maximal monotone operators in wasserstein spaces of

Maximal Monotone Operators in WassersteinSpaces of Probability Measures

Igor Stojkovic

Delft Institute of Applied MathematicsDelft University of Technology

Thursday, November 10th, 2011Dutch - Japanese Workshop Analysis of non-equilibrium evolution problems:

selected topics in material and life sciencesEurandum, Eindhoven

Contents of the Talk

I Review of Maximal Monotone Operators in Hilbert andBanach spaces

I Review of Wasserstein spaces and Gradient Flows there

I Maximal Monotone Operators in Waserstein spaces

Some history of Maximal Monotone Operators (2)

I Introduced by G. Minty (1960’s+-) in the context of studyingnetworks.

I Applications to elliptic, hyperbolic and parabolic PDE’s (insuch context introduced by F. Browder and his school).

I Applications in Optimization theory.

I Proximal point algorithm, Krasnoselskii algorithm (equivalentin Hilbert spaces).

By 1975 maximal monotone operators on reflexive Banach spaceshave been well understood.These objects have gained a status of a fundamental mathematicalobject; a lot of literature is available.

Some history of Maximal Monotone Operators (2)

I Introduced by G. Minty (1960’s+-) in the context of studyingnetworks.

I Applications to elliptic, hyperbolic and parabolic PDE’s (insuch context introduced by F. Browder and his school).

I Applications in Optimization theory.

I Proximal point algorithm, Krasnoselskii algorithm (equivalentin Hilbert spaces).

By 1975 maximal monotone operators on reflexive Banach spaceshave been well understood.These objects have gained a status of a fundamental mathematicalobject; a lot of literature is available.

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On a Banach space X a monotone operator T is defined as asubset of X × X ∗ such that

< x∗ − y∗, x − y >> 0, ∀ [x , x∗], [y , y∗] ∈ T .

In Hilbet spaces situation becomes more simple. For simplicity, Iwill continue reviewing the classical theory for Hibert spaces only.Nevertheless, little changes in reflexive Banach spaces, and by nowmuch thery has been constructed in non-reflexive spaces.

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On a Banach space X a monotone operator T is defined as asubset of X × X ∗ such that

< x∗ − y∗, x − y >> 0, ∀ [x , x∗], [y , y∗] ∈ T .

In Hilbet spaces situation becomes more simple. For simplicity, Iwill continue reviewing the classical theory for Hibert spaces only.Nevertheless, little changes in reflexive Banach spaces, and by nowmuch thery has been constructed in non-reflexive spaces.

Maximal monotone operators in Hilbert spaces (3 )

Let H be a Hilbert space.A subset (i.e. a multi-valued operator) A ⊂ H × H is monotone iff

< y1 − y2, x1 − x2 >> 0, ∀ [x1, y1], [x2, y2] ∈ A.

Such operator is maximal iff it is not a proper subset of anothermonotone subset A ⊂ H × H.

An equivalent characterization is give by the requirement that theresolvent operators

Jhx := (I + hA)−1x

are defined for all x ∈ H and h > 0.

< y1 − y2, x1 − x2 >> 0, ∀ [x1, y1], [x2, y2] ∈ A.

Jhx := (I + hA)−1x

< y1 − y2, x1 − x2 >> 0, ∀ [x1, y1], [x2, y2] ∈ A.

Jhx := (I + hA)−1x

‘Elliptic’ problem–Resolvents (2 )

Due to monotonicity assumption the resolvent operator Jh is singlevalued for each x ∈ H: if [x1, y1], [x2, y2] ∈ A and

x1 + hy1 = x , x2 + hy2 = x ,

then equating these expressions and due to monotonicity

|x1 − x2|2 = h < x1 − x2, y2 − y1 >6 0.

Therefore x1 = x2.

However, one must explicitly require existence of the solutions ofthe ‘elliptic’ problem

x1 + hAx1 = x

for x ∈ H and h > 0.

‘Elliptic’ problem–Resolvents (2 )

Due to monotonicity assumption the resolvent operator Jh is singlevalued for each x ∈ H: if [x1, y1], [x2, y2] ∈ A and

x1 + hy1 = x , x2 + hy2 = x ,

then equating these expressions and due to monotonicity

|x1 − x2|2 = h < x1 − x2, y2 − y1 >6 0.

Therefore x1 = x2.

