  A mean-field-type limit from stochastic moderately interacting many-particle
systems with singular Riesz potential is performed, leading to nonlocal
porous-medium equations in the whole space. The nonlocality is given by the
inverse of a fractional Laplacian, and the limit equation can be interpreted as
a transport equation with a fractional pressure. The proof is based on
Oelschl\"ager's approach and a priori estimates for the associated diffusion
equations, coming from energy-type and entropy inequalities as well as
parabolic regularity. An existence analysis of the fractional porous-medium
equation is also provided, based on a careful regularization procedure, new
variants of fractional Gagliardo--Nirenberg inequalities, and the div-curl
lemma. A consequence of the mean-field limit estimates is the propagation of
chaos property.
