扩散渐近下基于个体的细菌趋化模型
Individual-based models for bacterial chemotaxis in the diffusion asymptotics
AI总结：
本文利用概率方法证明具有内部状态的细菌趋化速度跳跃模型在扩散极限下收敛到平流扩散过程，并提出保持该极限的时间离散化技术。
摘要：
我们讨论了具有内部状态的细菌趋化速度跳跃模型，该内部状态允许速度跳跃率依赖于细菌运动路径上趋化剂浓度的记忆。利用概率技术，我们提供了一个路径结果，表明所考虑的过程在（长时间）扩散极限下收敛到平流扩散过程。我们还使用相同的方法（重新）证明，对于一种相关的、更简单的直接感知趋化剂梯度的过程，会产生相同的极限方程。此外，我们提出了一种时间离散化技术，该技术精确地保留了这些扩散极限，即没有依赖于时间离散化的误差。在配套论文\cite{variance}中，这些结果被用于构建一种耦合技术，该技术允许对具有内部状态的过程进行数值模拟，并实现渐近方差减少，即方差在扩散极限下消失。
英文摘要：
We discuss velocity-jump models for chemotaxis of bacteria with an internal state that allows the velocity jump rate to depend on the memory of the chemoattractant concentration along their path of motion. Using probabilistic techniques, we provide a pathwise result that shows that the considered process converges to an advection-diffusion process in the (long-time) diffusion limit. We also (re-)prove using the same approach that the same limiting equation arises for a related, simpler process with direct sensing of the chemoattractant gradient. Additionally, we propose a time discretization technique that retains these diffusion limits exactly, i.e., without error that depends on the time discretization. In the companion paper \cite{variance}, these results are used to construct a coupling technique that allows numerical simulation of the process with internal state with asymptotic variance reduction, in the sense that the variance vanishes in the diffusion limit.
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