However, one must explicitly require existence of the solutions ofthe ‘elliptic’ problem

x1 + hAx1 = x

for x ∈ H and h > 0.

Some basic properties of maximal monotone operators in H

I For each x ∈ Dom(A) the set Ax is a closed and convexsubset of H. A0x denotes unique element of minimal norm.

I For each x ∈ H we have Jhx → ProjDomAH x as h→ 0.

I Resolvent identity holds: Jhx = Jt( thx + (1− t

h )Jhx), forh > t > 0, x ∈ H.

Abstract Cauchy Problem

Find an absolutely continuous curve x : [0,+∞)→ H such that

dtx(t) ∈ −Ax(t), L1-a.e. t > 0

x(0) = x0 ∈ D(A)(1)

Solutions of the Cauchy Problem—’Exponential’ formula

Solutions of the CP are obtained by

Stx0 = x(t) : = limn→∞

(J tn)nx0

= limn→∞

(I +tA

n)−nx0

“ = ”e−tAx0, ∀x0 ∈ D(A).

In general (2) does not hold for x0 ∈ D(A) \ D(A)

Contraction Semigroup of Solutions of CP (4)

I For each x0 ∈ D(A) the curve 0 6 t → Stx0 is differentiablefor L1-a.e. t > 0 and

dtStx0 ∈ −AStx0, L1 -a.e. t > 0.

I The mapping S : [0,+∞)× D(A)→ D(A) is a contractionsemigroup which thus can be extended in a unique way to[0,+∞)× D(A)→ D(A)

I For each x0 ∈ D(A) and t > 0, 0 6 t → Stx0 is differentiablefrom the right and

dt+Stx0 = −A0Stx0

I For each x0 ∈ D(A) the fucntion 0 6 t → | ddt+

Stx0| isnon-increasing.

dt+Stx0 = −A0Stx0

Special case: A = ∂ϕ (3 )If A = ∂ϕ for a convex lower semi-continuous (l.s.c.) functionalϕ : H → (−∞,+∞] which is moreover proper (i.e. ϕ 6≡ +∞),then the associated Frechet sub-differential

∂ϕ := [x , ξ] ∈ H × H| < ξ, z − x > +ϕ(x) 6 ϕ(z)

is a maximal monotone operator.

The resolvents are given by

Jhx = (I +h∂ϕ)−1x = argminy∈H1

2h|y−x |2 +ϕ(x), h > 0, x ∈ H,

and the associated semigroup exibits (in addition) the regularizingeffect property:

e−t∂ϕx ∈ D(∂ϕ), ∀ t > 0, ∀ x ∈ D(∂ϕ).

Consequently, for each x ∈ D(∂ϕ)

dtStx0 ∈ −∂ϕ(Stx0), L1-a.e. t > 0!

∂ϕ := [x , ξ] ∈ H × H| < ξ, z − x > +ϕ(x) 6 ϕ(z)

2h|y−x |2 +ϕ(x), h > 0, x ∈ H,

e−t∂ϕx ∈ D(∂ϕ), ∀ t > 0, ∀ x ∈ D(∂ϕ).

dtStx0 ∈ −∂ϕ(Stx0), L1-a.e. t > 0!

∂ϕ := [x , ξ] ∈ H × H| < ξ, z − x > +ϕ(x) 6 ϕ(z)

2h|y−x |2 +ϕ(x), h > 0, x ∈ H,

e−t∂ϕx ∈ D(∂ϕ), ∀ t > 0, ∀ x ∈ D(∂ϕ).

dtStx0 ∈ −∂ϕ(Stx0), L1-a.e. t > 0!

Some elliptic PDE’s where monotone operators are useful (2 )

I Let Ω ⊂ Rd be bouneded, open with suff. smooth boundary,let a(x , u), i = 0, 1, .., n be ’reasonable’ functions

−∆u(x) + λ

[ n∑j=1

aj(x , u(x)) + a0(x , u(x))

]= 0, x ∈ Ω.

Solutions in L2(Ω). The relevant max. monotone operator is∆ : W 2,2(Ω)

⋂W 1,2

0 (Ω)→ L2(Ω).

