Let $\Delta$ be a convex body (i.e. a compact convex subset) or a convex polytope in $\mathbb{R}^n$. Let $x$ be a point inside $\Delta$ and consider a (uniform) random walk starting at $x$ inside $\Delta$. I am interested in the probability distribution of the first time the random walk hits a point in boundary $\partial \Delta$. We consider a uniform random walk were $x$ with equal probability can go to any point in a small neighborhood of radius $\delta$ of $x$. We are interested in the probability distribution (on the boundary) that the random walk hits a boundary point.

Is there a "nice" formula for this distribution on the boundary points (in terms of $x$ and $\Delta$)?

If $\Delta$ is a polytope one expects that this distribution should be related to the distance of $x$ to different facets of $\Delta$.

In general are there nice formula relating geometry of $\Delta$ and relative position of $x$ in $\Delta$ to probability distributions (associated to random walks)?

I am from algebraic geometry and don't know much about random walks. I thought this problem might be known to the experts.