Existence-Uniqueness for nonlinear integro-differential equations with drift in ${\mathbb{R}}^d$

In this article we consider a class of nonlinear integro-differential equations of the form $$\underset{\tau \in \mathcal{T}}{\inf }\{{\int }_{{\mathbb{R}}^d}(u(x+y)+u(x-y)-2u(x))\frac{k_{\tau }(x,y)}{\mathrm{\mid }y{\mathrm{\mid }}^{d+2s}}dy+b_{\tau }(x)\cdot \mathrm{\nabla }u(x)+g_{\tau }(x)\}-{\lambda }^{\ast }=0\text{in}{\mathbb{R}}^d,$$ where $0<\lambda (2-2s)\le k_{\tau }\le \mathrm{\Lambda }(2-2s)$ , $s\in (\frac{1}{2},1)$. 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 $k_{\tau }\mathrm{\mid }y{\mathrm{\mid }}^{-d-2s}$ and the drift $b_{\tau }$. Under a Foster-Lyapunov condition, we establish the existence of a unique solution pair $(u,{\lambda }^{\ast })$ 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)].