I The p-Laplacians: X := W 1,p(Ω), and de max. monotoneoperator is

∆pu := −div(|∇u|p−2∇u) ∈W−1,q(Ω),

( 1p + 1

q = 1), and for λ > 0 one can solve

λu + ∆pu = 0

Some elliptic PDE’s where monotone operators are useful (2 )

I Let Ω ⊂ Rd be bouneded, open with suff. smooth boundary,let a(x , u), i = 0, 1, .., n be ’reasonable’ functions

−∆u(x) + λ

[ n∑j=1

aj(x , u(x)) + a0(x , u(x))

]= 0, x ∈ Ω.

Solutions in L2(Ω). The relevant max. monotone operator is∆ : W 2,2(Ω)

⋂W 1,2

0 (Ω)→ L2(Ω).

I The p-Laplacians: X := W 1,p(Ω), and de max. monotoneoperator is

∆pu := −div(|∇u|p−2∇u) ∈W−1,q(Ω),

( 1p + 1

q = 1), and for λ > 0 one can solve

λu + ∆pu = 0

The wave equation

dt)2u(t, x) = ∆u(t, x), (t, x) ∈ [0,+∞)× Ω, Ω ⊂ Rd ,

u(x , t) = 0 for x ∈ ∂Ω, u(x , 0) := u0(x),d

dtu(x , 0) := u1(x)

Maximal monotone operator on W 1,20 (Ω)× L2(Ω)

A(u, v) := (v ,−∆u),

Dom(A) = (W 1,20 (Ω)

⋂W 2,2(Ω))×W 1, 20(Ω).

NB: This operator is not a sub-differential of a convex functional.

The wave equation

dt)2u(t, x) = ∆u(t, x), (t, x) ∈ [0,+∞)× Ω, Ω ⊂ Rd ,

u(x , t) = 0 for x ∈ ∂Ω, u(x , 0) := u0(x),d

dtu(x , 0) := u1(x)

Maximal monotone operator on W 1,20 (Ω)× L2(Ω)

A(u, v) := (v ,−∆u),

Dom(A) = (W 1,20 (Ω)

⋂W 2,2(Ω))×W 1, 20(Ω).

NB: This operator is not a sub-differential of a convex functional.

Wasserstein spaces of probability measures (P2(Rd),W2)1 (4 )

P(Rd) ⊃ P2(Rd) := µ|∫|x |2dµ(x) < +∞.

For µ0, µ1 ∈ P2(Rd) consider Γ(µ0, µ1) ⊂ P2(R2d) which consistsof measures γ such that

πjγ = µj , j = 0, 1, (πj(x0, x1) := xj).

Define the Wasserstein distance W2 (of order 2) by:

W 22 (µ0, µ1) := inf

γ∈Γ(µ0,µ1)∫R2d

|x0 − x1|2dγ(x0, x1)

Infimum always attained, and the set of optimizing couppling plansdenoted by Γo(µ0, µ1)—either a one point set or uncountablyinfinite.

(P2(Rd)) is a Polish space. Moreover, this space is positivelycurved in the sense of Alexandrov.

1Another term for these distances that can be found in the literature, andquite frankly a more appropriate one , is the Monge-Kantorovich distances.

P(Rd) ⊃ P2(Rd) := µ|∫|x |2dµ(x) < +∞.

πjγ = µj , j = 0, 1, (πj(x0, x1) := xj).

W 22 (µ0, µ1) := inf

|x0 − x1|2dγ(x0, x1)

P(Rd) ⊃ P2(Rd) := µ|∫|x |2dµ(x) < +∞.

πjγ = µj , j = 0, 1, (πj(x0, x1) := xj).

W 22 (µ0, µ1) := inf

|x0 − x1|2dγ(x0, x1)

P(Rd) ⊃ P2(Rd) := µ|∫|x |2dµ(x) < +∞.

πjγ = µj , j = 0, 1, (πj(x0, x1) := xj).

W 22 (µ0, µ1) := inf

|x0 − x1|2dγ(x0, x1)

The mass transport idea

The formulation from the previous slide is due to Kantorovich(1950’s). The original formulation of the problem is due to Monge(around 1800): given µ0, µ1 ∈ P2(Rd find a map r = rµ1

µ0 such thatr#µ0 = µ1 which mimizes the cost functional

C(r) :=

c(x , r(x))dµ0,

among al such maps r . In our case c(x , y) := |x − y |2. This is aprototype problem in the popular Optimal Transportation theory.More generaly µ0 and µ1 can be defined on different metric spacesand the cost function c may be quite general (convex, concave,etc.)

