(Local) self-similarity is a seminal concept, especially for Euclidean random fields. We study in this paper the extension of these notions to manifold indexed fields. We give conditions on the (local) self-similarity index that ensure the existence of fractional fields. Moreover, we explain how to identify the self-similar index. We describe a way of simulating Gaussian fractional fields.
(Local) self-similarity is a seminal concept, especially for Euclidean random fields. We study in this paper the extension of these notions to manifold indexed fields. We give conditions on the (local) self-similarity index that ensure the existence of fractional fields. Moreover, we explain how to identify the self-similar index. We describe a way of simulating Gaussian fractional fields.
Let Z=(X, Y) be a planar brownian motion, the filtration it generates, andBa linear brownian motion in the filtration . One says thatB(or its filtration) is maximal if no other linear -brownian motion has a filtration strictly bigger than that ofB. For instance, it is shown in [In Séminaire de Probabilités XLI 265–278 (2008) Springer] that B is maximal if there exists a linear brownian motion C independent of B and such that the planar brownian motion (B, C) generates the same filtration asZ....
For the Markov property of a multivariate process, a necessary and suficient condition on the multidimensional copula of the finite-dimensional distributions is given. This establishes that the Markov property is solely a property of the copula, i.e., of the dependence structure. This extends results by Darsow et al. [11] from dimension one to the multivariate case. In addition to the one-dimensional case also the spatial copula between the different dimensions has to be taken into account. Examples...