概率论：2025–2026 年度综述

An annual survey of the math.PR arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 144 papers and organised into 40 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Stochastic Differential Equations and Numerical Methods (15.7%) — Focuses on the analysis, discretization, and machine-learning-assisted solution of stochastic differential equations and related dynamical systems.
• Markov Chains and Convergence Analysis (12.8%) — Investigates mixing times, convergence rates, and structural properties of finite and infinite Markov chains.
• Random Matrices and Random Variables (11.7%) — Studies spectral properties, universality classes, and distributional limits of random matrices and discrete random variables.
• Random Graphs and Percolation Theory (9.3%) — Examines phase transitions, connectivity, and geometric structures in sparse and inhomogeneous random graph models.
• Point Processes and Gaussian Fields (9.1%) — Analyzes branching processes, Gaussian processes, and point process interactions in continuous and discrete spaces.
• Optimal Transport and Stochastic Control (8.7%) — Develops theoretical frameworks for optimal transport problems, martingale couplings, and stochastic control strategies.
• Mean Field Games and Interacting Systems (7.6%) — Explores mean-field limits, spin glass theory, and collective dynamics in large interacting particle systems.
• Ising Models and Phase Transitions (7.1%) — Investigates critical phenomena, free energy asymptotics, and phase transitions in lattice spin models.
• Contact Processes and First Passage Times (6.0%) — Focuses on survival probabilities, boundary crossing, and first passage properties of contact and diffusion processes.
• Central Limit Theorems and Asymptotics (5.2%) — Establishes limit theorems and asymptotic behaviors for high-dimensional and non-linear stochastic functionals.
• Brownian Motion and Fractional Processes (4.0%) — Studies geometric properties, absorption times, and fractal dimensions of Brownian and fractional Brownian motions.
• Random Walks on Groups and Trees (2.8%) — Analyzes self-reinforcing, self-avoiding, and branching random walks on trees, groups, and lattices.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.