What does this have to with PDE’s?? (2 )

During the last (over a) decade many PDE’s that satisfyconservation law and preserve positivity have been interpreted as agradient flow in a Waserstein-2 space (absolutely continuousmeasures are identified with their densities).

Initiating seminal papers:

I The Variational Formulation of the Fokker-Planck Equations.R. Jordan, D. Kinderlehrer, F. Otto

I The Geometry of Dissipative Equations: the Porous MediumEquation. F. Otto

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The whole theory is jointly called the Optimal TransporationTheory and it is a rather multi-disciplinary field joining:

I Probability,

I Analysis,

I Geometry, Geometric inequalities

I PDE’s.

Most (though not all!) of the PDE’s that have been interpreted asa gradient flow in P2(Rd) have already been ’solved’ by moreclassical means before. However, the new theory sheds much newlight and provides new insights into many different equations.

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The whole theory is jointly called the Optimal TransporationTheory and it is a rather multi-disciplinary field joining:

I Probability,

I Analysis,

I Geometry, Geometric inequalities

I PDE’s.

Most (though not all!) of the PDE’s that have been interpreted asa gradient flow in P2(Rd) have already been ’solved’ by moreclassical means before. However, the new theory sheds much newlight and provides new insights into many different equations.

Gradient flows in (P2(Rd),W2) (2)

∂tρ+∇ · (ρv) = 0 (Continuity Equation)

v = ∇δΦ

δρ(Non-linear Variational Condition)

ρ(0, .) = ρ0 ρ0 ∈ L1(Rd), ρ0 > 0.

Here Φ is an integral functional and δΦδu is it’s Euler-Lagrange first

variation:

Φ(u) :=

ϕ(x , u,Du)dx ,δΦ

δu= ϕu(x , u,Du)−divϕp(x , u,Du),

for a function ϕ = ϕ(x , u, p), with (x , u, p) ∈ R× R>0 × Rd .Applications: existence and asymptotic behaviour of solutions,contracton properties, Logarithmic Sobolev Inequalities,approximation algorithms, ...

Gradient flows in (P2(Rd),W2) (2)

∂tρ+∇ · (ρv) = 0 (Continuity Equation)

v = ∇δΦ

δρ(Non-linear Variational Condition)

ρ(0, .) = ρ0 ρ0 ∈ L1(Rd), ρ0 > 0.

Here Φ is an integral functional and δΦδu is it’s Euler-Lagrange first

variation:

Φ(u) :=

ϕ(x , u,Du)dx ,δΦ

δu= ϕu(x , u,Du)−divϕp(x , u,Du),

for a function ϕ = ϕ(x , u, p), with (x , u, p) ∈ R× R>0 × Rd .Applications: existence and asymptotic behaviour of solutions,contracton properties, Logarithmic Sobolev Inequalities,approximation algorithms, ...

The Continuity equation (2)

Theorem(Ambrosio-Gigli-Savare) A curve µ : (0,T )→ P2(Rd) is absoluteycontinuous w.r.t. W2 iff there is a vector fieldvt(x) ∈ TanµtP2(Rd) such that

∂µt +∇ · (vtµt) = 0, and∫ T

0|vt |2L2(µt ;Rd )dt < +∞.

(the first equation is interpreted in distributional sence). Moreoversuch vector field vt is t-a.e. unique

The tangent spaces TanµP2(Rd), for µ ∈ P2(Rd) are defined by

∇ϕ|ϕ ∈ C∞c (Rd)L2(µ;Rd )

The Continuity equation (2)

Theorem(Ambrosio-Gigli-Savare) A curve µ : (0,T )→ P2(Rd) is absoluteycontinuous w.r.t. W2 iff there is a vector fieldvt(x) ∈ TanµtP2(Rd) such that

∂µt +∇ · (vtµt) = 0, and∫ T

0|vt |2L2(µt ;Rd )dt < +∞.

(the first equation is interpreted in distributional sence). Moreoversuch vector field vt is t-a.e. unique

The tangent spaces TanµP2(Rd), for µ ∈ P2(Rd) are defined by

∇ϕ|ϕ ∈ C∞c (Rd)L2(µ;Rd )

Why is the Continuity equation in P2(Rd) true?

