Optimal transport, Euler equations, Mather and DiPerna-Lions theories

4.2. THE STOCHASTIC EXTENSION 43
This is exaclty what DiPerna-Lions and Ambrosio were able to do in [51, 26]. To state a
precise result, we need to introduce the concept of Regular Lagrangian Flows (RLF). The idea is
that, if there exists a flow which produces solutions in L 1 ∩L ∞ , it cannot concentrate. Therefore
we expect that, if such a flow exists, it must be a RLF in the sense of the following definition:
Definition 4.1.2 We say that X(t, x) is a RLF (starting at time 0), if:
(i) for L d -a.e. x, X(·, x) is an integral curve of the ODE starting from x (at time 0);
(ii) there exists a nonnegative constant C such that, for any t ∈ [0, T ],
X(t)#L d ≤ CL d .
It is not hard to show that, because of condition (ii), this concept is indeed invariant under
modifications of b, and so it is appropriate to deal with vector fields belonging to L p spaces.
As proved in [26], the following existence and uniqueness result for RLF holds:
Theorem 4.1.3 Assume that, for any µ0 ∈ L 1 (R d ) ∩ L ∞ (R d ) there exists a unique solution of
the PDE in L ∞ ([0, T ], L 1 (R d ) ∩ L ∞ (R d )). Then there exists a unique RLF. Moreover the RLF
is stable by smooth approximations.
The well-posedness of the PDE in L ∞ ([0, T ], L 1 (R d ) ∩ L ∞ (R d )) has been shown by DiPerna-
Lions [51] under the assumption
b ∈ W 1,p (R d ), [divb] − ∈ L ∞ (R d ),
and then generalized by Ambrosio [26] assuming only
b ∈ BV (R d ), divb ∈ L 1 (R d ), [divb] − ∈ L ∞ (R d ).
This theory presents still many open interesting questions, like to understand better whether
uniqueness holds under the above hypotheses in bigger classes like L ∞ ([0, T ], L 1 (R d )) (so that
the solution can be unbounded). Or at the level of the ODE to see whether, under one of the
above assumptions on the vector field, one can prove a statement like: there exists a set A ⊂ R n ,
with |A| = 0, such that for all x ∈ A the solution of the ODE is unique. These are problems
that I would like to attack in the future.
4.2 The stochastic extension
In the stocastic case, the continuity equation becomes the Fokker-Planck equation
∂tµt +
i
∂i(b i µt) − 1
2
∂ij(a ij µt) = 0,
ij