The cone of positive harmonic functions for scale-invariant diffusions

Abstract

Consider a scale invariant diffusion whose state space is a closed cone in R ^d , minus the vertex. Then the process is either recurrent, transient to ∞ or transient to the vertex of the cone. In the latter case, the diffusion has finite lifetime (a.s.) and converges to the vertex at the lifetime. The Martin boundary consists of two points, and the corresponding minimal harmonic functions are of the form 1 and |x| ^α ψ(x/|x|).

Keywords:

• Positive harmonic functions
• Martin boundary
• Recurrence classification
• Scale-invariance
• 60J60
• 60G17

Acknowledgements

I thank the referee for pointing out that Brownian scaling is not at all essential; any scaling is enough. I am also grateful for the suggestion to state the example given in §5 early on in the introduction. This very much improves the presentation.