Let's say I have a point-like Brownian particle undergoing two-dimensional diffusion on an infinite plane with the caveat is that the particle can never return to a coordinate that it previously visited (these coordinates become fully reflecting to the particle).  As such, the particle may trap itself if its trajectory generates a closed loop.  

Assuming the Brownian particle has some diffusion coefficient, $D$, is there a known mean and variance for the maximum distance the particle will travel from its origin before trapping itself?    

Is there a known result for the mean maximum distance from origin (and associated variance) for a random walker on an infinite two-dimensional lattice of some degree $m$?