  We consider an evolution equation similar to that introduced by Vese and
whose solution converges in large time to the convex envelope of the initial
datum. We give a stochastic control representation for the solution from which
we deduce, under quite general assumptions that the convergence in the
Lipschitz norm is in fact exponential in time. We then introduce a
non-autonomous gradient flow and prove that its trajectories all converge to
minimizers of the convex envelope.
