Section: New Results

Local self-similarity properties and stable or Gaussian random fields

Participants: Hermine Biermé, Jacques Istas, Céline Lacaux, Renaud Marty, Hans-Peter Scheffler.

$•$ Recently, an important class of anisotropic random fields called operator scaling random fields has been studied in [30] . To be more specific, the classical self-similarity property is replaced in [30] by the following operator scaling property:

The Hölder regularity properties of operator scaling Gaussian or stable harmonizable random fields have been studied in [30] and can be expressed in terms of the matrix $E$. In particular, they do not vary along the trajectories, which can be too restrictive for some applications (see our osteoporosis project at the Application Domains Section). In order to obtain some anisotropic random fields whose Hölder regularity properties are allowed to vary, we introduce in [1] a local version of the operator scaling property (similar to the local version of the classical self-similarity property defined in [27] ). This local property is illustrated in [1] , where we also define and study harmonizable multi-operator scaling stable randoms fields. For such a multi-operator random field, we obtain an accurate upper bound of both the modulus of continuity and global and directional Hölder regularities at any point $x$. As expected, the Hölder regularity properties vary along the trajectories.

$•$ In [24] , we study the sample paths properties of an anisotropic random field, which is defined as limit of an invariance principle and is of the same type as a multifractional Brownian sheet. Our first aim was to generalize [37] , that is to obtain some multifractional random fields indexed by ${ℝ}^d$ with $d\ge 2$ and to allow Hurst indices to be lower than $1/2$. To overcome the problem of the values of the Hurst indices which characterize the limit field, we focus on stationary sequences ${\left(X_n\left(H\right)\right)}_{n\in \mathbb{N}}$, where $H\in {(0,1)}^d$, defined by an harmonizable representation. Then, our limit field $S_h$ is defined as the limit of

for some suitable families ${\left(h_n^N\right)}_{n,N}$ and ${\left(r_n^N\right)}_{n,N}$. We then study the sample paths property of this limit field. In particular, we obtain some local self-similarity properties for its increments of order $k$ and its pointwise global and directional Hölder exponents. We also define (and obtain) some pointwise multi-Hölder exponents which characterize the Hölder property satisfied by the increments of order $d$ of $S_h$.

$•$ We are also interested in self-similar processes indexed by manifolds in [23] . This study is motivated by the fact various spatial data are indexed by a manifold and not by the Euclidean space ${ℝ}^d$ in practical situations such as image analysis.