We study Brownian zeroes in the neighborhood of which one can observe a non-typical growth rate of Brownian excursions. We interpret the multifractal curve for the Brownian zeroes calculated in [6] as the Hausdorff dimension of certain sets. This provides an example of the multifractal analysis of a statistically self-similar random fractal when both the spacing and the size of the corresponding nested sets are random.
We define multifractional Brownian fields indexed by a metric space, such as a manifold with its geodesic distance, when the distance is of negative type. This construction applies when the Brownian field indexed by the metric space exists, in particular for spheres, hyperbolic spaces and real trees.
Multifractional Processes with Random Exponent (MPRE) are obtained by replacing the Hurst parameter of Fractional Brownian Motion (FBM) with a stochastic process. This process need not be independent of the white noise generating the FBM. MPREs can be conveniently represented as random wavelet series. We will use this type of representation to study their Hölder regularity and their self-similarity.
By using a wavelet method we prove that the harmonisable-type N-parameter multifractional brownian motion (mfBm) is a locally nondeterministic gaussian random field. This nice property then allows us to establish joint continuity of the local times of an (N, d)-mfBm and to obtain some new results concerning its sample path behavior.
Let P1, ..., Pd be commuting Markov operators on L∞(X,F,μ), where (X,F,μ) is a probability measure space. Assuming that each Pi is either conservative or invertible, we prove that for every f in Lp(X,F,μ) with 1 ≤ p < ∞ the averagesAnf = (n + 1)-d Σ0≤ni≤n P1n1 P2n2 ... Pdnd f (n ≥ 0)converge almost everywhere if and only if there exists an invariant and equivalent finite measure λ for which the Radon-Nikodym derivative v = dλ/dμ is in the dual space Lp'(X,F,μ). Next we study the case in...
We compute the moments and free cumulants of the measure , where denotes the free Poisson law with parameter t > 0. We also compute free cumulants of the symmetrization of . Finally, we introduce the free symmetrization of a probability measure on ℝ and provide some examples.