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In this article we consider a class of nonlinear integro-differentialequations of the form $$\inf_{\tau \in\mathcal{T}} \bigg\{\int_{\mathbb{R}^d}(u(x+y)+u(x-y)-2u(x))\frac{k_{\tau}(x,y)}{|y|^{d+2s}} \,dy+ b_{\tau}(x) \cdot\nabla u(x)+g_{\tau}(x) \bigg\}-\lambda^*=0\quad \text{in} \hspace{2mm}\mathbb{R}^d,$$ where $0<\lambda(2-2s)\leq k_{\tau}\leq \Lambda (2-2s)$ , $s\in(\frac{1}{2},1)$. The above equation appears in the study of ergodic controlproblems in $\mathbb{R}^d$ when the controlled dynamics is governed bypure-jump L\'evy processes characterized by the kernels $k_{\tau}\,|y|^{-d-2s}$and the drift $b_\tau$. Under a Foster-Lyapunov condition, we establish theexistence of a unique solution pair $(u, \lambda^*)$ satisfying the aboveequation, provided we set $u(0)=0$. Results are then extended to cover the HJBequations of mixed local-nonlocal type and this significantly improves theresults in [Arapostathis-Caffarelli-Pang-Zheng (2019)].

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