Benassi, Albert; Roux, Daniel Elliptic self-similar stochastic processes. (English) Zbl 1055.60037 Rev. Mat. Iberoam. 19, No. 3, 767-796 (2003). Let \(M\) be a random measure and let \(L\) be an elliptic psuedo-differential operator on the \(d\)-dimensional space. The authors study the solution of the stochastic problem \(LX=M\), \(X(0)=0\), when some homogeneity and integrability conditions are assumed. The authors characterize the solutions \(X\) which are self-similar and with stationary increments in terms of the deriving measure \(M\). Then they use appropriate wavelet bases to expand these solutions and give regularity results. Reviewer: Tian You Hu (Green Bay) Cited in 5 Documents MSC: 60G18 Self-similar stochastic processes 60G20 Generalized stochastic processes Keywords:elliptic processes; self-similar processes with stationary increments; elliptic pseudo-differential operator; wavelet basis; regularity of sample paths; percolation tree; intermittency × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Benassi, A., Jaffard, S., Roux, D.: Elliptic Gaussian random pro- cesses. Rev. Mat. Iberoamericana 13 (1997), no. 1, 19-90. · Zbl 0880.60053 · doi:10.4171/RMI/217 [2] Benassi, A., Cohen, S., Deguy, S., Istas, J.: Self-similarity and inter- mittency. In Wavelets and Signal Processing. Birkhäuser, 2003. · Zbl 1043.60031 [3] Cramer, H., Leadbetter, M. R.: Stationary and Related Stochastic Processes. Sample function properties and their applications. John Wiley and Sons, Inc., New York-London-Sydney, 1967. · Zbl 0162.21102 [4] Daubechies, I.: Ten Lectures on Wavelets. Regional Conference Series in Applied Mathematics 61. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992. · Zbl 0776.42018 [5] Dobrushin, R. L.: Gaussian and their subordinated self-similar random generalised fields. Ann. Probab. 7 (1979), no. 1, 1-28. · Zbl 0392.60039 · doi:10.1214/aop/1176995145 [6] Gelfand, I. M., Vilenkin, N. Y.: Generalised functions, volume 4. Moscow, 1961. [7] Jaffard, S.: Old friends revisited: the multifractal nature of some classical functions. J. Fourier Anal. Appl. 3 (1997), no. 1, 1-22. · Zbl 0880.28007 · doi:10.1007/BF02647944 [8] Lyons, R.: Random walk and percolation on a tree. Ann. Probab. 18 (1990), 937-958. · Zbl 0714.60089 · doi:10.1214/aop/1176990730 [9] Mandelbrot, B. B.: Fractals: form, chance, and dimension.. W. H. Free- man and Co., San Francisco, Calif., 1977. · Zbl 0376.28020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.