Random walks on multidimensional landscapes are important to many areas of science and engineering.
In particular, properties of adaptive trajectories on fitness landscapes determine population fates
and thus play a central role in evolutionary biology.  To this end we have developed a path-based approach to continuous-time random walks
on discrete state spaces.  I will describe an efficient numerical algorithm for calculating statistical properties
of the stochastic path ensemble, including distributions of path lengths, times, and spatial structures.
This approach, based on general techniques from statistical physics, is applicable to state spaces and landscapes
of arbitrary complexity and structure.  It is especially well-suited to quantifying the diversity of stochastic trajectories
and repeatability of evolutionary events.  After demonstrating our approach on two reaction rate problems, I will present a biophysical model
that describes how proteins evolve new functions while maintaining thermodynamic stability.
Our methodology reveals several distinct modes of adaptation depending on key biological and biophysical
properties of the protein, reproducing important observations from directed evolution experiments.