Prof. Dr. Martina Zähle

   Recent publications and preprints

1. Rataj, M. Zähle: Curvature measures of singular sets, Springer Monographs in Mathematics, 2019.
2. Martina Zähle: Lectures on Fractal Geometry,  Fractals and Dynamics in Mathematics, Science, and the Arts: Theory and Applications 8, World Scientific, 2024.

• J. Rataj, S. Winter, M. Zähle: Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals, Preprint, http://arxiv.org/abs/2505.04801.
• A. Rozanova-Pierrat, A. Teplyaev, S. Winter, M. Zähle: Fractal curvatures and short-time asymptotics of heat content, Preprint.
• M. Zähle: Mean Minkowski and s-contents of V-variable random fractals. Asian J. Mathematics 27, No. 6 (2023), 955-970.
• J.Rataj, S.Winter, M. Zähle: Mean Lipschitz-Killing curvatures for homogeneous random fractals, J. Fractal Geometry 10 (2023), 1-42.
• M.Zähle: The mean Minkowski content of homogeneous random fractals. Mathematics 2020, 8, 883.
• M. Zähle, E. Schneider: Forward integrals and SDE with fractal noise. In: Horizons of Fractal Geometry and Complex Dimensions (2019), Contemporary Mathematics 73 
• M. Zähle: (S)PDE on fractals and Gaussian noise. In: Recent Developments in Fractals and Related Fields,  Eds. J. Barral and S. Seuret, Trends in Mathematics, Springer Internat. Publishing 2017, 295-312.
• E. Issoglio, M. Zähle: Regularity of the solutions to SPDEs in metric measure spaces. Stoch. PDE: Anal. Comp. 3 (2015), 372-389. (http://arxiv.org/abs/1409.3399Externer Link)
• M. Hinz, E. Issoglio, M. Zähle: Elementary pathwise methods for nonlinear parabolic and transport type SPDE with fractal noise. In: Modern Stochastics and Applications, Springer Optimiz. Applic. 90, 123-141.
• J. Rataj, M. Zähle: Legendrian cycles and curvature. J. Geometric Analysis. 25 (2015), 2133-2147.
• M. Zähle: Stability properties of fractal curvatures. in: Geometry and Analysis of Fractals,  Eds. Feng, De-Jun and Lau, Ka-Sing,  Springer Proc. Math. & Stat.  (2014), 343-354.
• M. Zähle: Curvature measures of fractal sets. Contemp. Math. 600 (2013), 381-399.
• T. Bohl, M. Zähle: Curvature-direction measures of self-similar sets. Geom. Dedicata. 167 (2013), 215-231. (http://arxiv.org/abs/1111.4457)Externer Link
• V. Radchenko, M. Zähle: Heat equation with a general stochastic measure on nested fractals. Stat. Probab. Letters 82 (2012), 699-704.
• J. Rataj, M. Zähle: Curvature densities of self-similar sets. Indiana Univ. Math. J. 61 (2012), 1425-1449.
• S. Winter, M. Zähle: Fractal curvature measures of self-similar sets. Advances Geom. 13 (2013), 229-244. (http://arxiv.org/abs/1007.0696Externer Link)
• M. Hinz, M. Zähle: Semigroups, potential spaces and applications to (S)PDE. Potential Anal. 36 (2012), 483-515. 
• H. Fink, C. Klüppelberg, M. Zähle: Conditional characteristic functions of processes related to fractional Brownian motion. J. Appl.Probab. 50 (2013), 166-183.
• M. Zähle: Lipschitz-Killing curvatures of self-similar random fractals. Trans. Amer. Math. Soc. 363 (2011), 2663-2684.
• J. Hu, M. Zähle: Schrödinger equations and heat kernel upper bounds on metric spaces. Forum Math. 22 (2010), 1213-1234.
• M. Hinz, M. Zähle: Gradient type noises II: Systems of partial differential equations. J. Funct. Anal. 256 (2009), 3192-3235.
• M. Hinz, M. Zähle: Gradient type noises I: Partial and hybrid integrals. Complex Variables and Elliptic Equations 54 (2009), 561-583.
• J. Hu, M. Zähle: Generalized Bessel and Riesz potentials on metric measure spaces. Potential Anal. 30 (2009), 315-340.
• M. Zähle: Potential spaces and traces of Lévy processes on h-sets. J. Contemp. Math. Anal. 44 (2009), 117-145.
• V. Knopova, M. Zähle: Spaces of generalized smootness on h-sets and related Dirichlet forms. Studia Math. 174 (2006), 277-308.
• W. Hansen, M. Zähle: Restricting isotropic stable Lévy processes from R^n to fractal sets. Forum Math. 18 (2006), 171-191.
• J. Rataj, M. Zähle: General normal cycles and Lipschitz manifolds of bounded curvature. Annals Global Anal. Geom. 27 (2005), 135-156.
• J. Hu, M. Zähle: Potential spaces on fractals. Studia Math. 170 (2005), 259-281.
• J. Hu, M. Zähle: Jump processes and nonlinear fractional heat equations. Math. Nachr. 279 (2006), 1-14.
• J. Rataj, M. Zähle: Normal cycles of Lipschitz manifolds by approximation with parallel sets. Diff. Geom. Appl. 19 (2003), 113-126.
• M. Zähle: Riesz potentials and Liouville operators on fractals. Potential Anal. 21 (2004), 193-208.
• M. Zähle: Long range dependence, no arbitrage, and the Black-Scholes formula. Stochastics and Dynamics 2 (2002), 265-280.
• M. Zähle: Riesz potentials and Besov spaces on fractals. In: Fractals in Graz 2001, Trends in Mathematics, Birkhäuser 2002, 271-276.
• M. Zähle: Stochastic differential equations with fractal noise. Math. Nachr. 278 (2005), 1097-1106.
• M. Zähle: Harmonic calculus on fractals - A measure geometric approach II. Trans. Amer. Math. Soc. 357 (2005), 3407-3423.

