On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model

In this work, we first prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in <i>ℝ</i>^<i>d</i> for <i>d</i>≥3. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Second, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in <i>ℝ</i>^<i>d</i>. We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.

本文首先证明了d≥3维欧几里得空间（ℝᵈ）中，带奇异相互作用核的麦克凯恩-弗拉索夫（McKean-Vlasov）随机微分方程对应的非线性鞅问题的适定性。本研究设定的独特之处在于，所研究的麦克凯恩-弗拉索夫过程在每一时刻，均通过奇异时空核与该过程所有过往时刻的边缘分布发生相互作用。其次，我们证明了该随机过程可作为d维欧几里得空间中抛物-抛物型凯勒-塞格尔（Keller-Segel）系统的概率解释。据此，我们在模型参数满足显式小性条件的前提下，得到了该系统的适定性结果。