  We develop a stochastic target representation for Ricci flow and normalized
Ricci flow on smooth, compact surfaces, analogous to Soner and Touzi's
representation of mean curvature flow. We prove a verification/uniqueness
theorem, and then consider geometric consequences of this stochastic
representation. Based on this stochastic approach, we give a proof that, for
surfaces of nonpositive Euler characteristic, the normalized Ricci flow
converges to a constant curvature metric exponentially quickly in every
$C^k$-norm. In the case of $C^0$ and $C^1$-convergence, we achieve this by
coupling two particles. To get $C^2$-convergence (in particular, convergence of
the curvature), we use a coupling of three particles. This triple coupling is
developed here only for the case of constant curvature metrics on surfaces,
though we suspect that some variants of this idea are applicable in other
situations and therefore be of independent interest. Finally, for $k\ge3$, the
$C^k$-convergence follows relatively easily using induction and coupling of two
particles. None of these techniques appear in the Ricci flow literature and
thus provide an alternative approach to the field.

Probability;Analysis of PDEs;Differential Geometry