Boundary Harnack principle for the absolute value of a one-dimensional subordinate Brownian motion killed at 0. (English) Zbl 1354.60088

Wagner, Vanja

Electron. Commun. Probab. 21, Paper No. 84, 12 p. (2016).

Summary: We prove the Harnack inequality and boundary Harnack principle for the absolute value of a one-dimensional recurrent subordinate Brownian motion killed upon hitting 0, when 0 is regular for itself and the Laplace exponent of the subordinator satisfies certain global scaling conditions. Using the conditional gauge theorem for symmetric Hunt processes we prove that the Green function of this process killed outside of some interval \((a,b)\) is comparable to the Green function of the corresponding killed subordinate Brownian motion. We also consider several properties of the compensated resolvent kernel \(h\), which is harmonic for our process on \((0,\infty )\).

MSC:

60J65	Brownian motion
60G51	Processes with independent increments; Lévy processes
60J45	Probabilistic potential theory
60J57	Multiplicative functionals and Markov processes

Keywords:

subordinate Brownian motion; Green functions; harmonic functions; Harnack inequality; boundary Harnack principle; Feynman-Kac transform; conditional gauge theorem