak solution via tightness arguments, and then apply a restricted Yamada-Watanabe theorem to obtain a strong solution. Our existence proof, different from the classical truncation-patching method, is interesting in its own right. We also provide an explicit example to which our criterion applies, while none of the Lipschitz, Osgood, or monotonicity conditions is satisfied.

math.PR↗

arXiv2026

Continuity and Discontinuity of McKean-Vlasov Phase Transitions via Bifurcation Theory

It is well known that the McKean-Vlasov stochastic differential equation with a symmetric double-well potential exhibits a continuous phase transition. In contrast, for an asymmetric double-well potential, the system undergoes a discontinuous phase transition, in which an invariant measure abruptly appears in the shallower well and subsequently splits into two as the temperature decreases. In this work, we systematically investigate the types of phase transitions driven by non-convex confining potentials and cooperative interactions in $\mathbb{R}^n$. Our main results consist of a pitchfork bifurcation theorem characterizing continuous phase transitions and a saddle-node bifurcation theorem governing discontinuous phase transitions. In addition, despite the dimension-dependent nature of phase transitions, we are able to extend Dawson's criterion for phase transition points -- originally formulated for quadratic interactions in $1$-dimensional space -- to $n$-dimensional space. This extension directly relates the critical parameter and bifurcation direction to the eigenvalue and eigenvector of the critical covariance matrix, respectively. Finally, we apply our theoretical results to the aforementioned double-well models, to a class of four-well models in two-dimensional Euclidean space that exhibit multiple phase transitions, and to an attractive Gaussian interaction model.

math.DS↗

arXiv2026

Convergence rates of Wasserstein gradient flows for nonlinear Fokker-Planck equations with mobility and related inequalities

For nonlinear Fokker-Planck equations with mobility, the Wasserstein gradient flow structure is described by the generalized relative entropy as the energy functional and the modified Wasserstein metric $W_h$ as the associated metric structure. This work investigates the nonlinear effects induced by mobility and establishes the corresponding inequalities. For nonlinear diffusion, we establish a logarithmic Sobolev inequality, which yields the convergence rate of the free energy functional and the Talagrand inequality. By further exploiting the relationship between the weighted homogeneous Sobolev norm and the $W_h$ metric, we derive an HWI inequality relating the relative entropy, the $W_h$ metric, and the Fisher information. In the case of mobility dependent drift and linear diffusion, the convergence rate in the $W_h$ metric is also obtained by applying the Girsanov theorem.

math.AP↗

arXiv2026

Semimartingale Optimal Transport with Jumps: A General Framework and Equivalent Formulations

We study a semimartingale optimal transport (SOT) problem where the cost depends on the full differential characteristics, and the minimisation is over all semimartingale laws with given marginals whose absolutely continuous characteristics lie in a prescribed closed convex set. Under only minimal assumptions of measurability, convexity, and lower semicontinuity on the cost function, we prove the existence of an optimal plan for SOT and establish a Kantorovich-type duality without time-regularity conditions. We further prove that SOT admits four equivalent formulations: (i) a Kantorovich duality formulation, (ii) a viscosity solution formulation of the Hamilton--Jacobi--Bellman equation, (iii) a martingale solution formulation via Markovian projection, (iv) a PDE formulation via weak solutions of a non-local Fokker--Planck--Kolmogorov equation. The framework simultaneously generalises classical optimal transport, martingale optimal transport, Schrödinger bridge problems, and barycentric weak optimal transport.

math.PR↗

arXiv2026

Nonlinear Vlasov-Fokker-Planck equations: From generalized Wasserstein gradient flow to GENERIC structure

We study the GENERIC (General Equation for Non-Equilibrium Reversible Irreversible Coupling) formulation of the nonlinear Vlasov-Fokker-Planck equation from the perspective of gradient flows along trajectories. After pulling back the reversible component, the evolution can be recast as a generalized Wasserstein gradient flow. The associated free energy functional consists of an entropy term and a conservative energy term, while the metric is induced by the Onsager operator. This trajectory based viewpoint shows that the trajectorial rate of free energy dissipation captures the influence of the nonlinear term, an effect that is not directly apparent at the macroscopic level. Finally, partial degeneracy yields a decomposition of the underlying metric space, which in turn enables the derivation of a partial HWI inequality.

math.AP↗

arXiv2025

Random Attractors for McKean-Vlasov SDEs

In this paper, we mainly focus on the existence of random attractors for McKean-Vlasov stochastic differential equations on a separable Hilbert space $H$. A significant challenge arises from the distribution-dependence of the coefficients, thereby causing the lack of the stochastic flow property on $H$. To address this issue, we first transform the original equation into a system on the product space $H \times \mathcal{P}(H)$ and consider the existence of random attractors on this space. We then analyze cocycles associated with two parametric dynamical systems. Within this framework, we define the corresponding pullback random attractor and develop a general theory for the existence of random attractors for such cocycles. Finally, we apply our theoretical results to McKean-Vlasov stochastic ordinary differential equations, McKean-Vlasov stochastic reaction-diffusion equations, and McKean-Vlasov stochastic 2D Navier-Stokes equations. In the case where the attractor reduces to a singleton set $\mathcal{A}(ω):=(ξ(ω),μ_\infty)$, we show that $ξ$ corresponds to the stationary solution for the decoupled SPDE,satisfying $\mathbb{P}\circ[ξ]^{-1}=μ_\infty$.

math.DS↗

arXiv2025

PicoPose: Progressive Pixel-to-Pixel Correspondence Learning for Novel Object Pose Estimation

