The conformal dimension of a metric space is the infimum of the Hausdorff dimension among all its quasisymmetric images. We develop tools related to the Fuglede modulus to study the conformal dimension of stochastic spaces. We first apply our techniques to construct minimal (in terms of conformal dimension) planar graphs. We further develop this line of inquiry by proving that a "natural" stochastic object, the graph of the one dimensional Brownian motion, is almost surely minimal. This is a joint work with Ilia Binder and Hrant Hakobyan.