相互作用扩散的交换机制：框架、与切换的比较及一个精确可解示例
The Swapping Mechanism for Interacting Diffusions: Framework, Comparison with Switching, and an Exactly Solvable Example
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中文总结 AI 辅助
研究两个相互作用跳跃扩散过程的交换机制，开发严格框架，建立相关性质，通过对称恒等式联系交换与状态切换，对比两者差异，给出精确可解示例并分析其性质，揭示不对称性、扩散率修正及收敛率等。
中文摘要 AI 辅助
我们为两个相互作用的跳跃扩散过程的交换机制开发了一个严格框架：每个粒子独立演化，但它们的位置在随机时间交换，这为复制交换蒙特卡罗提供了连续时间描述。在温和假设下，我们建立了强存在性、路径唯一性以及转移密度的前向柯尔莫哥洛夫方程。一个对称恒等式将交换与更标准的状态切换联系起来：交换密度是状态切换扇区的重新标记和，所以这些过程的置换不变可观测量一致，而反对称的则不同。我们证明在细致平衡和可逆性下，交换过程不会降低谱隙，并且在标签不变性下，反对称隙相对于解耦过程有所改善。我们还表明，对于\(N\)个粒子，交换在\(\mathbb{R}^N\)上保持具有多项式矩闭包的马尔可夫性，而切换需要跟踪整个对称群，突出了一个基本的复杂性差异。我们通过给出两个具有恒定漂移和交换率的布朗运动的精确可解基准来结束这项工作。对于这个模型，我们根据修正贝塞尔函数、精确矩和二次相关性得到了封闭形式的转移密度。渐近分析揭示了持续的不对称性、有效的扩散率修正以及在快速交换极限（其中\(s>0\)是恒定交换率）下\(s^{-1/2}\)阶的收敛率。
英文摘要
We develop a rigorous framework for the swapping mechanism for two interacting jump-diffusion processes: each particle evolves independently but their positions are exchanged at random times, providing a continuous-time description of the replica-exchange Monte Carlo. Under mild hypotheses we establish strong existence, pathwise uniqueness, and a forward Kolmogorov equation for the transition density. A symmetrisation identity links swapping to the more standard regime-switching: swapping densities are relabelled sums of switching sectors, so permutation-invariant observables of these processes coincide, while antisymmetric ones differ. We prove that under detailed balance and reversibility the swapping process never degrades the spectral gap, and that, under label invariance, the antisymmetric gap improves compared to the decoupled process. We also show that, for $N$ particles, swapping remains Markov on $\mathbb{R}^N$ with polynomial moment closures, whereas switching requires tracking the full symmetric group, highlighting a fundamental complexity difference. We conclude the work presenting an exactly solvable benchmark of two Brownian motions with constant drifts and swap rate. For this model we obtain closed-form transition densities in terms of modified Bessel functions, exact moments, and two-time correlations. The asymptotic analysis reveals persistent asymmetries, an effective diffusivity correction, and a convergence rate of order $s^{-1/2}$ in the fast-swap limit (where $s>0$ is the constant swap rate).
