By a variant of the techniques introduced by the first two authors in [DF] to
prove that second derivatives of solutions to the Monge-Ampere equation are
locally in $L\log L$, we obtain interior $W^{2,1+\varepsilon}$ estimates.

Motivated by the "tug-of-war" game studied in [12], we consider a "non-local"
version of the game which goes as follows: at every step two players pick
respectively a direction and then, instead of flipping a coin in order to
decide which direction to choose and then moving of a fixed amount $\epsilon>0$
(as is done in the classical case), it is a $s$-stable Levy process which
chooses at the same time both the direction and the distance to travel.
Starting from this game, we heuristically we derive a deterministic non-local
integro-differential equation that we call "infinity fractional

Consider transportation of one distribution of mass onto another, chosen to
optimize the total expected cost, where cost per unit mass transported from x
to y is given by a smooth function c(x,y).

We obtain various new well-posedness results for continuity and transport
equations, among them an existence and uniqueness theorem (in the class of
strongly continuous solutions) in the case of nearly incompressible vector
fields, possibly having a blow-up of the BV norm at the initial time. We apply
these results (valid in any space dimension) to the k x k chromatography system
of conservation laws and to the k x k Keyfitz and Kranzer system, both in one
space dimension.