Evolution of the Wasserstein distance between the
marginals of two Markov processes
Jacopo Corbetta
(École des Ponts - ParisTech)
Joint work with: Aurélien Alfonsi and Benjamin Jourdain
January 10, 2017
Jacopo Corbetta
Wasserstein distance evolution
January 10, 2017
1/9
The Wasserstein distance
Definition
The %−Wasserstein distance between two probability measures P
e on Rd is given by
and P
!1
%
Z
e =
W% (P, P)
Jacopo Corbetta
inf
%
|x − y | π(dx, dy )
e Rd ×Rd
π∈Π(P,P)
Wasserstein distance evolution
January 10, 2017
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The Wasserstein distance
Definition
The %−Wasserstein distance between two probability measures P
e on Rd is given by
and P
!1
%
Z
e =
W% (P, P)
inf
%
|x − y | π(dx, dy )
e Rd ×Rd
π∈Π(P,P)
Dual Representation
e
W%% (P, P)
Z
= sup −
Z
φ(x)P(dx) −
Rd
Rd
e
e
φ(y )P(dy )
e obtaining the sup is called Kantorovich potentials.
A couple (ψ, ψ)
Jacopo Corbetta
Wasserstein distance evolution
January 10, 2017
2/9
A generic heuristic formula
et }t≥0 be two Rd -valued Markov processes.
Let {Xt }t≥0 and {X
Then
Z
Z
et (dy )
et ) = −
ψet (y )P
W%% (Pt , P
ψt (x)Pt (dx) −
Rd
Jacopo Corbetta
Wasserstein distance evolution
Rd
January 10, 2017
3/9
A generic heuristic formula
et }t≥0 be two Rd -valued Markov processes.
Let {Xt }t≥0 and {X
Then
Z
Z
et (dy )
et ) = −
ψet (y )P
W%% (Pt , P
ψt (x)Pt (dx) −
Rd
Rd
For all 0 ≤ t
d %
et ) = −
W (Pt , P
dt %
Jacopo Corbetta
Z
Z
Lψt (x)Pt (dx) −
Rd
Wasserstein distance evolution
Rd
e
et (dx) .
L ψet (x)P
January 10, 2017
3/9
A generic heuristic formula
et }t≥0 be two Rd -valued Markov processes.
Let {Xt }t≥0 and {X
Then
Z
Z
et (dy )
et ) = −
ψet (y )P
W%% (Pt , P
ψt (x)Pt (dx) −
Rd
Rd
For all 0 ≤ t
d %
et ) = −
W (Pt , P
dt %
Z
Z
Lψt (x)Pt (dx) −
Rd
Rd
e
et (dx) .
L ψet (x)P
Integral formulation: for all 0 ≤ s ≤ t
et ) − W % (Ps , P
es ) =
W%% (Pt , P
%
Z t Z
Z
−
Lψr (x)Pr (dx) +
s
Jacopo Corbetta
Rd
Rd
Wasserstein distance evolution
e
e
e
L ψr (x)Pr (dx) dr .
January 10, 2017
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Formal proof
For every s, t ≥ 0
es )
W%% (Ps , P
Jacopo Corbetta
Z
Z
≥−
ψt (x) Ps (dx) −
Rd
Wasserstein distance evolution
Rd
es (dx) .
ψet (x) P
January 10, 2017
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Formal proof
For every s, t ≥ 0
es )
W%% (Ps , P
Z
Z
≥−
ψt (x) Ps (dx) −
Rd
Rd
es (dx) .
ψet (x) P
In particular
Z
Z
ψs (x)(Ps (dx) − Pt (dx)) +
es (dx) − P
et (dx))
ψes (x)(P
Rd
Rd
Z
et ) − W % (Ps , P
es )
≤ W%% (Pt , P
%
Z
es (dx) − P
et (dx)) .
ψt (x)(Ps (dx) − Pt (dx)) +
ψet (x)(P
≤
Rd
Jacopo Corbetta
Rd
Wasserstein distance evolution
January 10, 2017
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Formal proof (2)
Z tZ
Z
ψt (x)(Ps (dx) − Pt (dx)) = −
Rd
Jacopo Corbetta
Lψt (x)Pr (dx)dr
s
Wasserstein distance evolution
Rd
January 10, 2017
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Formal proof (2)
Z tZ
Z
ψt (x)(Ps (dx) − Pt (dx)) = −
Rd
Lψt (x)Pr (dx)dr
s
Rd
1 %
et+h ) − W % (Pt , P
et ) ≥
W% (Pt+h , P
%
h
Z t+hZ
Z t+hZ
1
e
e
−
Lψt (x)Pr (dx)dr
Lψt (x)Pr (dx)dr −
≥
h
t
Rd
t
Rd
Jacopo Corbetta
Wasserstein distance evolution
January 10, 2017
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Formal proof (2)
Z tZ
Z
ψt (x)(Ps (dx) − Pt (dx)) = −
Rd
Lψt (x)Pr (dx)dr
s
Rd
1 %
et+h ) − W % (Pt , P
et ) ≥
W% (Pt+h , P
%
h
Z t+hZ
Z t+hZ
1
e
e
−
Lψt (x)Pr (dx)dr
Lψt (x)Pr (dx)dr −
≥
h
t
Rd
t
Rd
Taking the limit for h → 0+
Z
Z
d
%
e
e
et (dx)
W (Pt , Pt ) ≥ −
Lψt (x)Pt (dx) −
L ψet (x)P
dt + %
d
d
R
R
In the same way:
d
et ) ≤ −
W % (Pt , P
dt − %
Jacopo Corbetta
Z
Z
Lψt (x)Pt (dx) −
Rd
Wasserstein distance evolution
Rd
e
et (dx) .
L ψet (x)P
January 10, 2017
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Main Issues
Technical problems:
I
I
ψt , Lψt integrability with respect to Ps .
er ) differentiability.
r 7→ W%% (Pr , P
Jacopo Corbetta
Wasserstein distance evolution
January 10, 2017
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Pure jump: Lf (x) = λ(x)
R
Rd
k(x, dy ) (f (y ) − f (x))
Theorem
Assume that
I
I
e
supx∈Rd max(λ(x), λ(x))
<∞
et |%(1+ε) ] is locally bounded.
t 7→ E[|Xt |%(1+ε) + |X
Then
I
R
R
et (dx) is locally
t 7→ Rd |Lψt (x)|Pt (dx) + Rd |e
L ψet (x)|P
bounded
et ) is locally Lipschitz on (0, +∞) and for
t 7→ W%% (Pt , P
I
almost every t ∈ (0, ∞)
Z
Z
d %
e
e
et (dx) .
W (Pt , Pt ) = −
Lψt (x)Pt (dx) −
L ψet (x)P
dt %
Rd
Rd
for every t ≥ 0 the integral formula holds true.
I
Jacopo Corbetta
Wasserstein distance evolution
January 10, 2017
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Piecewise Deterministic Markov Processes
Z
Lf (x) = V (x)∇f (x) + λ(x)
k(x, dy ) (f (y ) − f (x)) .
Rd
The result still holds true
Jacopo Corbetta
Wasserstein distance evolution
January 10, 2017
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Piecewise Deterministic Markov Processes
Z
Lf (x) = V (x)∇f (x) + λ(x)
k(x, dy ) (f (y ) − f (x)) .
RdC
The result still holds true but:
I
we have to restrict on the real line;
I
more regularity on the marginals is required.
Jacopo Corbetta
Wasserstein distance evolution
January 10, 2017
8/9
Thank you for your attention
Jacopo Corbetta
Wasserstein distance evolution
January 10, 2017
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