The purpose of this work is to provide an explicit construction of a strong
Feller semigroup on the space of probability measures over the real line that
additionally maps bounded measurable functions into Lipschitz continuous
functions, with a Lipschitz constant that blows up in an integrable manner in
small time. Our construction relies on a rearranged version of the stochastic
heat equation on the circle driven by a coloured noise. Formally, this
stochastic equation writes as a reflected equation in infinite dimension, a
topic that is known to be challenging. Under the action of the rearrangement,
the solution is forced to live in a space of quantile functions that is
isometric to the space of probability measures on the real line. We prove the
equation to be solvable by means of an Euler scheme in which we alternate flat
dynamics in the space of random variables on the circle with a rearrangement
operation that projects back the random variables onto the subset of quantile
functions. A first challenge is to prove that this scheme is tight. A second
one is to provide a consistent theory for the limiting reflected equation and
in particular to interpret in a relevant manner the reflection term. The last
step in our work is to establish the aforementioned Lipschitz property of the
semigroup by adapting earlier ideas from the Bismut-Elworthy-Li formula in
stochastic analysis