具有非均匀翻滚核的游走与翻滚方程的趋势至平衡态
Trend to equilibrium for run and tumble equations with non-uniform tumbling kernels
AI总结：
研究了细菌游走与翻滚模型在非均匀翻滚核下的长时间行为，针对依赖速度夹角和无界速度空间两种情况，证明其概率密度收敛至平衡态并给出显式收敛速率。
摘要：
我们研究游走与翻滚模型（run and tumble model）的长时间行为，该模型是一个描述细菌在化学刺激影响下运动的动理学-输运方程。实验表明，非均匀翻滚核具有物理相关性，而文献中广泛考虑的均匀翻滚核是为了降低数学分析的复杂性。我们考虑两种情况：(i) 翻滚核依赖于翻滚前后的速度夹角；(ii) 速度空间无界，且翻滚后速度服从 Maxwellian 速度分布。我们证明，对于任意概率测度初始数据，细菌的概率密度分布均收敛至平衡分布，并具有显式收敛速率（情况(i)为指数收敛，情况(ii)为代数收敛）。据我们所知，我们的结果是关于具有非均匀翻滚核的游走与翻滚方程长时间行为的首次结果。
英文摘要：
We study the long-time behaviour of a run and tumble model which is a kinetic-transport equation describing bacterial movement under the effect of a chemical stimulus. The experiments suggest that the non-uniform tumbling kernels are physically relevant ones as opposed to the uniform tumbling kernel which is widely considered in the literature to reduce the complexity of the mathematical analysis. We consider two cases: (i) the tumbling kernel depends on the angle between pre- and post-tumbling velocities, (ii) the velocity space is unbounded and the post-tumbling velocities follow the Maxwellian velocity distribution. We prove that the probability density distribution of bacteria converges to an equilibrium distribution with explicit (exponential for (i) and algebraic for (ii)) convergence rates, for any probability measure initial data. To the best of our knowledge, our results are the first results concerning the long-time behaviour of run and tumble equations with non-uniform tumbling kernels.
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