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We introduce an extension of the frog model to Euclidean space and prove properties for the spread of active particles. The new geometry introduces a phase transition that does not occur for the frog model on the lattice. Fix $r>0$ and place a particle at each point $x$ of a unit intensity Poisson point process $\mathcal P \subseteq \mathbb R^d - \mathbb B(0,r)$. Around each point in $\mathcal{P}$, put a ball of radius $r$. A particle at the origin performs Brownian motion. When it hits the ball around $x$ for some $x \in \mathcal P$, new particles begin independent Brownian motions from the centers of the balls in the cluster containing $x$. Subsequent visits to the cluster do nothing. This waking process continues indefinitely. For $r$ smaller than the critical threshold of continuum percolation, we show that the set of activated sites in $\mathcal P$ behaves like a linearly expanding ball. Moreover, in any fixed ball the set of active particles converges to a unit intensity Poisson point process. Lastly, we prove that the model expands at rate at least $t^{2- \epsilon}$ when $d=2$ and $r$ equals the critical threshold in continuum percolation.

Asymptotic behavior of the Brownian frog model