Dynamic and Stochastic Brenier Transport via Hopf-Lax formulae on Was

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1 Dynamic and Stochastic Brenier Transport via Hopf-Lax formulae on Wasserstein Space With many discussions with Yann Brenier and Wilfrid Gangbo Brenierfest, IHP, January 9-13, 2017

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7 ain points of the talk Optimal transport problem for the ballistic cost function", which is defined on phase space by, T b T (v, x) := inf{ v, γ(0) + L(t, γ(t), γ(t)) dt; γ C 1 ([0, T ), ); γ(t ) = x}, 0 where L : [0, T ] R {+ } is a suitable Lagrangian. Why this as opposed to the "fixed-state space cost"? Existence of Optimal maps Duality and Hamilton-Jacobi equations Corresponding Benamou-Brenier type formulas Hopf-Lax Type formulae on Wasserstein space Hamilton-Jacobi equations on Wasserstein space Connection to mean field games Stochastic mass transport with ballistic cost.

8 Why the ballistic cost? For a given function g, the value function { V g(t, x) = inf g(γ(0)) + V (0, x) = g(x), t is formally a solution of the Hamilton-Jacobi equation, 0 } L(s, γ(s), γ(s)) ds; γ C 1 ([0, T ), ); γ(t) = x, (HJ) tv + H(t, x, xv ) = 0 on [0, T ], where H is the associated Hamiltonian on [0, T ] T, i.e., Both the ballistic cost, b T (v, x) := inf{ v, γ(0) + and the fixed-end cost c T (y, x) := inf{ T H(t, y, x) = sup { v, x L(t, y, v)}. v T T are formally solutions to (HJ). 0 0 L(t, γ(t), γ(t)) dt; γ C 1 ([0, T ), ); γ(t ) = x}, L(t, γ(t), γ(t)) dt; γ C 1 ([0, T ), ); γ(0) = x, γ(t ) = y}

9 Hopf-Lax and Dual Hopf-Lax formula Both costs can be seen as "Kernels" that can be used to generate general solutions for (HJ). General Hopf-Lax Lower kernel: V g(t, x) = inf{g(y) + c(t, y, x); y } General Dual Hopf-Lax formula Upper kernel: V g(t, x) = sup{b(t, v, x) g (v); v } provided the Lagrangian L is jointly convex and the initial function g is convex. Classical Hopf-Lax and Dual Hopf-Lax formulae If L(x, v) = L 0 (v) and L 0 convex, then and c t(y, x) = tl 0 ( 1 t x y ) and bt(v, x) = v, x th 0(v). V g(t, x) = inf{g(y) + tl 0 ( 1 t x y ); y } = (g + th 0 ). When defined, the upper kernel is much better than the lower kernel.

10 Hopf-Lax and Dual Hopf-Lax formula Both costs can be seen as "Kernels" that can be used to generate general solutions for (HJ). General Hopf-Lax Lower kernel: V g(t, x) = inf{g(y) + c(t, y, x); y } General Dual Hopf-Lax formula Upper kernel: V g(t, x) = sup{b(t, v, x) g (v); v } provided the Lagrangian L is jointly convex and the initial function g is convex. Classical Hopf-Lax and Dual Hopf-Lax formulae If L(x, v) = L 0 (v) and L 0 convex, then and c t(y, x) = tl 0 ( 1 t x y ) and bt(v, x) = v, x th 0(v). V g(t, x) = inf{g(y) + tl 0 ( 1 t x y ); y } = (g + th 0 ). When defined, the upper kernel is much better than the lower kernel.

11 Another example If L(x, v) = L 0 (v Ax), where L 0 is convex, lsc and A a matrix, then b(t, v, x) = e ta x, v ψ(t, v), where Ψ(t, v) = t 0 H 0(e sa v) ds. While c(t, y, x) = Ψ (t, e ta y x). The value function is then, using the fundamental kernel While by using the dualizing kernel V g(t, x) = inf y {g(y) + Ψ (t, e ta y x)}. V g(t, x) = (g + Ψ(t, ) (e ta x).

