The mathematics of systems that evolve under randomness, modeled as families of random variables indexed by time — from Markov chains and martingales to Brownian motion and stochastic differential equations. The field spans stochastic analysis, filtering and optimal control under uncertainty, ergodic behavior of random dynamics, and concentration of measure, with models reaching across physics, engineering, finance, and biology.
Missions
🏆Completed
Captain: tianyipeng
Markov Entanglement: Value Decomposition Error in Multi-agent MDPsResearch Paper
Value decomposition — approximating the value of a joint state by a sum of per-agent local values — is a staple of multi-agent dynamic programming and reinforcement learning, from index policies for restless bandits to modern MARL architectures, yet it is normally used without justification. Chen and Peng (arXiv:2506.02385) supply one. They show a multi-agent MDP admits an exact value decomposition precisely when its transition matrix is not *entangled* — a notion built in direct analogy with quantum entanglement — and then turn that qualitative characterisation into a quantitative one: a measure of Markov entanglement bounds the decomposition error in general. This mission formalizes that core theory. The goal is Theorem 6, the general N-agent bound in the occupancy-weighted norm; the milestones are the equivalence between separability and exact decomposition, the perturbation machinery that carries a one-step transition error into a value-function error, and the extensions to shared global state and shared rewards. The paper's restless-bandit application, which needs mean-field machinery of its own, is left to a second mission in the series.
25 thms3 active usersReviewed
🏆Completed
Captain: tianyipeng
Markov Entanglement: Decomposition Error via Agent-wise TV DistanceResearch Paper
Multi-agent reinforcement learning approximates a global value function by summing per-agent local value functions learned independently — a trick that works surprisingly well in practice (ride-hailing dispatch, restless bandits) but had no general theoretical justification. Chen and Peng (arXiv:2506.02385) explain why: they define a *Markov entanglement* measure for the joint transition dynamics of a multi-agent MDP, directly analogous to quantum entanglement of a two-party state, and show it controls exactly how much error this value-decomposition trick incurs. This mission formalizes their sharpest quantitative bound (Theorem 4): the error of decomposing the global Q-function into per-agent local Q-functions is controlled, entrywise, by the agent-wise total-variation measure of Markov entanglement.
4 thms3 active usersReviewed
🏆Completed
Captain: wenxinzhang
Single-Server Queueing Convergence via Forward CouplingTextbook
Formalize sample-path stability for a continuous-time, unit-rate, infinite-buffer single-server queue. Starting from cumulative arriving service work, define the reflected transient workload, the workload constructed from the infinite past, long-run offered load, and two-time stationarity. Prove that subcritical load forces finite-time coupling and consequently that every finite initial workload converges in its two-time finite-dimensional distributions to the stationary workload law.