Zero temperature versus long time for diffusions driven by metric projections onto convex sets

For a closed subset C of Euclidean space, we study the diffusion whose drift pulls the state towards its nearest point in C, with noise intensity T. We let T tend to zero and time tend to infinity. Convexity of C is equivalent to a pathwise bound of the distance from C by the norm of the process of Ornstein and Uhlenbeck. For arbitrary closed sets the bound holds in law. For compact convex C, we describe the invariant law by the curvature measures and the intrinsic volumes of C. The limit points of the law of the state form the orbit of reflected Brownian motion in C started from the projection of the initial law. So the two limits commute only when this projection is the uniform law on C. Unless C is a point, the mixing time is of order $1/T$ up to a logarithm. For the control problem the two limits of the value function commute for every compact convex set and every bounded Lipschitz terminal cost. A companion paper treats finite disjoint unions of convex sets.