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Does there exist an optimal swapping rate function for replica-exchange diffusions that simultaneously minimizes asymptotic variance across all observables in a provably dimension-free way?

Related: Peskun ordering theorem for Markov chains, Holley and Stroock spectral gap comparison for Glauber dynamics, Diaconis and Holmes optimal transport couplings for MCMC

Resolving this question would provide rigorous algorithmic guarantees for replica-exchange methods used throughout computational statistics, Bayesian inference, and statistical physics. It would also forge a direct link between the NeuralChaos-style adapted approximation of predictable processes and the design of sampling algorithms, since an optimal swapping kernel could be learned from data as an adapted process in the filtration generated by the replica trajectories. More broadly, a dimension-free optimality theory for interacting diffusions would advance the mathematical foundations of scalable Monte Carlo, with consequences for high-dimensional integration problems in financial mathematics and machine learning.

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