11 February 2020
Analysis, Probability and Mathematical Physics on Fractals
A post-critically finite (p.c.f.) fractal with a regular harmonic structure admits an associated Dirichlet form, which is itself associated with a Laplacian. This Laplacian enables us to give an analog of the damped stochastic wave equation on the fractal.We show that a unique function-valued solution exists, which has an explicit formulation in terms of the spectral decomposition of the Laplacian. We then use a Kolmogorov-type continuity theorem to derive the spatial and temporal Hölder exponents of the solution. Our results extend the analogous results on the stochastic wave equation in one-dimensional Euclidean space. It is known that no function-valued solution to the stochastic wave equation can exist in Euclidean dimension 2 or higher. The fractal spaces that we work with always have spectral dimension less than 2, and show that this is the right analog of dimension to express the “curse of dimensionality” of the stochastic wave equation. Finally, we prove some results on the convergence to equilibrium of the solutions.
Submitted to ORA: