  We study Langevin dynamics for recovering the planted signal in the spiked
matrix model. We provide a "path-wise" characterization of the overlap between
the output of the Langevin algorithm and the planted signal. This overlap is
characterized in terms of a self-consistent system of integro-differential
equations, usually referred to as the
Crisanti-Horner-Sommers-Cugliandolo-Kurchan (CHSCK) equations in the spin glass
literature. As a second contribution, we derive an explicit formula for the
limiting overlap in terms of the signal-to-noise ratio and the injected noise
in the diffusion. This uncovers a sharp phase transition -- in one regime, the
limiting overlap is strictly positive, while in the other, the injected noise
overcomes the signal, and the limiting overlap is zero.
