论文标题
通过相对熵求解McKean-Vlasov SDE
Solving McKean-Vlasov SDEs via relative entropy
论文作者
论文摘要
在本文中，我们探讨了相对熵在证明McKean-Vlasov SDES和SPDE的较弱的相对熵的优点，从而扩展了缺少ARXIV中引入的技术：2105.02983。在SDE设置中，当相互作用取决于路径并且仅假定具有线性生长时，我们证明存在弱的存在和独特性。同时，当交互作用具有Krylov的$ l_t^q-l_x^p $ type singularity时，我们恢复并扩展了当前结果。我们将传统上不同的两个案例连接起来，并形成了一种解决方案理论，该理论足够强大，可以在奇异性的存在下驱动sublerear的增长，从而产生了麦基恩 - 维拉索夫SDE的新家族的良好性。我们的策略自然会扩展到所有$ h \ in \ weft（0,1 \ right）$的分数布朗驾驶噪声$ b^h $的情况，在每个单独的情况下，在\ left \ left（0，\ frac {1} {1} {2} {2} {2} \ right）$和$ h \ in \ weft（1）$（\ frac}的每个单独情况下，获得了新的结果。在SPDE设置中，我们从空间尺寸一号的随机热方程的原型中构造了McKean-Vlasov型SPDE，并具有有界的可测量系数，并且我们对随机波方程式进行相同的结构，并且具有仅在边界上作用的白噪声的SPDE。此外，我们将SDE的混乱结果的一些定量传播推广到SPDE设置中。
In this paper we explore the merit of relative entropy in proving weak well-posedness of McKean-Vlasov SDEs and SPDEs, extending the technique introduced in Lacker arxiv:2105.02983. In the SDE setting, we prove weak existence and uniqueness when the interaction is path dependent and only assumed to have linear growth. Meanwhile, we recover and extend the current results when the interaction has Krylov's $L_t^q-L_x^p$ type singularity for $\frac{d}{p}+\frac{2}{q}<1$, where $d$ is the dimension of space. We connect the aforementioned two cases which are traditionally disparate, and form a solution theory that is sufficiently robust to allow perturbations of sublinear growth at the presence of singularity, giving rise to the well-posedness of a new family of McKean-Vlasov SDEs. Our strategy naturally extends to the cases of a fractional Brownian driving noise $B^H$ for all $H\in\left(0,1\right)$, obtaining new results in each separate case $H\in\left(0,\frac{1}{2}\right)$ and $H\in\left(\frac{1}{2},1\right)$. In the SPDE setting, we construct McKean-Vlasov type SPDEs with bounded measurable coefficients from the prototype of stochastic heat equation in spatial dimension one, and we do the same construction for the stochastic wave equation and a SPDE with white noise acting only on the boundary. In addition, we generalize some quantitative propagation of chaos results for SDEs into the SPDE setting.
