On time scale invariance of random walks in confined space

- We consider the effects of time scale on Brownian random walks in confined space.
- For reflective boundaries the drift and diffusion rates are usually unaffected.
- For other impenetrable boundaries these properties vary with the time step.
- Similar effects are observed for absorbing boundaries.

Animal movement is often modelled on an individual level using simulated random walks. In such applications it is preferable that the properties of these random walks remain consistent when the choice of time is changed (time scale invariance). While this property is well understood in unbounded space, it has not been studied in detail for random walks in a confined domain. In this work we undertake an investigation of time scale invariance of the drift and diffusion rates of Brownian random walks subject to one of four simple boundary conditions. We find that time scale invariance is lost when the boundary condition is non-conservative, that is when movement (or individuals) is discarded due to boundary encounters. Where possible analytical results are used to describe the limits of the time scaling process, numerical results are then used to characterise the intermediate behaviour.