论文标题
具有量度作为初始数据和McKean-Vlasov方程的非线性Fokker-Planck方程的解决方案
Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations
论文作者
论文摘要
一个人证明了非线性fokker-planck方程（fpe）\ begin \ begin {align*}＆u_t-δ（β（u））+{\ mathrm {div}}（D（d div}}）（d（x）b（x）b（x）= 0， x \ in \ Mathbb {r}^d，\ d \ ne2，\\＆u（0，\ cdot）= u_0，\ mbox {in} \ mathbb {r}^d，\ end \ end \ end {align {align {align*} C^2（\ Mathbb {r}）$是一个无折叠功能，$ b \ in C^1 $，有限，$ b \ ge0 $，$ d \ in {l^\ infty}（\ Mathbb {r}^d; l^2（\ mathbb {r}^d）+l^\ infty（\ mathbb {r}^d），$ at $ {（{\ rm div} \，d）^ - } \ in L^\ infty（\ mathbb {r}^d）$，$β$，如果$ bigrantimed in l^\ infty（\ mathbb {rmathbb {rmathbb {rmathbb {rmathbb）此外，$ t \ to u（t，u_0）$是$ l^1（\ mathbb {r}^d）$中的收缩半群，它使不变的概率密度函数集中在$ \ mathbb {r}^d $中。如果$ {\ rm div} \，d \ ge0 $，$β'（r）\ ge a | r |^{α-1} $，以及$ |β（r）| \ le c r^α$，$α\ ge1，$α\ ge1，$ d \ ge3 $，然后d {d+（α-1）d}} \ | U_0 |^{\ frac2 {2+（m-1）d}}，$ $ $ t> 0 $，以及，如果$ d \ in l^2（\ Mathbb {r}^d; \ m} $ \ Mathbb {r}^d $中有界度量的$ \ MATHCAL {M} _b $。结果是任意初始定律的结果，我们获得了一类McKean-Vlasov SDE的解决方案，其系数对时间边际定律具有单一的依赖性。
One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker-Planck equation (FPE) \begin{align*} &u_t-Δ(β(u))+{\mathrm{ div}}(D(x)b(u)u)=0, \quad t\geq0,\ x\in\mathbb{R}^d,\ d\ne2, \\ &u(0,\cdot)=u_0,\mbox{in }\mathbb{R}^d, \end{align*} where $u_0\in L^1(\mathbb{R}^d)$, $β\in C^2(\mathbb{R})$ is a nondecreasing function, $b\in C^1$, bounded, $b\ge0$, $D\in {L^\infty}(\mathbb{R}^d;\mathbb{R}^d)$, ${\rm div}\,D\in L^2(\mathbb{R}^d)+L^\infty(\mathbb{R}^d),$ with ${({\rm div}\, D)^-}\in L^\infty(\mathbb{R}^d)$, $β$ strictly increasing, if $b$ is not constant. Moreover, $t\to u(t,u_0)$ is a semigroup of contractions in $L^1(\mathbb{R}^d)$, which leaves invariant the set of probability density functions in $\mathbb{R}^d$. If ${\rm div}\,D\ge0$, $β'(r)\ge a|r|^{α-1}$, and $|β(r)|\le C r^α$, $α\ge1,$ $d\ge3$, then $|u(t)|_{L^\infty}\le Ct^{-\frac d{d+(α-1)d}}\ |u_0|^{\frac2{2+(m-1)d}},$ $t>0$, and, if $D\in L^2(\mathbb{R}^d;\mathbb{R}^d)$, the existence extends to initial data $u_0$ in the space $\mathcal{M}_b$ of bounded measures in $\mathbb{R}^d$. As a consequence for arbitrary initial laws, we obtain weak solutions to a class of McKean-Vlasov SDEs with coefficients which have singular dependence on the time marginal laws.
