Large deviation principle for invariant distributions of memory gradient diffusions

In this paper, we consider a class of diffusions based on a memory gradient descent, i:e: whose drift term is built as the average all along the past of the trajectory of the gradient of a coercive function U. Under some classical assumptions on U, this type of diffusion is ergodic and admits a unique invariant distribution. With the view to optimization applications, we want to understand the behaviour of the invariant distribution when the diffusion coefficient goes to 0. In the non-memory case, the invariant distribution is explicit and the so-called Laplace method shows that a Large Deviation Principle (LDP) holds with an explicit rate function. In particular, such a result leads to a concentration of the invariant distribution around the global minima of U. Here, except in the linear case, we have no closed formula for the invariant distribution but we prove that a LDP can still be obtained. Then, in the one-dimensional case and under some assumptions on the second derivative of U, we get some bounds for the rate function that lead to the concentration around the global minima.