12 A dual cost function Introduce another cost functional c T (u, v) := inf{ T 0 L(t, γ(t), γ(t)) dt; γ C 1 ([0, T ), ); γ(0) = u, γ(t ) = v}, The new Lagrangian L is defined on by L(t, x, p) := L (t, p, x) = sup{ p, y + x, q L(t, y, q); (y, q) }. The corresponding Hamiltonian is H L is then given by H L(x, y) = H(y, x). Recall Bolza s duality: (P) = ( P), where (P) and its dual inf{ T 0 L(γ(s), γ(s)) ds + l(γ(0), γ(t )) over all γ C1 ([0, T ), )} ( P) inf{ T 0 L(γ(s), γ(s)) ds + l (γ(0), γ(t )) over all γ C 1 ([0, T ), ).} This has several consequences

13 Ballistic cost satisfies two H-J equations One consequence is that the Legendre transform of the value functional x V g(t, x) := inf{g(y) + c(t, y, x); y } is another value functional which yields that Ṽ g (t, w) = inf {g (v) + c(t, v, w); v }, b(t, v, x) = inf{ v, y +c(t, y, x); y } = sup{ w, x c(t, v, w); w }. So, x b(v, x) was a "solution" of the HJ equation tb + H(t, x, xb) = 0 on [0, T ], b 0 (x) = v, x. Now v b(v, x) is also a solution for another H-J equation: tb H(t, vb, v) = 0 on [0, T ], b T (v) = v, x.

14 The Ballistic Optimal Transport Problem The associated transport problems will be B T (µ 0, ν T ) := sup{ b T (v, x) dπ; π K(µ 0, ν T )}, B T (µ 0, ν T ) := inf{ b T (v, x) dπ; π K(µ 0, ν T )}, where µ 0 (resp., ν T ) is a probability measure on (resp., ), and K(µ 0, ν T ) is the set of probability measures π on whose marginal on (resp. on ) is µ 0 (resp., ν T ) (the transport plans). Note that when T = 0, we have b 0 (x, v) = v, x, which is exactly the case considered by Brenier, that is W (µ 0, ν 0 ) := sup{ v, x dπ; π K(µ 0, ν 0 )}, W (µ 0, ν 0 ) := inf{ v, x dπ; π K(µ 0, ν 0 )}, This is the dynamic version of the Wasserstein distance.

15 The Ballistic Optimal Transport Problem The associated transport problems will be B T (µ 0, ν T ) := sup{ b T (v, x) dπ; π K(µ 0, ν T )}, B T (µ 0, ν T ) := inf{ b T (v, x) dπ; π K(µ 0, ν T )}, where µ 0 (resp., ν T ) is a probability measure on (resp., ), and K(µ 0, ν T ) is the set of probability measures π on whose marginal on (resp. on ) is µ 0 (resp., ν T ) (the transport plans). Note that when T = 0, we have b 0 (x, v) = v, x, which is exactly the case considered by Brenier, that is W (µ 0, ν 0 ) := sup{ v, x dπ; π K(µ 0, ν 0 )}, W (µ 0, ν 0 ) := inf{ v, x dπ; π K(µ 0, ν 0 )}, This is the dynamic version of the Wasserstein distance.

16 One of the Kantorovich Potentials is nice By standard Kantorovich duality, B T (µ 0, ν 0 ) : = inf { b(v, x)) dπ; π K(µ 0, ν T ) } = sup { φ 1 (x) dν T (x) φ 0 (v) dµ 0 (v); φ 1, φ 0 K(b) }, where K(b) is the set of functions φ 1 L 1 (, ν T ), φ 0 L 1 (, µ 0 ) such that φ 1 (x) φ 0 (v) b(v, x) for all (v, x). Kantorovich functions in K(c) can be assumed to satisfy φ 1 (x) = inf b(v, x) + φ 0(v) and φ 0 (v) = sup φ 1 (x) b(v, x). v x Say that φ 0 (resp., φ 1 ) is an initial (resp., final) Kantorovich potential.

17 ain advantages of the ballistic cost b T (v, x) is concave in v and convex in x. It is also Lipschitz continuous. In the case of B T (µ 0, ν T ), the initial Kantorovich potential φ 0 is convex, though nothing can be said about φ 1. For B T (µ 0, ν T ), the final potential is convex and nothing can be said about φ 1. Even though c(y, x) is jointly convex, nothing can be said about the Kantorovich potentials of C T (µ 0, µ T ) := inf{ c T (y, x) dπ; π K(µ 0, µ T )}, including the case where L(x, v) = v p (p 1), that is when c 1 (y, x) = x y p. Gangbo-cCann worked with c-convexity in order to deal with the regularity of Kantorovich potentials.

