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The modulator of the local time
Author(s) -
Berman Simeon M.
Publication year - 1988
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.3160410108
Subject(s) - mathematics , smoothness , function (biology) , measure (data warehouse) , interval (graph theory) , gaussian , locally integrable function , integrable system , mathematical analysis , combinatorics , physics , computer science , quantum mechanics , database , evolutionary biology , biology
Let x(t ), 0 ≦ t ≦ 1, be a real measurable function having a local time α( x, t ) which is a continuous function of t for almost all x . It is also assumed that, for some m ≧ 2 and some real interval B , α m ( x , 1) is integrable over B . The modulator is a function M m ( t, B ), t > 0, denned in terms of α. It is shown that the modulator serves as a measure of the smoothness of the L m (B )‐valued function α(., t ) with respect to t . Then it is shown that the modulator plays a central role in precisely describing certain irregularity properties of x(t ). The results are applied to the case where x(t ) is the sample function of a real stochastic process. In this way new results are obtained for large classes of Gaussian and Markov processes.