  Stochastic embeddings of finite metric spaces into graph-theoretic trees have
proven to be a vital tool for constructing approximation algorithms in
theoretical computer science. In the present work, we build out some of the
basic theory of stochastic embeddings in the infinite setting with an aim
towards applications to Lipschitz free space theory. We prove that proper
metric spaces stochastically embedding into $\mathbb{R}$-trees have Lipschitz
free spaces isomorphic to $L^1$-spaces. We then undergo a systematic study of
stochastic embeddability of Gromov hyperbolic metric spaces into
$\mathbb{R}$-trees by way of stochastic embeddability of their boundaries into
ultrametric spaces. The following are obtained as our main results: (1) Every
snowflake of a compact, finite Nagata-dimensional metric space stochastically
embeds into an ultrametric space and has Lipschitz free space isomorphic to
$\ell^1$. (2) The Lipschitz free space over hyperbolic $n$-space is isomorphic
to the Lipschitz free space over Euclidean $n$-space. (3) Every infinite,
finitely generated hyperbolic group stochastically embeds into an
$\mathbb{R}$-tree, has Lipschitz free space isomorphic to $\ell^1$, and admits
a proper, uniformly Lipschitz affine action on $\ell^1$.
