A needle-shaped Brownian microswimmer in a channel

We consider a microswimmer modeled as a one-dimensional line segment -- a needle -- with a fixed swimming velocity.  The direction of swimming changes according to a Brownian process, and the swimmer is confined to an infinite channel.  This is a standard model for a simple microswimmer, or a confined wormlike chain polymer.  Using natural assumptions about reflection of the swimmer at boundaries, we compute the invariant distribution across the channel, and the statistics of spreading in the longitudinal direction.  When the needle length is longer than the channel width, we compute the mean drift velocity of the swimmer.  Otherwise, we examine the time it takes for the swimmer to reverse direction, and the effective diffusion constant of its large scale motion.