Consider a Brownian motion $B_t$ in $\mathbb{R}^n, n\geq 3$ and the ball $B(0, r)$ of radius $r$ around the origin. Let $\overline{C}$ be a compact set inside $B(0, r)$ such that $C$ is open in $B(0, r)$.
Denote the probability of the Brownian particle hitting $\overline{C}$ within time $t$ by $P_t\left({\overline{C}}\right)$. I recently heard in a talk that as long as $c_1r^2 \leq t \leq c_2r^2$, we have that $$C_1\text{cap}_M\left({\overline{C}}\right) \leq P_t\left({\overline{C}}\right) \leq C_2\text{cap}_M\left({\overline{C}}\right),$$
where $\text{cap}_M\left({\overline{C}}\right)$ is the Martin capacity for $\overline{C}$. I would really appreciate a reference for this result.
In context:
1) the closest such result I can find on my own is Theorem 8.24 from the book on Brownian motion by Morters and Peres, which says that for a compact $A \subset B(0, r)$, $$\frac{1}{2}\text{cap}_{M}(A)\leq P_{\tau_B}\left({\overline{C}}\right)\leq \text{cap}_{M}(A),$$ where $\tau_B$ is the first exit time from $B(0, r)$. May be these two results are related?
2) Related: also see Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$).