Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 33 / 5

Let $R=\{B_t:0\le t\le 1\}$ and $h=1/2$. For planar Brownian motion, the process $h^{-1/2}{B}_{ht}$ is again standard Brownian motion. Thus its time-$h$ range has law $\sqrt{h}R$. Two-dimensional Lebesgue measure scales by the square of the spatial factor, givingThe increments $B_{h+t}-B_h$, $0\le t\le h$, form another standard Brownian motion. Its range has the same law as $A_1$, and translation by $B_h$ does not change its area. HenceThe equality concerns the laws of the areas. No independence between these two original areas is needed here.

Path continuity makes each time-segment range compact and therefore Borel measurable. The membership event is jointly measurable in the sample and spatial variable: distance to the range equals the infimum of $|B_t-y|$ over a countable dense set of times. We may consequently apply the Tonelli theorem to its indicator.

Put $m=\mathbb{E}\lambda (R)<\infty$, using the allowed integrability hypothesis. Since $R=A_1\cup A_2$, finite-measure inclusion-exclusion and the preceding expectation identities giveFinally Tonelli theorem givesFiniteness of $m$ is what justifies the subtraction of expectations; an infinite-minus-infinite calculation would not establish this conclusion.

Set $W_t^{-}=B_{h-t}-B_h$ and $W_t^{+}=B_{h+t}-B_h$, $0\le t\le h$, with $h=1/2$. Both start at zero and are continuous. Increments of $W^{-}$ over consecutive disjoint time intervals are negatives of Brownian increments over corresponding disjoint past intervals, read in reverse order. They are independent centered normal random variables with covariance equal to the interval length times the identity. The same properties hold for $W^{+}$ using future increments. Every past increment is independent of all the future increments after deterministic time $h$, so the two whole processes are independent, by finite-dimensional independence and generation by rational-time evaluations. They are the independent Brownian arms at a deterministic time.

Let $R^{-}$ and $R^{+}$ be their ranges. Pathwise,Although $W^{-}$ need not be independent of $B_h$, this last equality uses only translation invariance of Lebesgue measure. Put $p(y)=\mathbb{P}(T_y\le h)$ for a standard Brownian motion started at zero. Independence of the centered processes and Tonelli theorem giveSince $p\ge 0$, it follows that $p(y)=0$ for Lebesgue-almost every $y$. ThereforeThe scaling relation in part (a) gives $\mathbb{E}\lambda (R)=2\mathbb{E}\lambda (A_1)=0$. A nonnegative random variable of expectation zero vanishes almost surely, proving the area of a planar Brownian path resultThe integral argument establishes the needed almost-everywhere spatial assertion; it does not require a prior theorem that individual points are polar.