Quantitative boundary H\"{o}lder estimates for the inhomogeneous Poisson problem through a probabilistic approach

In this paper we derive quantitative boundary H\"older estimates, with explicit constants, for the inhomogeneous Poisson problem in a bounded open set $D\subset \mathbb{R}^d$. Our approach has two main steps: firstly, we consider an arbitrary $D$ as above and prove that the boundary $\alpha$-H\"older regularity of the solution the Poisson equation is controlled, with explicit constants, by the H\"older seminorm of the boundary data, the $L^ \gamma$-norm of the forcing term with $\gamma>d/2$, and the $\alpha/2$-moment of the exit time from $D$ of the Brownian motion. Secondly, we derive explicit estimates for the $\alpha/2$-moment of the exit time in terms of the distance to the boundary, the regularity of the domain $D$, and $\alpha$. Using this approach, we derive explicit estimates for the same problem in domains satisfying exterior ball conditions, respectively exterior cone/wedge conditions, in terms of simple geometric features. As a consequence we also obtain explicit constants for pointwise estimates for the Green function and for the gradient of the solution. The obtained estimates can be employed to bypass the curse of high dimensions when aiming to approximate the solution of the Poisson problem using neural networks, obtaining polynomial scaling with dimension, which in some cases can be shown to be optimal.