Abstract
We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped with regular harmonic structures. We show that if the spectral dimension of the set is less than two, then function-valued "random-field" solutions to these SPDEs exist and are jointly H\"older continuous in space and time. We calculate the respective H\"older exponents, which extend the well-known results on the H\"older exponents of the solution to SHE on the unit interval. This shows that the "curse of dimensionality" of the SHE on $\mathbb{R}^n$ depends not on the geometric dimension of the ambient space but on the analytic properties of the operator through the spectral dimension. To prove these results we establish generic continuity theorems for stochastic processes indexed by these p.c.f.s.s. sets that are analogous to Kolmogorov's continuity theorem. We also investigate the long-time behaviour of the solutions to the fractal SHEs.
Highlights
The stochastic heat equation on Rn, n ∈ N is a stochastic partial differential equation which can be expressed formally as∂u (t, x) = Lu(t, x) + W (t, x), ∂t u(0, ·) = u0 for (t, x) ∈ [0, ∞) × Rn, where L is the Laplacian on Rn, u0 is a function on Rn and Wis a space-time white noise on R × Rn
One of the aims of the present paper is to investigate what happens regarding these two properties in the setting of finitely ramified fractals, which behave in many ways like spaces with dimension between one and two
In [14] it is shown that solutions to some nonlinear stochastic heat equations on more general metric measure spaces have Hölder continuous paths when considered as a random map from a “time” set to some space of functions
Summary
The stochastic heat equation (or SHE) on Rn, n ∈ N is a stochastic partial differential equation which can be expressed formally as. For examples of some previous work in this area, in [6] it is shown that on certain fractals a stochastic heat equation can be defined which yields a random-field solution, that is, a solution which is a random map [0, T ] × F → R In [14] (see [12]) it is shown that solutions to some nonlinear stochastic heat equations on more general metric measure spaces have Hölder continuous paths when considered as a random map from a “time” set to some space of functions.
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