Publications 1996-2002

• M. Zähle, H. Ziezold: Fractional derivatives of Weierstrass-type functions. J. Comput. Appl. Math. 76 (1996), 265-275.
• M. Zähle: Fractional differentiation in the self-affine case. V - The local degree of differentiability. Math. Nachr. 185 (1997), 279-306.
• F. Klingenhöfer, M. Zähle: Ordinary differential equations with fractal noise. Proc. Amer. Math. Soc. 127 (1999), 1021-1028.
• M. Zähle: Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Relat. Fields  111 (1998), 333-374.
• M. Zähle: Integration with respect to fractal functions and stochastic calculus II. Math. Nachr. 225 (2001), 145-183.
• M. Zähle: On the link between fractional and stochastic calculus. in: Stochastic dynamics, (Eds. H. Crauel and M. Gundlach) Springer, New York 1999.
• M. Zähle: Non-osculating sets of positive reach. Geom. Dedicata 76 (1999), 183-187.
• J. Rataj, M. Zähle: Curvatures and currents for unions of sets with positive reach II. Ann. Global Anal. Geom. 20 (2001), 1-21.
• M. Zähle: Local dimensions, average densities and self-conformal measures. Period. Math. Hung. 37 (1998), 217-225.
• M. Zähle: The average density of self-conformal measures. J. London Math. Soc. 63 (2001), 721-734.
• U. Freiberg, M. Zähle: Harmonic calculus on fractals - A measure - geometric approach I. Potential Anal. 16 (2002), 265-277.
• M. Zähle: Measure-theoretic Laplace operators on fractals. Canadian Math. Soc. Conference Proc. 29 (2000), 625-631.
• M. Zähle: Fractional integrals and derivatives with respect to a measure. Fract. Calculus Appl. Anal. 2 (1999), 537-542.
• M. Zähle: Forward integrals and stochastic differential equations. In: Seminar on Stochastic Analysis, Random Fields and Applications III, Eds. R.C. Dalang, M. Dozzi, F. Russo, Progress in Probab., Birkhäuser 2002, 293-302.
• J. Rataj, M. Zähle: A remark on mixed curvature measures for sets with positive reach. Beiträge Algebra Geom. 43 (2002), 171-179.
• M. Zähle: Riesz potentials of fractal measures. (Abstract). Real Anal. Exchange, 24th Summer Symp. Conf. Rep., May 2000, 109-110.

