论文标题
非线性integro-differention方程的存在，drift in $ \ m athbb {r}^d $
Existence-Uniqueness for nonlinear integro-differential equations with drift in $\mathbb{R}^d$
论文作者
论文摘要
在本文中，我们考虑了$ \ inf_ \ inf_ {τ\ in \ mathcal {t}} \ bigG \ \ \ {\ int {\ int {\ mathbb {r}^d} （u（x+y）+u（x-y）-2u（x））\ frac {k_τ（x，y）} {| y |^{d+2s}}} \，dy+b_τ（x）\ cdot \ cdot \ cdot \ nabla U（x） \ hspace {2mm} \ mathbb {r}^d，$$其中$ 0<λ（2-2S）\ leqk_τ\ leqleqλ（2-2s）$，$ s \ in（\ frac {1} {1} {2} {2} {2}，1）$。上面的方程式出现在$ \ mathbb {r}^d $中的Ergodic控制问题的研究中，当受控动力学受纯jumpLévy流程的控制时，由内核$k_τ\，| y | y |^{ - d-2-s} $和drift $b_τ$。在寄养lyapunov条件下，我们建立了满足上述方程式的唯一解决方案对$（u，λ^*）$，只要我们设置$ u（0）= 0 $。然后将结果扩展以覆盖混合局部非局部类型的HJB方程，这显着改善了[Arapostathis-Caffarelli-Pang-Zheng（2019）]的结果。
In this article we consider a class of nonlinear integro-differential equations of the form $$\inf_{τ\in\mathcal{T}} \bigg\{\int_{\mathbb{R}^d} (u(x+y)+u(x-y)-2u(x))\frac{k_τ(x,y)}{|y|^{d+2s}} \,dy+ b_τ(x) \cdot \nabla u(x)+g_τ(x) \bigg\}-λ^*=0\quad \text{in} \hspace{2mm} \mathbb{R}^d,$$ where $0<λ(2-2s)\leq k_τ\leq Λ(2-2s)$ , $s\in (\frac{1}{2},1)$. The above equation appears in the study of ergodic control problems in $\mathbb{R}^d$ when the controlled dynamics is governed by pure-jump Lévy processes characterized by the kernels $k_τ\,|y|^{-d-2s}$ and the drift $b_τ$. Under a Foster-Lyapunov condition, we establish the existence of a unique solution pair $(u, λ^*)$ satisfying the above equation, provided we set $u(0)=0$. Results are then extended to cover the HJB equations of mixed local-nonlocal type and this significantly improves the results in [Arapostathis-Caffarelli-Pang-Zheng (2019)].
