Conclusions for Hybrid Jump Diffusions and Related Problems
The "Workshop on Hybrid Jump-Diffusion Systems and Related Problems" was successfully held at the Kunming Tianyuan Mathematics International Center (TYMRC) from October 5th to 11th, 2025. The workshop focused on cutting-edge topics in stochastic differential equations, stochastic integration, and financial mathematics, covering several active research areas such as functional inequalities, heat kernel estimates, and backward stochastic differential equations.
The program featured 36 presentations by renowned probability theorists, including Chen Zhenqing, Xiang Kainan, Mao Xuerong, Wu Fuke, He Hui, and Li Juan, among other distinguished domestic and international scholars. Participants not only presented their latest research findings but also proposed numerous open problems, thereby helping to outline promising directions for future work.
A notable feature of the schedule was the dedicated time slots for informal discussion, arranged each afternoon and evening. These sessions effectively fostered in-depth communication among the participants and led to several concrete plans for future collaboration on topics of shared interest.
Conclusions of the Lectures in Oct 6th
Professor Kainan Xiang from Xiangtan university describe the following probabilistic approach to the Jacobian conjecture, which is a famous conjecture listed as one of 18 mathematical problems
for the 21st century by Steve Smale in 1998. This report offers a profound analysis of the deep-seated relationships between probability, combinatorics, and algebra. Professor Jinghai Shao at the Tianjing University considers stochastic functional differential equations with Markovian regime-switching on an countable state space, develop a truncation method based on a new construction of order preserving coupling for Markov chains to generalize the estimation of exponential functionals for Markov chains with infinite states. Professor Shihu Li from Jiangsu Normal University studies mean field stochastic partial differential equations with nonlinear kernels, establish the convergence of the empirical laws of interacting systems to the
law of solutions of mean field equations. Wujun Lv from Donghua university considers continuous-time observations of the Fourier coefficients of the solutions, investigates the
natural time discretization of the LSE and establish its weak consistency and asymptotic normality
under certain assumptions. Professor Yingqiu Li from the Changsha TechnologyUniversity gives the Yaglom-type theorem of Mandelbrot cascade in random environment, establishes the Berry-Essen boundary of the CLT of the model. Professor Xian Chen from Xiamen university studies the risk-sensitive average reward criterion for discrete-time Markov control processes. Developing a new approach to obtain a variational formula for the risk-sensitive average reward criterion without the compactness condition on the state space in the existing literature. Hua Zhang from Jiangxi finance and economic University gives the Varadhan Estimates for the Densities of Wiener Poisson Functionals and Applications, which focus on the advance of the theorey of Poisson Functionals. Fenfen Yang of Shanghai University establish the existence and uniqueness of the solution for distribution dependent SDEs with critical drift coefficients by utilizing fixed point theorem and Krylov's estimate, which generalize those results for distribution dependent SDEs with singular coefficients to critical conditions.
Conclusions of the Lectures in Oct 7th
Professor Zhenqing Chen at the University of Washington presents quantitative homogenization results for stable-like long range random walks in time-dependent random conductance models, where the conductance is bounded above but can be degenerate. Professor Hui He from Beijing Normal University investigates reaction–diffusion equations whose first-order spatial term includes a random drift. The behavior of the solution’s leading edge is analyzed under appropriate conditions. Professor Bingchang Wang of Shandong University reports the“decoupled forward–backward equation”method, overcoming the difficulties of solving fixed-point equations, and established conditions for consistency between decentralized asymptotically optimal strategies and mean-field control systems, with applications in economics and power systems. Shuaiqi Zhang from China University of Mining and Technology presents stochastic control problems for systems driven by sub-diffusions, establishing the existence and uniqueness of fully coupled FBSDEs, and investigating the stochastic maximum principle and dynamic programming principle, highlighting the system’s combined deterministic and stochastic characteristics. Professor Xiaobin Sun from Jiangsu Normal University studies a class of time-inhomogeneous SDEs with slow and fast time-scales, proving strong or weak convergence of the slow component under different conditions and characterizing the limiting process via the martingale problem approach. Linlin Tian from Donghua University presented the optimal investment problem for a renewal risk model with observable Erlang-distributed interarrival times, establishing the concavity of the value function via the HJB equation and deriving an explicit expression for the optimal investment strategy. Professor Xinghu Jin from Hefei University of Technology presented the Euler–Maruyama discretization for $\mathbb{R}^d$-valued ergodic stochastic differential equations with Markovian switching, establishing quantitative error bounds between the original process and its discrete approximation under a specially designed metric using the Lindeberg principle.
