论文标题
沿瓦斯恒星的衍生物沿密度的曲线
Derivative over Wasserstein spaces along curves of densities
论文作者
论文摘要
在本文中，如果在概率空间$（ω，\ Mathcal {f}，q）$上定义的任何随机变量$ξ$，我们专注于研究表单$ l \ mapsto f_q（l）的功能的衍生物：= f \ big（（lq）_接受的$ big（lq）_配置$ $ l \ in \ Mathcal {l}^q：= \ {l \ in l^1（ω，\ Mathcal {f}，q; \ m athbb {r} _+）：\ e^q [l] = 1 \} $ in $ $ \ MATHCAL {P}（\ Mathbb {r}^d）$ \ Mathbb {r}^d $上的概率定律，其Borel $σ$ -field $ \ field $ \ Mathcal $ \ Mathcal {b}（\ Mathbb {r}^d）$。上述形式的函数$ f_q $的可不同性问题的问题起源于对受控动力学仅承认弱解决方案的平均场控制问题的研究。受P.-L的启发狮子的结果[18]我们表明，如果给定$ l \ in \ Mathcal {l}^q $，$ l'\ mapsto f_ {lq} {lq}（l'）：\ Mathcal {lq}^{lq} {lq} {lq} \ rightArrow \ rightArrow \ rightArrow \ rightarl \ rythbb {r} $在$ l'= 1 $ n IS $，deriv is de deriv（ $ g：\ mathbb {r}^d \ rightArrow \ mathbb {r} $是一个borel函数，仅通过law $（lq）_ξ$取决于$（q，l，ξ）$。用$ \ partial_1f（（（lq）_ξ，x）：= g（x），\，x \ in \ mathbb {r}^d $，我们研究其属性，并将其与部分衍生物联系起来，最近在[6]中进行了调查，在[6]中进行了调查，以及在$ f $ f $限制的情况下， $ \ MATHCAL {P} _2（\ Mathbb {r}^d）$在P.-L.中是可区分的。狮子的感官和$（lq）_ξ\ in \ Mathcal {p} _2（\ Mathbb {r}^d）$，我们研究了衍生物与$ f_q（l）= f \ big（（lq）_配$ f $ fip $ f $的密度相对于$ f_q（l）= f _q（l）的密度与$ f $ fipbority的关系。我们的主要结果表明，$ \ partial_x \ partial_1f（（（lq）_ξ，x）= \partial_μf（（（（lq）_ _配给，x），\ x \ in \ mathbb {r}^d，$ where $ \partial_μf（（lq），x） $ f：\ MATHCAL {P} _2（\ Mathbb {r}^d）\ rightArrow \ Mathbb {r} $ at $（lq）_ξ$。
In this paper, given any random variable $ξ$ defined over a probability space $(Ω,\mathcal{F},Q)$, we focus on the study of the derivative of functions of the form $L\mapsto F_Q(L):=f\big((LQ)_ξ\big),$ defined over the convex cone of densities $L\in\mathcal{L}^Q:=\{ L\in L^1(Ω,\mathcal{F},Q;\mathbb{R}_+):\ E^Q[L]=1\}$ in $L^1(Ω,\mathcal{F},Q).$ Here $f$ is a function over the space $\mathcal{P}(\mathbb{R}^d)$ of probability laws over $\mathbb{R}^d$ endowed with its Borel $σ$-field $\mathcal{B}(\mathbb{R}^d)$. The problem of the differentiability of functions $F_Q$ of the above form has its origin in the study of mean-field control problems for which the controlled dynamics admit only weak solutions. Inspired by P.-L. Lions' results [18] we show that, if for given $L\in\mathcal{L}^Q$, $L'\mapsto F_{LQ}(L'):\mathcal{L}^{LQ}\rightarrow\mathbb{R}$ is differentiable at $L'=1$, the derivative is of the form $g(ξ)$, where $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is a Borel function which depends on $(Q,L,ξ)$ only through the law $(LQ)_ξ$. Denoting this derivative by $\partial_1F((LQ)_ξ,x):=g(x),\, x\in\mathbb{R}^d$, we study its properties, and we relate it to partial derivatives, recently investigated in [6], and, moreover, in the case when $f$ restricted to the 2-Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$ is differentiable in P.-L. Lions' sense and $(LQ)_ξ\in\mathcal{P}_2(\mathbb{R}^d)$, we investigate the relation between the derivative with respect to the density of $F_Q(L)=f\big((LQ)_ξ\big)$ and the derivative of $f$ with respect to the probability measure. Our main result here shows that $\partial_x\partial_1F((LQ)_ξ,x)=\partial_μf((LQ)_ξ,x),\ x\in \mathbb{R}^d,$ where $\partial_μf((LQ)_ξ,x)$ denotes the derivative of $f:\mathcal{P}_2(\mathbb{R}^d)\rightarrow \mathbb{R}$ at $(LQ)_ξ$.
