  This paper introduces a general class of Replicator-Mutator equations on a
multi-dimensional fitness space. We establish a novel probabilistic
representation of weak solutions of the equation by using the theory of
Fockker-Planck-Kolmogorov (FPK) equations and a martingale extraction approach.
The examples with closed-form probabilistic solutions for different fitness
functions considered in the existing literature are provided. We also construct
a particle system and prove a general convergence result to any solution to the
FPK equation associated with the extended Replicator-Mutator equation with
respect to a Wasserstein-like distance adapted to our probabilistic framework.