RGB-based novel object pose estimation is critical for rapid deployment in robotic applications, yet zero-shot generalization remains a key challenge. In this paper, we introduce PicoPose, a novel framework designed to tackle this task using a three-stage pixel-to-pixel correspondence learning process. Firstly, PicoPose matches features from the RGB observation with those from rendered object templates, identifying the best-matched template and establishing coarse correspondences. Secondly, PicoPose smooths the correspondences by globally regressing a 2D affine transformation, including in-plane rotation, scale, and 2D translation, from the coarse correspondence map. Thirdly, PicoPose applies the affine transformation to the feature map of the best-matched template and learns correspondence offsets within local regions to achieve fine-grained correspondences. By progressively refining the correspondences, PicoPose significantly improves the accuracy of object poses computed via PnP/RANSAC. PicoPose achieves state-of-the-art performance on the seven core datasets of the BOP benchmark, demonstrating exceptional generalization to novel objects. Code and trained models are available at https://github.com/foollh/PicoPose.

cs.CV↗

arXiv2025

Quenched invariance principle with a rate for random dynamical systems

In this paper, we consider the quenched invariance principle for random Young towers driven by an ergodic system. In particular, we obtain the Wassertein convergence rate in the quenched invariance principle. As a key ingredient, we derive a new martingale-coboundary decomposition for the random tower map, which provides a good control over sums of squares of the approximating martingale. We apply our results to a class of random dynamical systems that admit a random Young tower, such as independent and identically distributed (i.i.d.) translations of Viana maps, intermittent maps of the interval and small random perturbations of Anosov maps with an ergodic driving system.

math.DS↗

arXiv2025

Invariant measures for stochastic Burgers equation on unbounded domains

In this paper, we investigate the stochastic damped Burgers equation with multiplicative noise defined on the entire real line. We demonstrate the existence and uniqueness of a mild solution to the stochastic damped Burgers equation and establish that the solution is uniformly bounded in time. Furthermore, by employing the uniform estimates on the tails of the solution, we obtain the tightness of a family of probability distributions of the solution. Subsequently, by applying the Krylov-Bogolioubov theorem, we establish the existence of invariant measures.

math.DS↗

arXiv2025

On the averaging theorems for stochastic perturbation of conservative linear systems

For stochastic perturbations of linear systems with non-zero pure imaginary spectrum we discuss the averaging theorems in terms of the slow-fast action-angle variables and in the sense of Krylov-Bogoliubov. Then we show that if the diffusion matrix of the perturbation is uniformly elliptic, then in all cases the averaged dynamics does not depend on a hamiltonian part of the perturbation.

math.DS↗

arXiv2025

Ergodicity of inhomogeneous Markov processes under general criteria

This paper is concerned with ergodic properties of inhomogeneous Markov processes. Since the transition probabilities depend on initial times, the existing methods to obtain invariant measures for homogeneous Markov processes are not applicable straightforwardly. We impose some appropriate conditions under which invariant measure families for inhomogeneous Markov processes can be studied. Specifically, the existence of invariant measure families is established by either a generalization of the classical Krylov-Bogolyubov method or a Lyapunov criterion. Moreover, the uniqueness and exponential ergodicity are demonstrated under a contraction assumption of the transition probabilities on a large set. Finally, three examples, including Markov chains, diffusion processes and storage processes, are analyzed to illustrate the practicality of our method.

math.PR↗

arXiv2025

A trajectorial approach to the gradient flow of McKean-Vlasov SDEs with mobility

We establish the gradient flow representation of diffusion with mobility $b$ with respect to the modified Wasserstein quasi-metric $W_h$, where $h(r)=rb(r)$. The appropriate selection of the free energy functional depends on the specific form of the generalized entropy. Different from the JKO scheme, we derive the trajectorial version of the relative entropy dissipation identity for the McKean-Vlasov stochastic differential equation (SDE) with Nemytskii-type coefficients, utilizing techniques from stochastic analysis. Based on this, we demonstrate that the trajectorial average of the solution process to the McKean-Vlasov SDE, with respect to the underlying measure, corresponds to the rate of dissipation of the free energy. As an application, we present the energy dissipation of the Fermi-Dirac-Fokker-Planck equation, a model widely used in physics and biology to describe saturation effects. Inspired by numerical simulations, we propose two questions on condensation phenomena and non-exponential convergence rate.

math.PR↗

arXiv2025

Invariant measures for stochastic Burgers equation in unbounded domains with space-time white noise

In this paper, we investigate the stochastic damped Burgers equation with multiplicative space-time white noise defined on the entire real line. We prove the existence and uniqueness of a mild solution of the stochastic damped Burgers equation in the weighted space and establish that the solution is bounded in probability. Furthermore, by using the Krylov-Bogolioubov theorem, we obtain the existence of invariant measures.

math.DS↗

arXiv2024

Wasserstein convergence rates in the invariance principle for sequential dynamical systems

In this paper, we consider the convergence rate with respect to the Wasserstein distance in the invariance principle for sequential dynamical systems. We utilize and modify the techniques previously employed for stationary sequences to address our non-stationary case. Under certain assumptions, we can apply our result to a large class of dynamical systems, including sequential $β_n$-transformations, piecewise uniformly expanding maps with additive noise in one-dimensional and multidimensional case, and so on.

math.DS↗

arXiv2024

Continuous dependence for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions

In this paper, we consider the continuous dependence on initial values and parameters of solutions as well as invariant measures for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions. In contrast to the classical SDEs, the solutions for McKean-Vlasov SDEs do not converge in probability although the initial values converge in probability, which is due to the mismatch of the distances between measures. Finally, we give some examples to illustrate our theoretical results.

math.DS↗