research

2026

1. SDE & SPDE

   Connecting Orbits in Cooperative McKean-Vlasov SDEs

   Chunlin Liu, Baoyou Qu, Jinxiang Yao, and Yanpeng Zhi

   2026

   Abs arXiv

   In this work we extend the framework of monotone dynamical systems to a broad and important class of stochastic equations, namely cooperative McKean-Vlasov SDEs with multiplicative noise. Under a locally dissipative assumption, our main theorem establishes the existence of multiple order-related invariant measures in the the Wasserstein space together with monotone connecting orbits (heteroclinic orbits) between them, with respect to the stochastic order. The presence of such connecting orbits also reveals the unstable nature of those invariant measures appearing as their backward limits, a dynamical feature that has remained largely unexplored in stochastic equations. The framework applies to a wide range of classical models, including granular media equations in double-well and multi-well confining potentials with quadratic interaction, perturbed double-well landscapes, and interacting multi-species population models. Our method is based on building a monotone dynamical system that preserves the stochastic order, achieved through a cone compatible with this order and an extension of the classical Dancer-Hess connecting orbit theorem.

2. SDE & SPDE

   Global Dynamics of Granular Media Equations via Stochastic Order

   Baoyou Qu, Jinxiang Yao, and Yanpeng Zhi

   2026

   Abs arXiv

   This paper studies the rich dynamics of one-dimensional granular media equations with attractive quadratic interactions. Building on the monotone dynamical systems framework developed in an earlier work, we allow for multiplicative noise, in contrast to most existing results restricted to additive noise. Within this framework, we show that, in the one-dimensional setting, invariant measures are totally ordered with respect to the stochastic order. The basins of attraction of the minimal and maximal invariant measures contain unbounded open sets in the 2-Wasserstein space, which is vacant in previous research even for additive noises. Also, our main results address the global convergence to the order interval enclosed by the minimal and maximal invariant measures, and an alternating arrangement of invariant measures in terms of stability (locally attracting) and instability (as the backward limit of a connecting orbit). Our theorems cover a wide range of classical granular media equations, such as double-well and multi-well landscapes. Specific values for the parameter ranges, explicit descriptions of attracting sets and phase diagrams are provided.

3. SDE & SPDE

   Weak Coupling Limit, Triviality and Renormalized Solution of Elliptic Phi_4^4 Model with Explosive Boundary Condition

   Chunrong Feng, Huaizhong Zhao, and Yanpeng Zhi

   2026

   In preparation

2025

1. SO & SC

   Zero-Sum Linear-Quadratic Stochastic Ergodic Differential Games

   Jingrui Sun and Yanpeng Zhi

   2025

   Submitted
2. SO & SC

   Multi-Armed Bandit Approaches for Location Planning with Dynamic Relief Supplies Allocation Under Disaster Uncertainty

   Jun Liang, Zongjia Zhang, and Yanpeng Zhi

   Smart Cities, 2025

   Abs DOI

   Natural disasters (e.g., floods, earthquakes) significantly impact citizens, economies, and the environment worldwide. Due to their sudden onset, devastating effects, and high uncertainty, it is crucial for emergency departments to take swift action to minimize losses. Among these actions, planning the locations of relief supply distribution centers and dynamically allocating supplies is paramount, as governments must prioritize citizens’ safety and basic living needs following disasters. To address this challenge, this paper develops a three-layer emergency logistics network to manage the flow of emergency materials, from warehouses to transfer stations to disaster sites. A bi-objective, multi-period stochastic integer programming model is proposed to solve the emergency location, distribution, and allocation problem under uncertainty, focusing on three key decisions: transfer station selection, upstream emergency material distribution, and downstream emergency material allocation. We introduce a multi-armed bandit algorithm, named the Geometric Greedy algorithm, to optimize transfer station planning while accounting for subsequent dynamic relief supply distribution and allocation in a stochastic environment. The new algorithm is compared with two widely used multi-armed bandit algorithms: the ϵ-Greedy algorithm and the Upper Confidence Bound (UCB) algorithm. A case study in the Futian District of Shenzhen, China, demonstrates the practicality of our model and algorithms. The results show that the Geometric Greedy algorithm excels in both computational efficiency and convergence stability. This research offers valuable guidelines for emergency departments in optimizing the layout and flow of emergency logistics networks.