论文标题
具有可测量内核的McKean-Vlasov SDE的密度平滑度
Smoothness of the density for McKean-Vlasov SDEs with measurable kernel
论文作者
论文摘要
考虑McKean-Vlasov SDE $$ dx_t = \ langle b（x_t- \ cdot），μ_t\ rangle dt+dt+dw_t，\ quadμ_t= \ propatatorName {law}（law}（x_t），$ w $，其中$ w $是$ n $ n $ dimensional brownian Motion and $ brownian Motion and $ b：\ nath $ b：\ nathbb {可测量的功能。首先假设$ b \ in l^\ infty $，我们证明，相对于lebesgue度量，$ x_t $的法律$μ__t$具有密度$ p_t $，这是持续可区分的，梯度为$γ$-Hölder的$γ$-Hölder连续$γ\ in（0,1,1）$。进一步假设$ b \ in \ Mathcal {c} _b^1 $，我们证明密度$ p_t $是无限差异的。在通过噪声角度的正则化中，这表明McKean-Vlasov SDES在系数的规律性假设下，麦基恩 - 维拉索夫SDE往往比没有密度依赖性的SDE具有更平滑的密度函数。我们观察到了满足Krylov的可集成性条件的单数相互作用内核的类似现象，用于分配内核$ b \ in B _ {\ infty，\ infty，\ infty}^α$，$α\ in（-1,0）$，以及由$α$ stable nisival of a $α$ stable的噪声，以及$α$ stable的噪声。
Consider the McKean-Vlasov SDE $$ dX_t=\langle b(X_t-\cdot),μ_t\rangle dt+dW_t,\quad μ_t=\operatorname{Law}(X_t), $$ where $W$ is the $n$-dimensional Brownian motion and $b:\mathbb{R}^d\to\mathbb{R}^d$ is a measurable function. First assuming $b\in L^\infty$, we prove that the law $μ_t$ of $X_t$ has a density $p_t$ with respect to the Lebesgue measure, which is continuously differentiable with gradient being $γ$-Hölder continuous for each $γ\in(0,1)$. Assume further that $b\in \mathcal{C}_b^1$, we prove that the density $p_t$ is infinitely differentiable. In the regularization by noise perspective, this shows McKean-Vlasov SDEs tend to have a smoother density function than SDEs without density dependence, under the same regularity assumption of the coefficients. We observe similar phenomenon for singular interaction kernels satisfying Krylov's integrability condition, for distributional kernels $b\in B_{\infty,\infty}^α$, $α\in(-1,0)$, and for processes driven by an $α$-stable noise for $α\in(1,2)$.
