Diffusion method in field theories with fakeons

We adapt the diffusion method employed in fundamentally nonlocal field theories to determine the number of initial conditions for the classicized dynamics of unitary field theories with fakeons, characterized by inverse powers of the d’Alembertian operator <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mo>□</a:mo> </a:mrow> </a:math> . We show that this number is two and we recover all the results obtained with a direct calculation elsewhere, including explicit solutions of linear toy models. Possible applications to nonlocal gravity are discussed.