18 inimizing ap for Ballistic Cost Theorem (A): Under suitable assumptions on the Lagrangian L. Let µ 0, ν T be probabilities on, such that µ 0 is absolutely continuous with respect to Lebesgue measure. Then, 1. The following duality holds: { B T (µ 0, ν T ) = sup φ T (x) dν T (x) + φ 0 (v) dµ 0 (v); φ 0 concave & φ t solution of (HJ)}. { tφ + H(t, x, xφ) = 0 on [0, T ], φ(0, x) = φ 0 (x), 2. There exists a concave function φ 0 : R and a bounded locally Lipschitz vector field X(x, t) : ]0, T [ ) such that, if Φ t s, (s, t) ]0, T [ 2 is the flow of X from time s to time t, then B T (µ 0, ν T ) = b T (v, Φ T 0 φ 0 (v)dµ 0 (v).

19 inimizing ap for Ballistic Cost Theorem (A): Under suitable assumptions on the Lagrangian L. Let µ 0, ν T be probabilities on, such that µ 0 is absolutely continuous with respect to Lebesgue measure. Then, 1. The following duality holds: { B T (µ 0, ν T ) = sup φ T (x) dν T (x) + φ 0 (v) dµ 0 (v); φ 0 concave & φ t solution of (HJ)}. { tφ + H(t, x, xφ) = 0 on [0, T ], φ(0, x) = φ 0 (x), 2. There exists a concave function φ 0 : R and a bounded locally Lipschitz vector field X(x, t) : ]0, T [ ) such that, if Φ t s, (s, t) ]0, T [ 2 is the flow of X from time s to time t, then B T (µ 0, ν T ) = b T (v, Φ T 0 φ 0 (v)dµ 0 (v).

20 aximizing ap for Ballistic Cost Theorem (B): Under suitable assumptions on the Lagrangian L. Let µ 0, ν T be probabilities on, such that ν T is absolutely continuous with respect to Lebesgue measure. Then, 1. The following duality holds: { B T (µ 0, ν T ) = inf ψt (x) dν T (x) + ψ 0 (v) dµ 0 (v); ψ T convex & ψ t solution of (dual-hj)}. { tψ H( vψ, v) = 0 on [0, T ], ψ(t, v) = ψ T (v), 2. There exists a convex function ψ : R and a bounded locally Lipschitz vector field Y (x, t) : ]0, T [ ) such that, if Ψ t s, (s, t) ]0, T [ 2 is the flow of Y from time s to time t, then B T (µ 0, ν T ) = b T (v, ψ Ψ T 0 (v))dµ 0 (v).

21 aximizing ap for Ballistic Cost Theorem (B): Under suitable assumptions on the Lagrangian L. Let µ 0, ν T be probabilities on, such that ν T is absolutely continuous with respect to Lebesgue measure. Then, 1. The following duality holds: { B T (µ 0, ν T ) = inf ψt (x) dν T (x) + ψ 0 (v) dµ 0 (v); ψ T convex & ψ t solution of (dual-hj)}. { tψ H( vψ, v) = 0 on [0, T ], ψ(t, v) = ψ T (v), 2. There exists a convex function ψ : R and a bounded locally Lipschitz vector field Y (x, t) : ]0, T [ ) such that, if Ψ t s, (s, t) ]0, T [ 2 is the flow of Y from time s to time t, then B T (µ 0, ν T ) = b T (v, ψ Ψ T 0 (v))dµ 0 (v).

22 Our assumptions on the Lagrangian (A1) L : R is convex, proper and lower semi-continuous. (A2) The set F(x) := {v; L(x, v) < } is non-empty for all x, and for some ρ > 0, we have for all x, dist(0, F (x)) ρ(1 + x ). (A3) For all (x, v), we have L(x, v) θ(max{0, v α x }) β x, where α, β are constants, and θ is coercive, non-decreasing on [0, ). Equivalently, for the corresponding Hamiltonian H, (A1) H(x, y) is finite and concave in x convex in y. (A2) H(x, y) φ(y) + (α y + β) x, where α, β constants and φ convex. (A3) H(x, y) ψ(y) (γ x + δ) y, where γ, δ constants and ψ concave.

