Local nondeterminism and local times of general stochastic processes

Let X(t), 0 1, be a real stochastic process, and let g(t ), 0 ~ 1, be a nonnegative integrable function. X(t ) is said to be locally g-nondeterministic if for every k > 2, there exists c~ > 0 such that the joint density of the k 1 increments, X (t~ + 1 ) X (t J), j =1, ... , k -1, t1 ... tk, evaluated at the origin, is bounded above by This condition implies the validity of key estimates in the analysis of the local time of the process. The latter imply specific irregularity properties of the sample functions. Such properties have been studied for several years in the context of Gaussian processes. The contribution of this work is the demonstration that local nondeterminism can be usefully defined even for processes that are not necessarily Gaussian, and that the comprehensive theory of sample function irregularity for the latter processes can be extended to more general processes. Applications to Markov processes are exhibited. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Science Foundation, Grant MCS 8201119. Annales de l’Institut Henri Poincaré-Section B-Vol. XIX, 0020-2347/1983/189/ 5,00/ (~) Gauthier-Villars

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