Gareth G Roberts , Omar Wani 

Geometries of eroding landscapes contain important information about geologic, climatic, biotic and geomorphic processes. They are also characterised by variability, which makes disentangling their origins challenging. Observations and physical models of fluvial processes, which set the pace of erosion on most continents, emphasise complexity and variability. In contrast, the spectral content of longitudinal river profiles and similarity of geometries at scales $\gtrsim100$ km highlight relatively simple emergent properties. A general challenge then, addressed in this manuscript, is development of a theory of landscape evolution that embraces such scale-dependent insights. We do so by incorporating randomness and probability into a theory of fluvial erosion. First, we explore the use of stochastic differential equations of the Langevin type, and the Fokker-Planck equation, for predicting migration of erosional fronts. Second, analytical approaches incorporating distributions of driving forces, critical thresholds and associated proxies are developed. Finally, a linear programming approach is introduced, that, at its core, treats evolution of longitudinal profiles as a Markovian stochastic problem. The theory is developed essentially from first principles and incorporates physics governing fluvial erosion. We explore predictions of this theory, including the natural growth of discontinuities and scale-dependent evolution, including local complexity and emergent simplicity.