23 Hopf-Lax Formula for the minimal cost Theorem (B): Assume = R d and that L satisfies hypothesis (A1), (A2) and (A3), and let µ 0 (resp. ν T ) be a probability measure on (resp., ). If µ 0 is absolutely continuous with respect to Lebesgue measure, then 1. The following Hopf-Lax formula holds: B T (µ 0, ν T ) = inf{w (µ 0, ν) + C T (ν, ν T ); ν P()}. 2. The infimum is attained at some probability measure ν 0 on. 3. The initial Kantorovich potential for C T (ν 0, ν T ) is concave. Worth noting: If L(x, v) = 1 2 v 2 (i.e., c(y, x) = 1 2 x y 2 ), the initial Kantorovich potential for C T (ν 0, ν T ) is then of the form φ 0 (y) = g(y) 1 2 y 2 where g is a convex function. But φ 0 can still be concave if 0 D 2 g I, which is what occurs above in (3).

24 A consequence of Hopf-Lax formula By the Hopf-Lax inequality, there is ν 0 on such that B T (µ 0, ν T ) = C T (ν 0, ν T ) + W (µ 0, ν 0 ). Let g be the concave function on such that ( g) # µ 0 = ν 0 and W (µ 0, ν 0 ) = g(v), v dµ 0 (v). Let Φ T 0 be the flow such that C T (ν 0, ν T ) = c T (y, Φ T 0 y)dν 0 (y). Since b T (v, x) c T ( g(v), x) + g(v), v for all v B T (µ 0, ν T ) b T (v, Φ T 0 g(v))dµ 0 (v) {c T ( g(v), Φ T 0 g(v)) + g(v), v } dµ 0 (v) = c T (y, Φ T 0 y)dν 0 (y) + g(v), v dµ 0 (v) = C T (ν 0, ν T ) + W (µ 0, ν) = B T (µ 0, ν T ).

25 Reverse Hopf-Lax Formula However this formula doesn t lift: c(t, y, x) = sup{b(t, v, x) v, y ; v }. Theorem (D): Assume ν 0 and ν T are probability measures on such that ν 0 is absolutely continuous with respect to Lebesgue measure. Then, TFAE: 1. The initial Kantorovich potential of C T (ν 0, ν T ) is concave. 2. The following holds: C T (ν 0, ν T ) = sup{b T (µ, ν T ) W (ν 0, µ); µ P( )}. and the sup is attained at some probability measure µ 0 on. Corollary: Consider the cost c(y, x) = c(x y), where c is a convex function on and let ν 0, ν 1 be probability measures on such that the initial Kantorovich potential associated to C T (ν 0, ν T ) is concave. Then, there exist concave functions φ 0 : R and φ 1 : R such that C 1 (ν 0, ν 1 ) K = c( φ 1 φ 0 (y) y)dν 0 (y) = φ 1 (y) φ 0 (y), y dν 0 (y), where K = K (c) is a constant and φ is the concave Legendre transform of φ.

26 Brenier-Benamou Type formula Theorem (E): For fixed probability measures µ 0 on and ν T on, As a function of the end measure: { T B T (µ 0, ν T ) = inf W (µ 0, ρ 0 ) + L ( x, w ) } t(x) dϱ t(x)dt; (ϱ, w) P(0, T ; ν T ) where P(0, T ; ν T ) is the set of pairs (ϱ, w) such that t ϱ t P(), t w t R n are paths of Borel vector fields such that { tϱ + (ϱw) = 0 in D ( (0, T ) ) ϱ T = ν T. As a function of the initial measure: { B T (µ 0, ν T ) = sup W (ν T, ρ T ) 0 T where P(0, T ; µ 0 ) is the set of pairs (ϱ, w) such that { tϱ + (ϱw) = 0 in D ( (0, T ) ) ϱ 0 = µ 0. 0 L ( x, w ) } t(x) dϱ t(x)dt; (ϱ, w) P(0, T ; µ 0 )

27 Lifting a value function to a value function on Wasserstein space Started with φ 0 (y) = v, y and defined b v(t, x) as a Value functional { t b v(t, x) = inf φ 0 (γ(0)) + { = inf φ 0 (y) + c t(y, x); y } It satisfies the Hamilton-Jacobi equation on. 0 } L(s, γ(s), γ(s)) ds; γ C 1 ([0, T ), ); γ(t) = x (Hopf Lax formula). tb + H(t, x, xb) = 0 on [0, T ], We then lifted b v to Wasserstein space by defining B µ0 (t, ν) = B t (µ 0, ν). B µ0 (t, ν) = inf{w (µ 0, ν) + C t( ν, ν); ν P()} (Hopf Lax formula) = inf{u µ0 (ϱ 0 ) + t 0 L(ϱ, w)dt; (ϱ, w) P(0, t; ν)}(value functional) 1. Do they satisfy a Hamilton-Jacobi equation on Wasserstein space? 2. Do they provide solutions to mean field games?