Publications before 1996

• The average fractal dimension and projections of measures and sets in R^n. Fractals 3 (1995), 747-754.
• (with J. Rataj) Mixed curvature measures for sets of positive reach and a translative integral formula. Geom. Dedicata 57 (1995), 259-283.
• (with E. Arbeiter) Geometric measures for random mosaics in spherical spaces. Stoch. and Stoch. Reports 46 (1994), 63-77.
• (with N. Patzschke) Fractional differentiation in the self-affine case. IV - Random measures. Stoch. and Stoch. Reports 49 (1994), 87-98.
• (with N. Patzschke) Fractional differentiation in the self-affine case. III - The density of the Cantor set. Proc. Amer. Math. Soc. 117 (1993), 137-144.
• (with N. Patzschke) Fractional differentiation in the self-affine case. II - Extremal Processes. Stoch. Processes Appl. 45 (1993), 61-72.
• (with N. Patzschke) Fractional differentiation in the self-affine case. I - Random functions. Stoch. Processes Appl. 43 (1992), 165-175.
• (with N. Patzschke) Self-similar measures are locally scale invariant. Probab. Theory Relat. Fields 97 (1993), 559-574.
• (with L. Leistritz) Topological mean value relations for random cell complexes. Math. Nachr. 155 (1992), 57-72.
• (with E. Arbeiter) Kinematic relations for Hausdorff moment measures in spherical spaces. Math. Nachr. 153 (1991), 333-348.
• Approximation and characterization of generalized Lipschitz-Killing curvatures. Ann. Global Anal. Geom. 8 (1990), 249-260.
• Wicksell's corpuscle problem in sperical spaces. J. Appl. Prob. 27 (1990), 701-706.
• A kinematic formula and moment measures on random sets. Math. Nachr. 149 (1990), 325-340.
• (with W. Rother) Palm measures in homogeneous spaces. Math. Nachr. 149 (1990), 255-263.
• (with W. Rother) A short proof of principal kinematic formula and extensions. Trans. Amer. Math. Soc. 321 (1990), 547-558.

Publications before 1990

• Absolute curvature measures. Math. Nachr. 140 (1989), 83-90.
• Random cell complexes and generalized sets. Ann. Probab. 16 (1988), 1742-1766.
• (with V. Weiss) Geometric measures for random curved mosaics of R^d. Math. Nachr. 138 (1988), 313-326.
• Normal cycles and second order rectifiable sets. (unpublished manuscript).
• Polyhedron theorems for non-smooth cell complexes. Math. Nachr. 131 (1987), 299-310.
• Curvatures and currents for unions of sets with positive reach. Geom. Dedicata 23 (1987), 155-171.
• Integral and current representation of Federer's curvature measures. Arch. Math. 46 (1986), 557-567.
• Curvature measures and random sets II. Probab. Th. Rel. Fields 71 (1986), 37-58.
• Curvature measures and random sets I. Math. Nachr. 119 (1984), 327-339.
• Thick section stereology for random fibres. Math. Operationsforsch. Statist. 15 (1984), 429-435.
• Random set processes in homogeneous Riemannian spaces. Math. Nachr. 110 (1983), 179-193.
• Random processes of Hausdorff rectifiable closed sets. Math. Nachr. 108 (1982), 49-72.
• Ergodic properties of random fields and pattern with embedded point processes. Teor. Verojat. Primen. 27 (1982), 502-513.
• Ergodic properties of general Palm measures. Math. Nachr. 95 (1980), 93-106.
• On the Euler characteristic of excursions of random fields. Forschungserg. FSU, N/80/34 (1980).

Publications before 1980

• On common invariance properties of translation invariant measures and their Palm measures. Forschungserg. FSU, N/79/44 (1979).
• Ergodic theorems for Erlang pure loss systems in the case of dependent services times. Math. Nachr. 79 (1977), 317-323.