Fractional Gaussian fields on the Sierpinski gasket and related fractals

2020/03/09 by Fabrice Baudoin, Baudoin, Fabrice, Céline Lacaux +1
Computer Science · Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Topological and Geometric Data Analysis #advanced mathematical theories

paper · doi:10.48550/arxiv.2003.04408

openalex publication_date 2020/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define and study a fractional Gaussian field X with Hurst parameter H on the Sierpiński gasket K equipped with its Hausdorff measure μ. It appears as a solution, in a weak sense, of the equation (-Δ)^s X =W where W is a Gaussian white noise on L₀²(K,μ), Δ the Laplacian on K and s= (d_h+2H)/(2d_w), where d_h is the Hausdorff dimension of K and d_w its walk dimension. The construction of those fields is then extended to other fractals including the Sierpiński carpet.

Related

• Dirichlet fractional Gaussian fields on the Sierpinski gasket and their discrete graph approximations
• Fractional stable random fields on the Sierpiński gasket
• Fractional Gaussian fields: a survey
• The Kusuoka measure and the energy Laplacian on level-k Sierpiński gaskets
• Constructing fractional Gaussian fields from long-range divisible sandpiles on the torus