A decomposition of the Brownian path |
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| Affiliation: | 1. Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iaşi, Carol 1 Blvd., no. 11, Iaşi, Romania;2. Department of Mathematics, “Gheorghe Asachi” Technical University of Iaşi, Carol I Blvd., no. 11, Iaşi, 700506, Romania;3. “Octav Mayer” Mathematics Institute of the Romanian Academy, Iaşi Branch, Carol I Blvd., no. 8, Iaşi, 700506, Romania;4. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland;1. Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain;2. Departamento de Análisis Matemático, Universidad de la Laguna, 38271, La Laguna (Tenerife), Spain |
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| Abstract: | The Brownian path {ω(s); 0 ⩽ s ⩽ t} is dissected and then reassembled in such a way that - (i) the last visit γt at the origin, as well as the fragment {ω(s); γt ⩽ s ⩽ t}, are left invariant;
- (ii) on [0, γt], local time becomes maximum-to-date and occupation time of|R+ becomes location of maximum; and
- (iii) the resulting process is again Brownian.
Characterizations of conditional processes are employed to establish the result. Several consequences of the latter are discussed. |
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