论文标题
在多维McKean-Vlasov SDE上，具有与抛物线抛物线凯勒 - 塞格模型相关的纪念和奇异互动
On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model
论文作者
论文摘要
在这项工作中，我们首先证明了与McKean-vlasov随机微分方程有关的非线性martingale问题的良好性，并在$ \ mathbb {r}^d $ for $ d \ geq 3 $中具有单数相互作用内核。我们设置的特殊性是，我们研究的McKean-Vlasov过程每次都通过奇异的时空内核进行互动及其过去的所有时间边际定律。其次，我们证明我们的随机过程是$ \ Mathbb {r}^d $中的抛物线抛物线keller-segel系统的一种概率解释。因此，我们在模型参数的明确小条件下为后者获得了良好的结果。
In this work we firstly prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in $\mathbb{R}^d$ for $d\geq 3$. 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. Secondly, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in $\mathbb{R}^d$. We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.
