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The celebrated Fourier inversion formula provides a useful way to re-construct a regular enough, e.g. square-integrable, function via its own Fourier transform. In this article, we give the first probabilistic proof of this classical theorem, even for Euclidean spaces of arbitrary dimension. Particularly, our proof motivates why the one-half weight, for the one-dimensional case in Lemma 1 , comes naturally to play due to the inherent spatial symmetry; another similar interpretation can be found in the higher dimensional analogue. [ABSTRACT FROM AUTHOR]

We consider mean field games between a dominant leader and many followers, such that each follower is subject to a heterogeneous delay effect from the leader's action, who in turn can exercise governance on the population through this influence. The delay effects are assumed to be discretely distributed among the followers. Given regular enough coefficients, we describe a necessary condition for the existence of a solution for the equilibrium by a system of coupled forward–backward stochastic differential equations and stochastic partial differential equations. We provide a thorough study for the particular linear quadratic case. By adopting a functional approach, we obtain the time-independent sufficient condition, which warrants the unique existence of the solution of the whole mean field game problem. Several numerical illustrations with different time horizons and populations are demonstrated. [ABSTRACT FROM AUTHOR]