Year: 2010
Type: article
Abstract: Consider a diffusion process corresponding to the operator L=(1/2)a(d2/dx2) + b(d/dx) and that is transient to +∞. For c > 0, we give an explicit criterion in terms of the coefficients a and b that determines whether or not the diffusion almost surely eventually stops making down-crossings of length c. As a particular case, we show that, if a = 1, then the diffusion almost surely stops making down-crossings of length c if b(x) ⩾ (1/2c) log x + (γ/c) log log x for some γ > 1 and for large x, but makes down-crossings of length c at arbitrarily large times if b(x) ⩽ (1/2c) log x + (1/c) log log x for large x.
DOI: 10.1112/blms/bdq020
Author: Ross G. Pinsky
Institution: Technion – Israel Institute of Technology
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2089429729