  This paper introduces a probabilistic formulation for the isometric embedding
of a Riemannian manifold $(M^n,g)$ into Euclidean space $\mathbb{R}^q$. Given
$\alpha \in ]\tfrac{1}{2},1]$, we show that a $C^{1,\alpha}$ embedding $u: M
\to \mathbb{R}^q$ is isometric if and only if the intrinsic and extrinsic
constructions of Brownian motion on $u(M)\subset \mathbb{R}^q$ yield processes
with the same law. The equivalence is first established for smooth embeddings;
this is followed by a renormalization procedure for $C^{1,\alpha}$ embeddings.
In particular, we also construct extrinsic Brownian motion when $g \in C^2$ and
$u$ is a $C^{1,\alpha}$ isometric embedding.
  This formulation is based on a gedanken experiment that relates the intrinsic
and extrinsic constructions of Brownian motion on an embedded manifold to the
measurement of geodesic distance by observers in distinct frames of reference.
This viewpoint provides a thermodynamic formalism for the isometric embedding
problem that is suited to applications in geometric deep learning, stochastic
optimization and turbulence.
