论文标题
非线性fokker-planck方程和麦基恩 - 维拉索夫sdes的唯一性：退化案例
Uniqueness for nonlinear Fokker-Planck equations and for McKean-Vlasov SDEs: The degenerate case
论文作者
论文摘要
这项工作与（可能是堕落的）fokker-planck方程$ρ_t_t-Δβ（ρ）+{\ rm div}（db（ρ）= 0 $ in $ in $ in $（0，\ iftty）\ times \ nims \ ntimbb c^d， ρ_0（x）$。在$β：\ mathbb {r} \ to \ mathbb {r}，\，\，\，\，b：\ mathbb {r} \ to \ mathbb {r} $和$ d：\ mathbb {r MathBb {r}^d \ to \ to \ to \ toreeq w flof \ ge1下， $ρ（t）= s（t）ρ_0：[0，\ infty）\ to l^1（\ mathbb {r}^d）$作为非线性半群理论意义上的温和解决方案。该流在$ l^\ infty（（（0，t）\ times \ times \ times \ mathbb {r}^d）\ cap l^1（（（（0，t）\ times \ times \ mathbb {r}^d），$ $ \ $ $ \ forall t> 0 $，schwartz分布解决方案$（0，\ intip）\ ntiles）\ n $此外，对于$ρ_0\ in L^1（\ Mathbb {r}^d）\ Cap H^{ - 1}（\ Mathbb {r}^d）$，$ t \ to s（t）ρ_0$与$ [0，\ infty）$ in $ h^n n n n n n n o $ in $ [0，\ 1} $ n nord and py in $ [0，\ ind}（$ nord）。作为主要应用程序，证明了相应的麦克基 - 维拉索夫SDES的弱独特性。
This work is concerned with the existence and uniqueness of generalized (mild or distributional) solutions to (possibly degenerate) Fokker-Planck equations $ρ_t-Δβ(ρ)+{\rm div}(Db(ρ)ρ)=0$ in $(0,\infty)\times\mathbb{R}^d,$ $ρ(0,x) \equiv ρ_0(x)$. Under suitable assumptions on $β:\mathbb{R}\to\mathbb{R},\,b:\mathbb{R}\to\mathbb{R}$ and $D:\mathbb{R}^d\to\mathbb{R}^d$, $d\ge1$, this equation generates a unique flow $ρ(t)=S(t)ρ_0:[0,\infty)\to L^1(\mathbb{R}^d)$ as a mild solution in the sense of nonlinear semigroup theory. This flow is also unique in the class of $L^\infty((0,T)\times\mathbb{R}^d)\cap L^1((0,T)\times\mathbb{R}^d),$ $\forall T>0$, Schwartz distributional solutions on $(0,\infty)\times\mathbb{R}^d$. Moreover, for $ρ_0\in L^1(\mathbb{R}^d)\cap H^{-1}(\mathbb{R}^d)$, $t\to S(t)ρ_0$ is differentiable from the right on $[0,\infty)$ in $H^{-1}(\mathbb{R}^d)$-norm. As a main application, the weak uniqueness of the corresponding McKean-Vlasov SDEs is proven.
