Parabolic Bellman-Systems with Mean Field Dependence

We consider the necessary conditions for Nash-points of Vlasov-McKean functionals (Formula presented.) ((Formula presented.)). The corresponding payoffs (Formula presented.) depend on the controls (Formula presented.) and, in addition, on the field variable (Formula presented.). The necessary conditions lead to a coupled forward-backward system of nonlinear parabolic equations, motivated by stochastic differential games. The payoffs may have a critical nonlinearity of quadratic growth and any polynomial growth w.r.t. m is allowed as long as it can be dominated by the controls in a certain sense. We show existence and regularity of solutions to these mean-field-dependent Bellman systems by a purely analytical approach, no tools from stochastics are needed.