28 Under technical conditions (Ambrosio-Feng) (at least in a particular case): Value functionals on Wasserstein space yield a unique metric viscosity solution for { tb + H(t, ν, νb(t, µ)) = 0, B(0, ν) = W (µ 0, ν) Here the Hamiltonian on Wasserstein space is defined as H(ν, ζ) = sup{ ζ, ξ dν L(ν, ξ); ξ Tν (P())} (Gangbo-Swiech) Value functions on Wasserstein space with suitable initial data yield solutions to the so-called aster equation for mean field games without diffusion and without potential term. Theorem (Gangbo-Swiech) Assume U 0 : P() R, U 0 : P() R are such that qu 0 (q, µ) µu 0 (µ)(q) q µ P(), and consider the value functional, t U(t, ν) = inf L(ϱ, w)dt + U 0 (ϱ 0 ) (ϱ,w) P(0,t;ν) 0 Then, there exists U : [0, T ] P() R such that qu t(q, µ) µu t(µ)(q) q µ P(). and U satisfies the aster equation (but without diffusion)

29 ean Field Equation This yields the existence for any probabilities µ 0, ν T, a function β : [0, T ] P() R such that xβ(t, x, µ) µb µ0 (t, µ)(x) x µ P(). There exists ρ AC 2 ((0, T ) P()) such that tβ + µβ(t, x, µ) H(x, xβ) dµ + H(x, xβ(t, x, µ)) = 0, tρ + (ρ H(x, xβ)) = 0, β(0,, ) = β 0, ρ(t, ) = ν T, where β 0 (x, ρ) = φ ρ(x), where φ ρ is the convex function such that φ ρ pushes µ 0 into ρ. What about solutions to mean field games that include diffusions?

30 Stochastically dynamic mass transport B s T (µ 0, ν T ) := inf inf E P V µ 0 X A,X T ν T { T } V, X 0 + L(t, X(t), β(t, X)) dt, 0 where A is the class of all R d -valued continuous semimartingales (X t) 0 t T on a complete filtered probability space (Ω, F, (F t), P) such that there exists a Borel measurable β X : [0, T ] C([0, T ]) R d satisfying 1. w β X (t, w) is B(C([0, t])) +-measurable for all t, where B(C([0, t]) denotes the Borel σ-field on C([0, t]). 2. X(t) = X(0) + t 0 β X (s, X) ds + W X (t), where W X (t) is a σ[x(s); 0 s t]-brownian motion. The fixed end measures cost has been studied by ikami, Thieulin, Leonard. { T } CT s (ν 0, ν T ) := inf E P L(t, X(t), β(t, X)) dt; X A, X(0) ν 0, X(T ) ν T, 0

31 Theorem (F): Under suitable conditions on L 1. Duality: { B s T (µ 0, ν T ) = sup φ T (x) dν T (x) + φ 0 (v) dµ 0 (v); φ 0 concave & φ t solution of (HJB)}. { tφ + 1 φ + H(t, x, xφ) 2 = 0 on [0, T ], φ(0, x) = φ 0 (x), 2. For any probability measure ν on, we have { T B s T (µ 0, ν) = inf W (µ 0, ρ 0 ) + L ( t, x, b ) } t(x) dϱ t(x)dt; (ϱ, b) P(0, T ; ν), 0 where P(0, T ; ν T ) is the set of pairs (ϱ, b) such that t ϱ t P(), t b t R n are paths of Borel vector fields such that { tϱ 1 2 ρ + (ϱb) = 0 in D ( (0, T ) ) ϱ T = ν.

32 Diffusive ean Field Games There exists β : [0, T ] P() R and ρ AC 2 ((0, T ) P()) such that tβ 1 β + 2 µβ(t, x, µ) H(x, xβ) dµ + H(t, x, xβ(t, x, µ)) = 0, tρ 1 ρ + (ρ H(t, x, xβ)) = 0, 2 β(0,, ) = β 0, ρ(t, ) = ν T, where β 0 (x, ρ) = φ ρ(x), where φ ρ is the convex function such that φ ρ pushes µ 0 into ρ.

33 any Happy Returns Yann Brenier