The navigation system on the star-ship Enterprise is broken. There is nothing one can do but Chief Engineer Scott is able to fix a distance to be traveled, then the star-ship will make a space-hop with an uniformly random direction. Assume that the current distance from the star-ship to the Sun is and the radius of the Solar System is . The star-ship will make consecutive space-hops with independent uniformly random directions until it reaches Solar System. How large is the probability that the star-ship eventually reaches the Solar System after a finite number of space-hops? How to maximize the chance to accomplish this trip?
Assume that are i.i.d. uniformly random point on the unit sphere
which are also random directions that the star-ship will travel along. Let be the position of the Sun and be the position of the star-ship after space-hops. We can set being the initial position of the star-ship. Set . We have
where is the size of the -th space-hop which is a -measurable random variable. We call the random sequence a strategy.
Proposition 1. Set . Then is a supermartingale which converges with probability 1 to some as . It is a martingale if and only if for all .
Proof: We have
where is the Lebesgue measure on the unit sphere . Applying the mean value property to the harmonic function with , we have
If , by the maximum principle, we have
Hence is a positive supermartingale. By Doob’s convergence theorem, tends to some with probability 1. It is also obvious that is a martingale if and only if for all .
Denote by the first time that the star-ship reaches the Solar System.
Proposition 2. The probability that the star-ship ever reaches the Solar System is strictly smaller than , i.e.
Proof: Note that is also a non-negative supermartingale. We thus have
On the event , we have . Therefore,
It immediately follows that . Since defines a sequence of increasing events, we have
We now show that is actually a strict upper bound for . Notice that the distribution of is non-atomic for all . Indeed, assuming that for all , we have that conditioning on is uniformly distributed on the sphere . It follows that and thus for all . It is also clear that is non-atomic. By the principle of induction, is non-atomic for all . Therefore, on the event ,
It follows that . By Fatou’s lemma, we thus have
Let be a fixed integer. Set .
Proposition 3. Assume that with for all . Then for all , , i.e. the star-ship will reaches to a position with an arbitrary large distance from the Sun after a finite number of space-hops with probability 1.
Proof: Let the first coordinate of . Note that is a sum of i.i.d. random symmetric variables. We thus have
Proposition 4. For each , there exists a strategy such that
Proof: Set with for all . Note that is a positive martingale since for all . We also note that for , we have . Thus for all ,
Applying the optional stopping theorem to the martingale and the stopping time , we have
It immediately follows that
Taking , we obtain that .