Fractional differentiation in the self-affine case I – Random functions

AbstractThe invariance structure of self-affine functions and measures leads to the concept of fractional Cesáro derivatives and densities, respectively. In the present paper the case of random functions from Rp into Rq is considered. It is shown that the corresponding derivatives exist a.s. and equal a constant in the ergodic case. Part II will deal with the class of self-similar extremal processes and certain extensions. In Part III the fractional density of the Cantor measure will be evaluated, and arbitrary self-similar random measures will be treated in Part IV. There exists a deeper connection to fractional differentiation in the theory of function spaces which will be established elsewhere

Nonlinear Young integrals via fractional calculus

For H\"older continuous functions $W(t,x)$ and $\varphi_t$, we define nonlinear integral $\int_a^b W(dt, \varphi_t)$ via fractional calculus. This nonlinear integral arises naturally in the Feynman-Kac formula for stochastic heat equations with random coefficients. We also define iterated nonlinear integrals.Comment: arXiv admin note: substantial text overlap with arXiv:1404.758

Curvature-direction measures of self-similar sets

We obtain fractal Lipschitz-Killing curvature-direction measures for a large class of self-similar sets F in R^d. Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean curvatures of differentiable submanifolds. They decouple as independent products of the unit Hausdorff measure on F and a self-similar fibre measure on the sphere, which can be computed by an integral formula. The corresponding local density approach uses an ergodic dynamical system formed by extending the code space shift by a subgroup of the orthogonal group. We then give a remarkably simple proof for the resulting measure version under minimal assumptions.Comment: 17 pages, 2 figures. Update for author's name chang

L\'evy-areas of Ornstein-Uhlenbeck processes in Hilbert-spaces

In this paper we investigate the existence and some useful properties of the L\'evy areas of Ornstein-Uhlenbeck processes associated to Hilbert-space-valued fractional Brownian-motions with Hurst parameter $H\in (1/3,1/2]$. We prove that this stochastic area has a H\"older-continuous version with sufficiently large H\"older-exponent and that can be approximated by smooth areas. In addition, we prove the stationarity of this area.Comment: 18 page

Regularity of the Solutions to SPDEs in Metric Measure Spaces

In this paper we study the regularity of non-linear parabolic PDEs and stochastic PDEs on metric measure spaces admitting heat kernel estimates. In particular we consider mild function solutions to abstract Cauchy problems and show that the unique solution is Hölder continuous in time with values in a suitable fractional Sobolev space. As this analysis is done via a-priori estimates, we can apply this result to stochastic PDEs on metric measure spaces and solve the equation in a pathwise sense for almost all paths. The main example of noise term is of fractional Brownian type and the metric measure spaces can be classical as well as given by various fractal structures. The whole approach is low dimensional and works for spectral dimensions less than 4

The Euler-Maruyama approximation for the absorption time of the CEV diffusion

A standard convergence analysis of the simulation schemes for the hitting times of diffusions typically requires non-degeneracy of their coefficients on the boundary, which excludes the possibility of absorption. In this paper we consider the CEV diffusion from the mathematical finance and show how a weakly consistent approximation for the absorption time can be constructed, using the Euler-Maruyama scheme

Multifractal tubes

Tube formulas refer to the study of volumes of $r$ neighbourhoods of sets. For sets satisfying some (possible very weak) convexity conditions, this has a long history. However, within the past 20 years Lapidus has initiated and pioneered a systematic study of tube formulas for fractal sets. Following this, it is natural to ask to what extend it is possible to develop a theory of multifractal tube formulas for multifractal measures. In this paper we propose and develop a framework for such a theory. Firstly, we define multifractal tube formulas and, more generally, multifractal tube measures for general multifractal measures. Secondly, we introduce and develop two approaches for analysing these concepts for self-similar multifractal measures, namely: (1) Multifractal tubes of self-similar measures and renewal theory. Using techniques from renewal theory we give a complete description of the asymptotic behaviour of the multifractal tube formulas and tube measures of self-similar measures satisfying the Open Set Condition. (2) Multifractal tubes of self-similar measures and zeta-functions. Unfortunately, renewal theory techniques do not yield "explicit" expressions for the functions describing the asymptotic behaviour of the multifractal tube formulas and tube measures of self-similar measures. This is clearly undesirable. For this reason, we introduce and develop a second framework for studying multifractal tube formulas of self-similar measures. This approach is based on multifractal zeta-functions and allow us obtain "explicit" expressions for the multifractal tube formulas of self-similar measures, namely, using the Mellin transform and the residue theorem, we are able to express the multifractal tube formulas as sums involving the residues of the zeta-function.Comment: 122 page

Sixty Years of Fractal Projections

Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.Comment: Submitted to proceedings of Fractals and Stochastics

Holder exponents of irregular signals and local fractional derivatives

It has been recognized recently that fractional calculus is useful for handling scaling structures and processes. We begin this survey by pointing out the relevance of the subject to physical situations. Then the essential definitions and formulae from fractional calculus are summarized and their immediate use in the study of scaling in physical systems is given. This is followed by a brief summary of classical results. The main theme of the review rests on the notion of local fractional derivatives. There is a direct connection between local fractional differentiability properties and the dimensions/ local Holder exponents of nowhere differentiable functions. It is argued that local fractional derivatives provide a powerful tool to analyse the pointwise behaviour of irregular signals and functions.Comment: 20 pages, Late

Uniform convergence of discrete curvatures from nets of curvature lines

We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.Comment: 21 pages, 8 figure