## How many Brownian particles escape when you control with total drift = 1?

**Update 12/2015.**
This problem is now solved in the preprint
Optimal Surviving Strategy for Drifted Brownian Motions with Absorption
by Wenpin Tang and Li-Cheng Tsai.

First consider *K* particles performing independent
Brownian motion on the positive half-line, with state 0 absorbing, and
each particle started at state 1 at time 0.
Now suppose we have at our
disposal a unit quantity of positive drift, which we can
distribute amongst the non-absorbed particles at time *t*
according to any control policy we choose.
Let *F(K)* be the maximum, over all control policies, of the expectation
of the number of particles that survive forever.

It is a challenging exercise in stochastic calculus to show that *F(K)* grows as order
*K^{1/2}*.

**Problem.**
Prove *F(K) = (c + o(1))K^{1/2}* for some explicit constant *c*.

**Problem.**
Analyze the policy of assigning drift 1 to the leftmost particle.

Here are some unpublished 2002 notes on this problem.
Related work appears in One-dimensional Brownian
particle systems with rank dependent drifts by Soumik Pal and Jim Pitman,
and in The advantage of capitalism vs. socialism depends on the criterion
by H. P. Mckean and L. A. Shepp.

**History.**
The model was inspired by a (now discontinued) child's board game
*Up The River -- Race to the Harbor* by Ravensburger.
I gave a talk on this topic in February 2002 at a Berkeley meeting to honor
David Blackwell and Lester Dubins.
That talk prompted the "related work" above.