The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each area gets a winding number coefficient giving the net number of times it was traversed counterclockwise). But Douglas Zare's comment suggested an version which counts area without sign and without multiplicity.

For a compact set $K \subset \mathbb{R}^2$, let $E(K)$, the "exterior" of $K$, be the unique unbounded component of $\mathbb{R}^2 \setminus K$. Let $\mathcal{A}(K) = m(\mathbb{R}^2 \setminus E(K))$ be the area (Lebesgue measure) of the region "enclosed" by $K$.

Preliminary question: let $\mathcal{K}(\mathbb{R}^2)$ be the set of all compact subsets of $\mathbb{R}^2$, with the Vietoris topology. Is $\mathcal{A} : \mathcal{K}(\mathbb{R}^2) \to [0,\infty)$ a Borel function?

Now let $B(t)$ be a standard 2-dimensional Brownian motion. Consider the process $A_t = \mathcal{A}(B([0,t]))$ which gives the area enclosed by $B$ up to time $t$. If the answer to the preliminary question is yes, then I believe $A_t$ is a stochastic process, jointly measurable and adapted to the filtration of $B$. It is nondecreasing, and I think maybe it is right continuous, but I am not quite sure.

What is known about $A_t$? Is it equal (in distribution) to some well-known process? What are its moments? Any other interesting properties?