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.