论文标题
关于汉密尔顿 - 雅各比 - 贝尔曼方程的良好性和运输噪声
On the well-posedness of a Hamilton-Jacobi-Bellman equation with transport noise
论文作者
论文摘要
在本文中，我们考虑以下非线性随机偏微分方程（SPDE）：\ begin {align*} \ begin {cases} \ Mathrm {d} u（s，x）= \ sum^n_ {i = 1} \ mathscr {l} _i u（s，s，s，x） u（s，x）\ vert^2 \ right）\ mathrm {d} s，\ quad＆\ text {in}（0，t）\ times \ times \ mathbb {t}^n， u（0，x）= u_0（x），＆\ text {on} \ mathbb {t}^n， \ end {cases} \ end {align*}其中$ \ mathbb {t}^n $是$ n $ dimensional torus，函数$ u_0，v：\ mathbb {t}^n \ to \ to \ to \ to \ to \ mathbb {r} $ and undress and undress and undress and undress and $ \ n rdise an colection i colection线性操作员。这可以看作是汉密尔顿 - 雅各比 - 贝尔曼方程，并在任何空间维度中具有运输噪声的问题。我们介绍了PDE领域的强大解决方案的概念，并确定了最大解决方案的存在和独特性（在停止时间之前强的解决方案）。此外，对于特定类别的$ \ {\ Mathscr {l} _i \} _ i $，我们建立了强大解决方案的全局良好性。证据依靠研究原始SPDE的相关截短版本，并显示其在强大解决方案类别中的全球范围。
In this paper we consider the following non-linear stochastic partial differential equation (SPDE): \begin{align*} \begin{cases} \mathrm{d}u(s,x)=\sum^n_{i=1} \mathscr{L}_i u(s,x)\circ \mathrm{d}W_i(s)+\left(V(x)+μΔu(s,x)-\frac{1}{2}\vert\nabla u(s,x)\vert^2\right)\mathrm{d}s, \quad &\text{in } (0,T)\times \mathbb{T}^n, u(0,x)=u_0(x), & \text{on } \mathbb{T}^n, \end{cases} \end{align*} where $\mathbb{T}^n$ is the $n$-dimensional torus, the functions $u_0, V: \mathbb{T}^n \to \mathbb{R}$ are given and $\{\mathscr{L}_i\}_i$ is a collection of first order linear operators. This can be seen as a Cauchy problem for a Hamilton-Jacobi-Bellman equation with transport noise in any space dimension. We introduce the concept of a strong solution from the realm of PDEs and establish the existence and uniqueness of maximal solutions (strong solutions upto a stopping time). Moreover, for a particular class of $\{\mathscr{L}_i\}_i$ we establish global well-posedness of strong solutions. The proof relies on studying an associated truncated version of the original SPDE and showing its global well-posedness in the class of strong solutions.
