The question http://mathoverflow.net/questions/202456/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?