Reproducing Kernel Approach to Linear-Quadratic Mean Field Control Problems with Additive Noise

We show in this work how to develop a kernel approach to solve linear-quadratic mean field control problems. We use operator-valued kernels, which is consistent with the fact that we are dealing with an infinite dimensional control problem due to the mean-field term. But the stochastic aspect of the problem brings also a difficulty of a different nature. The kernel is defined over the time variable, and conversely to the deterministic case, information must be considered. Thus the kernel acts on random processes, even for ordinary stochastic control problems. This type of kernels has not appeared previously in the literature. Extensions, like partially observable systems or multiplicative noise, will be considered in the future.