I would like to know the expected value for the area covered by a disc of radius $R$ whose center undergoes Brownian motion (diffusion).

Specifically, let $\mathbf{X}_t$ represent a two-dimensional Brownian motion, and define the covered set $C_t = \bigcup_t B(\mathbf{X}_t,R)$ to be the union of discs centered at every point along the realized trajectory. Define $A_t$ to be the area of the set $C_t$. What is the expected value $\mathbb{E}[A_t]$ as a function of $t$?

I am specifically interested in this for the mathematical modeling of a chemical reaction and would be happy with an approximate solution that works in the short-to-moderate time regime, where $|x_t|$ is at most (say) $5R$. I think that this time regime is too short to use the lattice random walk result for the number of distinct sites ($t / \log t$ from Dvoretzky and Erdös).

Thank you.