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Occupation density and sample path properties of \(N\)-parameter processes. (English) Zbl 1042.60031
Merzbach, Ely (ed.), Topics in spatial stochastic processes. Lectures given at the C.I.M.E. summer school, Martina Franca, Italy, July 1–8, 2001. Berlin: Springer (ISBN 3-540-00295-2/pbk). Lect. Notes Math. 1802, 127-166 (2003).
The author surveys work done on the existence and regularity of occupation densities for multiparameter processes, i.e. stochastic processes with \(N\)-dimensional parameter and values in \(R^d\). In the first chapter he discusses general methods which do not depend on the particular distribution of the field (or indeed on its randomness), such as the underlying basic measure theory, Fourier transforms of multivariate measures and the relation of Hölder exponents and Hausdorff dimension. The second chapter is devoted to the Gaussian case, here the role of local nondeterminism in the analysis of Gaussian random fields is discussed, but also Brownian sheets (where LND fails) get a mention. The final chapter presents techniques for the analysis of stable Lévy sheets, and Fourier analytic methods are emphasised; the final section of this chapter is devoted to limit theorems for the occupation density and occupation integrals.
For the entire collection see [Zbl 1005.00040].

60G60 Random fields
60G17 Sample path properties
60J55 Local time and additive functionals