## Earthquake diffusion process on metric graphs

We consider a model of an earthquake source for rupture times as a stochastic load-state process (a Brownian relaxation oscillator). An earthquake happens when the process reaches a threeshold, and then it relaxes to a ground level and begin a new cycle. We develop this process on a star graph $S_k$, with $k\in\mathbb{N}$. $S_k$ is a tree having one internal node $v$ and $k$ leaves $l_i$ with $i=1,...,k$, each connected with $v$ by an edge $e_i$ with $i=1,...,k$. It is a metric graph, so we identfy each $e_i$ with an interval $[a,b]\in\mathbb{R}$. We can see how this structure represents a seismic source, propagating with probability $p$ on a different direction in the space, stopping when the critical threshold is reached. The process on $S_k$ starts from $v$ and diffuses at each time on a randomly chosen $e_i$, until some stopping a-priori defined requirements are achieved. We apply different boundary conditions on $v$ and $l_i$ (i.e. Neumann, Robin, Feller-Wentzell), to deduce how the behavior of the Brownian motion on $S_k$ can change. As a final goal, we are interested in the construction of a network made by the composition of many star graphs $S_k$ to deal with the described process on more complex structures.

Area: CS50 - Anomalous phenomena on regular and irregular domains (Mirko D'Ovidio)

Keywords: earthquakes, diffusion process, Brownian motion, star graph

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