Fractional Cauchy problem on random snowflakes

Abstract

We consider time-changed Brownian motions on random Koch (pre-fractal and fractal) domains where the time change is given by the inverse to a subordinator. In particular, we study the fractional Cauchy problem with Robin condition on the pre-fractal boundary obtaining asymptotic results for the corresponding fractional diffusions with Robin, Neumann and Dirichlet boundary conditions on the fractal domain.

Asymptotics for quasilinear obstacle problems in bad domains

“…Remark 6.8. We point out that it is possible to extend the present result to other domains with prefractal and fractal boundaries like, for example, quasi-filling fractal layers or random snowflakes (see [10] and the reference therein); the key tool is that the domains have good "extension" properties (see [33]). Moreover, it is possible to perform asymptotic analysis also in the so-called "Sobolev admissible domains" (see [17], [28]).…”

On the Non-Local Boundary Value Problem from the Probabilistic Viewpoint

“…In [33], we consider the sequence of time-changed process X L,n = X n • L, where X n is an elastic Brownian motion on Ω (ξ|n) with elastic coefficient c n . We study the asymptotic behavior of X L,n depending on the asymptotics for c n .…”

Asymptotics for Time-Fractional Venttsel’ Problems in Fractal Domains

“…Among others, we refer to [15][16][17][18][19][20], and the references therein, and to [21] for time-fractional Venttsel' problems in Lipschitz domains. For time-fractional equations in fractal domains, we refer to, for example, [22,23].…”