Brownian motions on a half line

The classical Brownian motions.Feller's Brownian motions.Outline.Standard Brownian motion: stopping times' and local times. Brownian motion on [0, -f-oo ).Special case: p+(0) < 1.Green operators and generators: p+(0) 1. Generator and Green operators computed: p+(0) 1.Special case-p2 0 < p3 and pC Special case" p2 > 0 p4. Increasing differential processes.Sample paths: pl p3 0 O/p---koo).Simple Markovian character: pl p3 0 (ps > O/p4 oo ).Local times: p p 0(ps> O/p -koo).Sample paths and Green operators: pl u(O) + pa(@u)(O) p2 u+(0) -k f+ (ps > O/p + ).16. Bounded interval: [-1, -k 1]. 17.Two-sided barriers.18. Simple Brownian motions.19.Feller's differential operators.20.Birth and death processes.[u(/) u(O)]p(dl)Numbering. 1 means formula 1 of the present section; 2.1 means formula 1 of Section 2, etc.; the numbering of the diagrams is similar. The classical Brownian motionsConsider the space of all (continuous) sample paths w'[0, -t-oo) --R