To give a hint consider the following equation on Rd :

dtXt(x) = vt(Xt(x), t ∈ [0,T ],

which has solutions if the vector field (vt)t is sufficiently nice. Fora µ0 given, define the curve µt := (Xt)#µ0. The one easilycomputes for ϕ ∈ C∞c (Rd) that

∫ϕdµt =

∫ϕ(Xt(x)dµ0(x)

∫< ∇ϕ(Xt(x)), vt(Xt(x)) > dµ0(x)

∫< ∇ϕ, vt > dµt

Some examples of gradient flows in Wasserstein spaces (2)(Recall the gradient flow structure from several slides back)1.: The heat equation:

∂ρt = ∆ρt = ∇ · (ρt ∇ρtρt)

Corresponding Wasserstein-2 functional is the logarithmic entropy:

Φ(ρ) :=

ρ logρdx ,δΦ

δρ= logρ+ 1, v = −∇ρ

ρ, ρv = −∇ρ.

2.: The Fokker-Planck quations:

∂ρt = ∆ρt +∇ · (ρ∇V ) = ∇ · (ρt(∇ρtρt+∇V ))

Here V : Rd → (−∞,+∞] is l.s.c. conex and proper.Corresponding Wasserstein-2 quantities:

Φ(ρ) :=

ρ (logρ+ V (x))dx =

ρ(log(ρ/e−V ))dx ,

δρ= logρ+ 1 + V , v = −∇(

δρ) = −∇ρ

ρ+∇V ,

ρv = −∇ρ+ ρ∇V .

Some examples of gradient flows in Wasserstein spaces (2)(Recall the gradient flow structure from several slides back)1.: The heat equation:

∂ρt = ∆ρt = ∇ · (ρt ∇ρtρt)

Corresponding Wasserstein-2 functional is the logarithmic entropy:

Φ(ρ) :=

ρ logρdx ,δΦ

δρ= logρ+ 1, v = −∇ρ

ρ, ρv = −∇ρ.

2.: The Fokker-Planck quations:

∂ρt = ∆ρt +∇ · (ρ∇V ) = ∇ · (ρt(∇ρtρt+∇V ))

Here V : Rd → (−∞,+∞] is l.s.c. conex and proper.Corresponding Wasserstein-2 quantities:

Φ(ρ) :=

ρ (logρ+ V (x))dx =

ρ(log(ρ/e−V ))dx ,

δρ= logρ+ 1 + V , v = −∇(

δρ) = −∇ρ

ρ+∇V ,

ρv = −∇ρ+ ρ∇V .

Some examples of non-linear (2)

3.: Non-linear (power like) equation:

∂ρt = ∇ · (∇ρβt ) = ∇ · ( ββ−1ρt∇ρ

β−1t )

Φ(ρ) := 1β−1

∫Rd ρ

βdx , δΦδρ = β

β−1ρβ−1, ρv = ρ∇ δΦ

δρ = ∆ρβ.

4.: Patlak-Keller-Sigel model:

∂tρ = ∆ρ+∇ · (ρ∇c), c(x) := − 12π

∫R2 log|x − y | ρ(y) dy

Φ(ρ) =∫ρlogρdx + 1

∫ ∫R2×R2 log|x − y |ρ(x)ρ(y)dxdy

δΦδρ = logρ+ 1 + 1

2π log|.| ∗ ρ = logρ+ 1− c

ρv = ρ∇ δΦδρ = ∇ρ+ ρ∇c [thus ∇ · (ρv) = ∆ρ+∇ · (ρ∇c) ]

Some examples of non-linear (2)

3.: Non-linear (power like) equation:

∂ρt = ∇ · (∇ρβt ) = ∇ · ( ββ−1ρt∇ρ

β−1t )

Φ(ρ) := 1β−1

∫Rd ρ

βdx , δΦδρ = β

β−1ρβ−1, ρv = ρ∇ δΦ

δρ = ∆ρβ.

