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Measure of the multiple self‐intersection set of a markov process
Author(s) -
Berman Simeon M.
Publication year - 1990
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160430502
Subject(s) - mathematics , measure (data warehouse) , combinatorics , hausdorff measure , bounded function , intersection (aeronautics) , discrete mathematics , probability measure , set function , hausdorff dimension , set (abstract data type) , mathematical analysis , database , computer science , engineering , programming language , aerospace engineering
Let X ( t ), t ≧ 0, be a Markov process in R m with homogeneous transition density p ( t; x, y ). For a closed bounded set B ⊂ R m , X is said to have a self‐intersection of order r ≧ 2 in B if there are distinct points t 1 < … < t r such that X ( t 1 ) ∈ B and X ( t j ) = X ( t 1 ), for j = 2,…, r . The focus of this work is the Hausdorff measure, suitably defined, of the set of such r ‐tuples. The main result is that under general conditions on p ( t; x, y ) as well as the specific conditionthere is a measure function M ( t ), defined in terms of the integral above, such that the corresponding Hausdorff measure of self‐intersection set is positive, with positive probability. The results are applied to Lévy and diffusion processes, and are shown to extend recent results in this area.