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  • 1
    Electronic Resource
    Electronic Resource
    Self-similar random measures are locally scale invariant (1993)
    Springer
    Probability theory and related fields 97 (1993), S. 559-574 
    Details
    ISSN: 1432-2064
    Keywords: 60G57 ; 60D05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In an earlier paper Patzschke and U. Zähle [11] have proved the existence of a fractional tangent measure at the typical point of a self-similar random measure Φ under rather special technical assumptions. In the present paper we remove the most restrictive one. Here we suppose the open set condition for the similarities, a constant positive lower bound for the random contraction ratios, and vanishing Φ on the boundary of the open set with probability 1. The tangent measure isD-scale-invariant, whereD is the similarity dimension of Φ. Moreover, we approximate the tangential distribution by means of Φ and use this in order to prove that the Hausdorff dimension of the tangent measure equalsD. Since the former coincides with the Hausdorff dimension of Φ we obtain an earlier result of Mauldin and Williams [9] as a corollary.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01192964
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  • 2
    Electronic Resource
    Electronic Resource
    Integration with respect to fractal functions and stochastic calculus. I (1998)
    Springer
    Probability theory and related fields 111 (1998), S. 333-374 
    Details
    ISSN: 1432-2064
    Keywords: Mathematical Subject Classification(1991): Primary 60H05; Secondary 26A33 ; 26A42
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. The classical Lebesgue–Stieltjes integral ∫ b a fdg of real or complex-valued functions on a finite interval (a,b) is extended to a large class of integrands f and integrators g of unbounded variation. The key is to use composition formulas and integration-by-part rules for fractional integrals and Weyl derivatives. In the special case of Hölder continuous functions f and g of summed order greater than 1 convergence of the corresponding Riemann–Stieltjes sums is proved. The results are applied to stochastic integrals where g is replaced by the Wiener process and f by adapted as well as anticipating random functions. In the anticipating case we work within Slobodeckij spaces and introduce a stochastic integral for which the classical Itô formula remains valid. Moreover, this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional Brownian motion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s004400050171
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  • 3
    Electronic Resource
    Electronic Resource
    Curvature measures and random sets II (1986)
    Springer
    Probability theory and related fields 71 (1986), S. 37-58 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In choosing models of stochastic geometry three general problems play a role which are closely connected with each other: 1) Construction of the random geometric objects under consideration 2) Measurabilities 3) Geometric behaviour In the present paper second order local geometric properties of random subsets of R d are of interest. These properties are described by signed curvature measures in a measure geometric context. The theory of point processes on general spaces (here on the space of subsets with positive reach) provides an appropriate framework for solving construction and measurability problems. Mean value relations for random curvature measures associated with such set processes are derived by means of invariance properties. Ergodic interpretations of the curvature densities are also given. The appendix provides auxiliary results for random signed Radon measures in locally compact separable Hausdorff spaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00366271
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  • 4
    Electronic Resource
    Electronic Resource
    Integral and current representation of Federer's curvature measures (1986)
    Springer
    Archiv der Mathematik 46 (1986), S. 557-567 
    Details
    ISSN: 1420-8938
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01195026
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  • 5
    Electronic Resource
    Electronic Resource
    Absolute curvature measures, II (1992)
    Springer
    Geometriae dedicata 41 (1992), S. 229-240 
    Details
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The absolute curvature measures for sets of positive reach in R d introduced in [7] satisfy the following kinematic relations: Their integrated values on the intersections with (or on the tangential projections onto) uniformly moved p-planes are constant multiples of the corresponding absolute curvature measures of the primary set. In the special case of convex bodies the first result is the so-called Crofton formula. An analogue for signed curvature measures is well known in the differential geometry of smooth manifolds, but the motion of absolute curvatures used there does not lead to this property. For the special case of smooth compact hypermanifolds our absolute curvature measures agree with those introduced by Santaló [4] with other methods. In the appendix, the section formula is applied to motion invariant random sets.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00182423
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  • 6
    Electronic Resource
    Electronic Resource
    Mixed curvature measures for sets of positive reach and a translative integral formula (1995)
    Springer
    Geometriae dedicata 57 (1995), S. 259-283 
    Details
    ISSN: 1572-9168
    Keywords: 53C65 ; 52A22
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Mixed curvature measures for sets of positive reach are introduced and a translative version of the principal kinematic formula from integral geometry is proved. This is an extension of a known result from convex geometry. Integral representations of the mixed curvature measures in various particular cases of dimension two and three are derived.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01263484
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  • 7
    Electronic Resource
    Electronic Resource
    Curvatures and currents for unions of sets with positive reach (1987)
    Springer
    Geometriae dedicata 23 (1987), S. 155-171 
    Details
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For locally finite unions of sets with positive reach in R d, generalized unit normal bundles are introduced in support of a certain set additive index function. Given an appropriate orientation to the normal bundle, signed curvature measures may be defined by means of associated locally rectifiable currents (with index function as multiplicity) and specially chosen differential forms. In the case of ‘regular’ sets this is shown to be equivalent to well-known classical concepts via former results. The present approach leads to unified methods in proving integral-geometric relations. Some of them are stated in this paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00181273
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  • 8
    Electronic Resource
    Electronic Resource
    Nonosculating Sets of Positive Reach (1999)
    Springer
    Geometriae dedicata 76 (1999), S. 183-187 
    Details
    ISSN: 1572-9168
    Keywords: sets of positive reach ; exceptional relative positions ; nonosculating condition.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For integral geometric and local topological properties of two sets of positive reach, one fixed and the other moving or translating, certain exceptional relative positions have to be excluded. We give a measure geometric justification in form of a Sard-type theorem which extends to more general singular sets. A slightly stronger result has been proved by Schneider for the special case of convex bodies.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005162207767
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  • 9
    Electronic Resource
    Electronic Resource
    Approximation and characterization of generalised Lipschitz-Killing curvatures (1990)
    Springer
    Annals of global analysis and geometry 8 (1990), S. 249-260 
    Details
    ISSN: 1572-9060
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A differential-geometric and measure-geometric analogue to Hadwiger's characterization of linear combinations of Minkowski functionals of convex bodies as continuous additive euclidean invariants is given. The equivalent of the quermassintegrals are generalised Lipschitz-Killing curvatures and measures. By means of polyhedral approximation with respect to flat seminorms of associated normal cycles the general problem may be reduced to the classical case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00127938
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  • 10
    Electronic Resource
    Electronic Resource
    Local Dimensions, Average Densities and Self-Conformal Measures (1998)
    Springer
    Periodica mathematica Hungarica 37 (1998), S. 217-225 
    Details
    ISSN: 1588-2829
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider subsets F of $$\mathbb{R}^n$$ generated by iterated function systems with contracting conformal C1+γ-diffeomorphisms whose Hausdorff dimension is s. The unique s-self-conformal probability measure μ agrees with the normalised s-dimensional Hausdorff measure on F. Using the associated dynamical system and ergodic theory we develop a potential theoretic representation of the average densities of μ at almost all points of F. Before we formulate some general relationships between average densities and local dimensions of measures.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1004746923670
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