Heat flow in metric measure spaces
´
Giuseppe Savare
(joint work with Luigi Ambrosio and Nicola Gigli)
It is well known that in a complete Riemannian manifold (X, g) the heat equation
∂t u − ∆g u = 0,
(1)
associated to the Laplace-Beltrami operator ∆g , is the gradient flow in the Hilbert
space L2m (X) of the Dirichlet energy
Z
1
|∇u|2g dm, u ∈ Wm1,2 (X, g),
(2)
D(u) :=
2 X
where m is the Riemannian volume measure and Wm1,2 (X, g) is the usual Sobolev
space of L2m (X) functions whose (distributional) gradient is square summable.
Entropy and Wasserstein distance. In their seminal paper [9], Jordan, Kinderlehrer, and Otto showed that the Heat equation in the Euclidean space
X := Rd (endowed with the canonical metric and the d-dimensional Lebesgue
measure) can also be viewed as the gradient flow of the Entropy functional
Z
(3) Entm (ρ) :=
u log u dm if ρ = um m,
Entm (ρ) = +∞ if ρ 6 m,
X
with respect to the so-called L2 -Wasserstein distance W2 in the space of the Borel
probability measures P(X): the (extended, since it can take the value +∞) distance W2 (µ, ν) between two measures in P(X) can be defined as
Z
2
(4)
W2 (µ, ν) := min d2 (x, y) dπ(x, y) ∈ [0, ∞],
where the minimum is taken among all the couplings π ∈ P(X × X) having
marginals µ and ν and d(x, y) denotes the Riemannian distance induced by g.
This new characterization of the heat flow (in fact stated for the more general
class of Fokker-Planck equations) has been further extended to Hilbert spaces
[1, 3], Riemannian manifolds [6, 13], Finsler [11] and Alexandrov spaces [7].
Metric-Measure spaces with lower Ricci curvature bounds. The talk has
been devoted to present the recent results of [2] concerning the identification of
these flows in a wide class of metric-measure spaces (X, d, m), including in particular those satisfying the lower Ricci curvature bound CD(K, ∞) recently introduced
by Sturm [12] and Lott-Villani [10].
Here (X, d) is a complete metric space and m is a Borel measure on X such that
Z
2
e−L d (x,x0 ) dm < ∞ for some x0 ∈ X, L ≥ 0.
X
CD(K, ∞) spaces naturally arise in the Wasserstein approach, since they are defined by assuming the geodesic K-convexity of the entropy functional (3). More
precisely, (X, d, m) has Ricci curvature bounded from below by K ∈ R according
to [12, 10] if every couple µ0 , µ1 ∈ D(Entm ) = {ρ ∈ P(X) : Entm (ρ) < ∞} with
1
W2 (µ0 , µ1 ) < ∞ can be connected by a minimal geodesic curve s 7→ µs ∈ D(Entm ),
s ∈ [0, 1], such that for every s, t ∈ [0, 1]
W2 (µs , µt ) = |s − t|W2 (µ0 , µ1 ),
(5)
K
s(1 − s)W22 (µ0 , µ1 ).
2
The gradient flow of the Entropy functional in the Wasserstein space.
Let us recall (see e.g. [1]) that a curve of maximal slope for the Entropy functional
in the Wasserstein space (P(X), W2 ) is a curve ρ : [0, ∞) → D(Entm ) satisfying
2
(a) ρ ∈ ACloc
([0, ∞); P(X), W2 ), i.e. there exists a function v ∈ L2loc (0, ∞) s.t.
Z t
v(r) dr whenever 0 ≤ s ≤ t < ∞;
W2 (ρs , ρt ) ≤
Entm (µs ) ≤ (1 − s)Entm (µ0 ) + sEntm (µ1 ) −
s
its metric velocity |ρ|
˙ is then defined by
W2 (ρt , ρt+h )
h→0
|h|
|ρ˙ t | := lim
for a.e. t ∈ (0, ∞).
(b) For every 0 ≤ s ≤ t < ∞ we have
Z
1 t
Entm (ρt ) +
|ρ˙ r |2 + |∇− Entm |2 (ρr ) dr = Entm (ρs ),
2 s
(ρ)−Entm σ)+
denotes the descending slope
where |∇− Entm |(ρ) := lim sup (EntmW
2 (ρ,σ)
σ→ρ
of the entropy.
If (X, d, m) is a CD(K, ∞) space, then adapting the arguments of [1, 8] it is possible
to prove [2] that for every initial datum ρ0 = u0 m in D(Entm ) there exists a unique
curve of maximal slope ρt = ut m for the Entropy functional according to (a) and
(b) above.
Cheeger energy in (X, d, m). The L2 -approach can also be settled in a metric
measure space (X, d, m), by introducing a weak notion of “modulus of the gradient”
and of Sobolev space Wm1,2 (X, d), inspired by Cheeger [5]. The Cheeger energy
is defined by
Z
Z
1
2
2
(6) Ch(u) := inf lim inf |∇uh | dm : uh ∈ Lip(X),
|uh − u| dm → 0 ,
h→∞
2
X
where for a Lipschitz function u : X → R we set
|u(y) − u(x)|
.
|∇u|(x) := lim sup
d(x, y)
y→x
It is not difficult to prove that Ch : L2m (X) → [0, ∞] is a convex (but no more
quadratic, in general) and lower semicontinuous functional, whose proper domain
{u ∈ L2m (X) : Ch(u) < ∞} provides a natural definition for the space Wm1,2 (X, d).
When Ch(u) < ∞, the Cheeger energy admits the integral representation
Z
1
(7)
Ch(u) =
|∇u|2w dm,
2 X
2
where |∇u|w denotes the relaxed gradient of u, a quantity that satisfies various
calculus properties similar to the classical ones for |∇u|g in a smooth Riemannian
setting; it also turns out to be extremely useful to estimate the derivative of u
along suitable collections of absolutely continuous curves in X.
The (possibily nonlinear) Laplace operator ∆m,d can be defined in a dense
subset of L2m (X) as the minimal selection of the subdifferential of Ch; it generates
a unique flow (see e.g. [4]) Ht : L2m (X) → Wm1,2 (X, d) such that ut := Ht (u0 ) satisfy
the Cauchy problem for the evolution equation
∂t u − ∆m,d u = 0,
(8)
lim ut = u0
t↓0
in L2m (X).
Ht is a contraction w.r.t. any Lpm (X)-norm and it is order and mass preserving: in
particular,
Z
Z
u0 ≥ 0,
u0 dm = 1 ⇒ ut ≥ 0,
ut dm = 1 for every t ≥ 0.
X
X
R
Theorem [2]. If ρ0 = u0 m ∈ P(X) with u0 ∈ L2m (X) and X d2 (x, x0 ) dρ0 < ∞,
the curve ρt = ut m is a curve of maximal slope for the Entropy functional in P(X)
if and only if ut = Ht u0 is the L2 -gradient flow of the Cheeger energy solving (8).
References
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urich. Birkh¨
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second edition, 2008.
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