## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Miniconference on Operator Theory and Partial Differential Equations. M. Cowling, C. Meaney, and W. Moran, eds. Proceedings of the Centre for Mathematical Analysis, v. 14. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1986), 326

### Some positive eigenfunctions for elliptic operators with oblique derivative boundary conditions and consequences for the stationary densities of reflected Brownian motions

#### Abstract

Positive eigenfunctions for elliptic operators with oblique derivative boundary conditions arise as the stationary densities of reflected Brownian motions, briefly RBM's. An RBM is a diffusion process that behaves like Brownian motion with a constant drift velocity ~ in the interior of a d-dimensional domain and is instantaneously reflected at the boundary in a direction specified by a non-tangential vector field v on the boundary. When the domain is bounded and smooth and the vector field is smooth, it is shown that the stationary density is of a simple exponential form for all ~ if and only if the vector field v satisfies a certain skew-symmetry condition. A formal analogue of this result for polyhedral domains, where v is constant on each face, will also be given. Consequences for the existence and uniqueness of an RBM with such non-smooth data will be drawn from this.

Details will appear elsewhere.

#### Article information

**Dates**

First available in Project Euclid:
18 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416336610

#### Citation

Williams, Ruth J. Some positive eigenfunctions for elliptic operators with oblique derivative boundary conditions and consequences for the stationary densities of reflected Brownian motions. Miniconference on Operator Theory and Partial Differential Equations, 326, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1986. https://projecteuclid.org/euclid.pcma/1416336610