4.: Patlak-Keller-Sigel model:

∂tρ = ∆ρ+∇ · (ρ∇c), c(x) := − 12π

∫R2 log|x − y | ρ(y) dy

Φ(ρ) =∫ρlogρdx + 1

∫ ∫R2×R2 log|x − y |ρ(x)ρ(y)dxdy

δΦδρ = logρ+ 1 + 1

2π log|.| ∗ ρ = logρ+ 1− c

ρv = ρ∇ δΦδρ = ∇ρ+ ρ∇c [thus ∇ · (ρv) = ∆ρ+∇ · (ρ∇c) ]

Metric gradient flow re-formulations in Hilbert

spaces—Ambrosio-Gigli-Savare point of veiw

dtx(t) ∈ −∂Φ(x(t)) ⇐⇒

<− d

dtx(t), z − x(t) > +Φ(x(t)) 6 Φ(z), ∀z ∈ Dom(Φ)

dtx(t), z − x(t) >=

dt|x(t)− z |2

Hence in Hilbert spaces GF curves associated to Φ are equivalentlyreformulated as the Evolution Variational Inequality (EVI)

dtd(x(t), z)2 + Φ(x(t) 6 Φ(z), ∀z ∈ Dom(Φ),

a problem which can be posed in any metric space!

dtx(t) ∈ −∂Φ(x(t)) ⇐⇒

<− d

dtx(t), z − x(t) >=

dt|x(t)− z |2

dtx(t) ∈ −∂Φ(x(t)) ⇐⇒

<− d

dtx(t), z − x(t) >=

dt|x(t)− z |2

’Necessary’ geomoetric properties for the existence of

resolvents and of solutions

Most importantly a non-positively curvature like condition: foreach v , x0, x1 ∈ X there is a curve t → x(t) connecting x0 with x1

such that

d2(v , x(t)) 6 (1− t)d2(v , x0) + td2(v , x1)− t(1− t)d2(x0, x1), and

Φ(x(t)) 6 (1− t)Φ(x0) + tΦ(x1), ∀t ∈ [0, 1].

Also Φ must be l.s.c., proper and bounded from below on a ball inX .

Generalized geodesics in (P2(Rd),W2)

For any tripple σ, µ0, µ1 ∈ P2(Rd),W2) there is a measureγ ∈ P2(R3d) whose 1,2 and 1,3 marginals are optimal couplings forσ, µ0 and σ, µ1 respectively, the curve

t → ((1− t)π1 + tπ2)#γ

does the trick! That is W 22 is (-1)-convex along such curves.

However we always need to prove convexity of the functional underconsideration along such curves (or alternatively prove that it’slevel sets are compact2).

2NB Even in the linear setting, GF solutions exist either if we are in aHilbert space—flat setting, or functional has (relatively )compact level sets wrtsuitable topology. Indeed no general results in Banach spaces!

Frechet Sub-differential of convex functionals in Wasserstein

spaces (AGS)

DEFINITION:

ξ ∈ ∂Φ(µ) ⊂ L2(µ,Rd) iff∫R2d

< ξ(x1), x2 − x1 > dγ + Φ(µ) 6 Φ(ν), ∀γ ∈ Γo(µ, ν)

The most immeadeate approach in generalizing to operators seemsto be ∫

< ξ(x1)− ξ(x2), x2 − x1 > dγ > 0

However this doesn’t account for the geometric assumptions, i.e.we need to make use of the generalized (-1)-convexity of W 2

Frechet Sub-differential of convex functionals in Wasserstein

spaces (AGS)

DEFINITION:

ξ ∈ ∂Φ(µ) ⊂ L2(µ,Rd) iff∫R2d

< ξ(x1), x2 − x1 > dγ + Φ(µ) 6 Φ(ν), ∀γ ∈ Γo(µ, ν)

The most immeadeate approach in generalizing to operators seemsto be ∫

< ξ(x1)− ξ(x2), x2 − x1 > dγ > 0

However this doesn’t account for the geometric assumptions, i.e.we need to make use of the generalized (-1)-convexity of W 2

Maximal Monotone Operators in Generalized Sense (MMG) in

P2(Rd)

DEFINITION (I.S. 2010): Operators consist of ordered pairs [µ, ξ],where ξ ∈ TanµP2(Rd). Monotone in generalized sense iff for each[µ1, ξ1, [µ2, ξ2] ∈ A and each pair of base measuresσ1, σ2 ∈ P2(Rd) there are γ1, γ2 ∈ P2(R3d), whose 1,2 and 1,3marginals are optimal for σ1, µ1, etc (draw a picture!) and suchthat∫

< ξ(x1), x2 − x1 > dγ1 +

∫R3d

< ξ2(x2), x1 − x2 > dγ2 6 0

Notion of Maximality

An MMG operator A is maximal if resolvents are defined for allsufficiently small h > 0 and µ ∈ DomA) as follows: given µ and hwe say that a measure µh ∈ P2(Rd) is the resolvent Jhµ if

[µh,rµµh − i

h] ∈ A (“⇐⇒′′ µ− µh

h∈ A(µh))

Off course, it is absolutely necessary to show that this in factextends the framework of Ambrosio-Gigli-Savare, for it has beenproven that many functionals fit into their framework, and manyPDE’s can be treated accordingly. In particuar, I had to prove thata part of sub-differetntial of a generalized convex functionalsatisfies my definition.

