Nonlinear Diﬀer. Equ. Appl.
2018 Springer International Publishing AG,
part of Springer Nature
Nonlinear Diﬀerential Equations
and Applications NoDEA
An entropy generation formula
on RCD(K, ∞) spaces
Abstract. J. Feng and T. Nguyen have shown that the solutions of the
Fokker–Planck equation in R
satisfy an entropy generation formula. We
prove that, in compact metric measure spaces with the RCD(K, ∞)prop-
erty, a similar result holds for curves of measures whose density is bounded
away from zero and inﬁnity. We use this fact to show the existence of min-
imal characteristics for the stochastic value function.
Mathematics Subject Classiﬁcation. 49L25, 58J35.
denote the d-dimensional torus and let P(T
) denote the set
of the Borel probability measures on T
. Let the two functions F :(−∞, 0] ×
→ R and U : P(T
) → R be continuous. The stochastic value function
U :(−∞, 0] ×P(T
) → R
is deﬁned as
− F (s, x)
where t ≤ 0andμ:[t, 0] →P(T
) is a weak solution of the Fokker–Planck
)=0,s∈ [t, 0]
The inf in (1.1) is over all “reasonable” vector ﬁelds X.
Work partially supported by the PRIN2009 Grant “Critical Point Theory and Perturbative
Methods for Nonlinear Diﬀerential Equations”.