Conclusions of the Lectures in Oct 8th
Professor Fuke Wu at the Huazhong University of Science and Technology focuses on systems of singularly perturbed forward-backward stochastic differential equations (FBSDEs) and control problems. These results provide insights into the convergence rate and extend existing results on the averaging principles for such stochastic control problems. Professor Fubao Xi from Beijing Institute of Technology considers a class of switching diffusion systems consisting of continuous and discrete components, in which the switching rates of discrete component depend on the value of the continuous component involving past history. Peisen Li of Beijing Institute of Technology study a stochastic Lotka–Volterra model, establishes sharp sufficient conditions for uniform ergodicity in total variation. Bin Qian from Changshu Institute of Technology presents reports the Bismut formula, which is obtained by Mallivan calculus and coupling. Moreover, he also
give Poincare inequality, Log-Sobolev inequality and Wang-Harnack inequality for the
associated semigroup P_t. Professor Dejun Luo from Chinese Academy of Sciences show that under diffusive rescaling, the fluctuations of the density converge to a Gaussian limit, described by an additive stochastic heat equation. Professor Longmin Wang from Nankai University
exhibited a relatively hyperbolic group with convergent Poincar\'e series generated by H_r(n).
This quantity appears naturally when studying asymptotic properties of branching random walks driven by \mu on \Gamma. Hailing Dong from Shenzhen University focused on a class of highly nonlinear stochastic differential delay equations (SDDEs) driven by Lévy noise and Markovian chain, where the drift and diffusion coefficients satisfy more general polynomial growth condition.
The key aim of her research is to investigate the stabilization problem by delay feedback controls.
Wei Liu from Shanghai Normal University study on this topic for a class of SDEs with periodic coefficient is reported. His work prompts to the case of non-autonomous SDEs, which bring technical challenges including time-inhomogeneity and periodicity.
Conclusions of the Lectures in Oct 9th
Professor Xuerong Mao from the University of Stratchclyde aims to determine whether or not a stochastic state feedback control can make a given nonlinear hybrid differential equation, which is not stable in distribution, to become stable in distribution. Professor Huacheng Zhou from Central
South Universitye study the event-triggered finite-dimensional output feedback stabilization for a onedimensional stochastic heat equation. He further obtain that the closed-loop system achieves almost surely exponential stability. Jing Wu from Sun Yet-sen Universityo discuss via the probabilistic approach the relations between the viscosity solution and the distribution solution to PDEs with Neumann boundary conditions. Professor Guohuan Zhao from Chinese Academy of Sciences discusses recent results on the well-posedness of McKean–Vlasov equations with
critical interaction kernels, motivated by the two-dimensional vorticity formulation of the Navier–
Stokes equations. Defei Zhang from the College of Red River proposed a new stochastic epidemic model driven by G-Brownian motion. He shows a unique positive solution in the sense of quasi surely. Xuekang Zhang from Anhui Polytechnic University discribes the asymptotic behavior of the improved trajectory fitting estimation for linear self-interacting diffusions, which includes a variety of deep results. Xiaowen Zhou from Concordia University solve the exit problem to oscillating Brownian Motion, and then adopt a perturbation approach to find an expression of potential measure for oscillating Brownian Motion. The transition density is found by inverting the Laplace transform. Lidan Wang from Nankai University consider a large class of anisotropic Markov processes, which, in contrast with isotropic Markov processes, only jump along the coordinate directions.
Conclusions of the Lectures in Oct 10th
Professor Juan Li from Shandong University study of Peng’s stochastic maximum principle (SMP) for a stochastic control problem composed of a controlled forward stochastic differential equation (SDE) as dynamics and a controlled backward SDE which defines the cost functional. Professor Xiaoyue Li from Tiangong University reports some results on the dynamical behaviors of McKean-Vlasov stochastic differential equations (SDEs) with common noise whose coefficients depend on both the state and the measure. Obtaining the existence and uniqueness of the invariant measure for McKean-Vlasov SDEs with common noise whose drift and diffusion coefficients grow polynomially. Wei Liu from University of Liverpool studies on this topic for a class of SDEs with periodic coefficient is reported. Compared with existing fruitful results of this topic, our work is devoted to the case of non-autonomous SDEs, which bring technical challenges including time-inhomogeneity and periodicity. Professor Zhenzhong Zhang from Donghua University consider the Dirichlet problem for weakly coupled non-local operators with Neumann boundary conditions. Under some conditions, we prove that the existence and uniqueness for a class exterior Dirichlet problem with Neumann boundary conditions. Besides, Some sufficient conditions for the maximum principle, which presents the advance in the related field.
Acknowledgements
We acknowledge the staff for Tianyuan Mathematics International Center. Thank you all for the time and energy for this workshop.