The trick is to ’cut down’ their definition of the sub-differential andshow that the whole construction can be reproduced in a differentway (thus also covering the case when one has a functional).

[µh,rµµh − i

h] ∈ A (“⇐⇒′′ µ− µh

h∈ A(µh))

[µh,rµµh − i

h] ∈ A (“⇐⇒′′ µ− µh

h∈ A(µh))

Main results of my extension

I Proved that my notion extends the AGS notion

I Proved anlogues of some basic proerties of maximal monotoneoperators in Hilbert spaces (proofs not that basic though),such as the Resolvent Identity and also a theorem stating thatfor any µ ∈ P2(Rd) for which resolvents are defined we have

Jhµ→ nearest point projection onto Dom(A)W2

I Constructed solutions of the associated abstract Cauchyproblem, with exactly the same proerties as in Hilbert spaces.This had to be done by a diferent technique then AGS did(due to lack of the full variational structure). The trick is to’connect’ the Euler backward aproximations along the timeaxes togehter with ’descrete derivatives’ into measures on ahigher dimensional domain, and do a fine analysis, there byminding the curvature carefully!

Applications

Most natural PDE to consider seemd to be the Fokker-Planckequation with a monotone but non-gradient drift:

∂tρ = ∆ +∇ · (ρB)(= ∇ · (ρ(∇ρρ

where B : Rd → Rd is a maximal monotone operator itself. In thisdirection I abtained the following results:

I Construct solutions directly in the Wasserstein-2 space bymeans of the product formula (w.r.t. W2): we have the heateq. semigroup, and also the natural transport (contraction)semigroup Rtµ := St#µ where (St)t is the semigroup on Rd

induced by B—without any deep thery of operators.This yealds contraction property of the Fokker-Plancksemigroup.

Unfortunately for me, Natile-Peletier-Savare were a bit fasterwith proving contraction proerty (although by differentmeans).

However I proved the absolute continuity of paths and eventhe same type of regularizing effect which holds in the GFcase (i.e. when B = ∇V ). I also proved that this equation isa GF only if B = ∇V , thus I am indeed need new theory fortreatment! The proof usses my notion of maximal monotoneoperators btw.

Infinite dimensional Brownian motion

Cameron Martin space associated to a Gaussian mesure on aBanach space.H a Hilbert space, (W n

t )t>0 an independant sequence of scalarBrownian motions, λn > 0,

∑n λn <∞, (en)n an o.n.b. in H,

define a trace class operator Qx :=∑λn < x , en > en, and

consider a stochastic process (i.e. a BM)

Bt :=∑n

√λnW n

t en, t > 0.

The law of each Bt is a Gaussian measure on H with zeroexpectation and covariance op. tQ. I consider convolutions of aninitial measure µ0 ∈ P(H) with νt := LawBt . This an analogue ofthe heat equation, although not quite same.

For in infinite dimensions there exists no translation invariantmeasure such as the Lebesgue measure. Moreover we do not havean invariant measure for this Markov semigroup. I am thereforeconvinced that this is not a GF. But I do expect to recover it as aflow generated by a max. mon. op in the (semi)metric Wassersteinspace arising by using the Cameron-Martin distance instead of theHdistance. 3.

3Kantorovich transport problem and also Fokker-Planck eqs have alreadybeen considered in such way by Feyel-Ustunel and Sturm et al

There is one other (very!) interesting PDE’s application that I amworking on right now. However this is not finished so hopefully Ican tell you next time about it.

In the meanwhile if anyone has any other idea about applying thisstuff in any way and likes to discuss it with me, please contact me:[email protected]@gmail.com

There is one other (very!) interesting PDE’s application that I amworking on right now. However this is not finished so hopefully Ican tell you next time about it.

In the meanwhile if anyone has any other idea about applying thisstuff in any way and likes to discuss it with me, please contact me:[email protected]@gmail.com

THATS ALL FOLKS,THANK YOU FOR LISTENING

maximal monotone operators in wasserstein spaces of

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