On some relations between Optimal Transport and Stochastic Geometric Mechanics

Transcription

1 Title On some relations between Optimal Transport and Stochastic Geometric Mechanics Banff, December 218 Ana Bela Cruzeiro Dep. Mathematics IST and Grupo de Física-Matemática Univ. Lisboa 1 / 2

2 Title Based on joint works with M. Arnaudon, X. Chen, J.-C. Zambrini Extensions of (some) results, with applications to fluids: with T. Ratiu, C. Léonard 2 / 2

3 Optimal transport problem Recalling the Monge-Kantorovich (MK) optimal transport problem (flat case) 1 inf π Π(µ,σ) x y 2 dπ(x, y) 2 R d R d where Π(µ, σ) = { joint distributions s.t. the marginals along x and y coordinates are µ and σ resp. }. Since the cost" 1 x y 2 = inf X Ẋ(t) 2 dt X {C([, 1]; R d ) : X() = x, X(1) = y}, the MK problem is equivalent to 3 / 2

4 Optimal transport problem 1 inf π 2 R d R d or, by desintegration, to ( 1 Ẋ(t) 2 dt dp x,y )dπ(x, y) inf P Ẋ(t) 2 dt dp with P prob. measure on C 1 ([, 1]; R d ) with marginals µ and σ at t =, t = 1 and P x,y the one with initial and final marginals δ x and δ y. 4 / 2

5 Optimal transport problem Eulerian (control) version of this problem: inf v v(t, x v (t)) 2 dtdp v continuous, ẋ v (t) = v(t, x v (t)) a.s., law of x v () = µ, law of x v (1) = σ. If dµ = ρ dx, dσ = ρ 1 dx given, law of x v (t) = ρ t dx, then (continuity equation) ρ t = div (ρv) 5 / 2

6 Optimal transport problem If ψ solves the Hamilton-Jacobi equation ψ t = 1 2 ψ 2 and ρ t = div (ρ ψ), with ρ(, x) = ρ (x), ρ(1, x) = ρ 1 (x), then v = ψ solves the problem. Moreover v t = (v )v 6 / 2

7 The Schrödinger problem On C([, 1]; R d ) consider the laws (Q) of diffusions dx(t) = ɛ dw (t) + Y (t)dt, Entropy functional of Q w.r.t. P: H(Q; P) = ( dq ) log dq dp Choose for P the law of the Wiener process law of X() = dx H(Q; P) = H(Q ; P ) Y (t) 2 dtdq, 2 where Q, P are the marginals of Q and P at time. This is a consequence of Girsanov s Theorem. 7 / 2

8 The Schrödinger problem Schrödinger problem: minimise the entropy functional, subject to given Q and Q 1. If Y (t) = v(t, X(t)), the density ρ t of X at time t satisfies the Fokker-Planck equation ρ t = div (ρv) + ɛ 2 ρ If, moreover, v = ϕ satisfies Hamilton-Jacobi-Bellman equation, then v = ϕ. So (Burgers equation) ϕ t v t = 1 2 ϕ 2 + ɛ 2 ϕ = (v )v + ɛ 2 v 8 / 2

9 The Schrödinger problem Remarks: 1. With the change of variable v = ϕ, ϕ = log η, η equation) t = ɛ 2 η (heat 2. When ɛ formally Schrödinger problem converges to the optimal transport problem (C. Léonard, etc) (delicate problem) 9 / 2

10 Brownian motion on Lie groups G Lie group with < > right (left) invariant metric Levi-Civita connection e identity element G T e (G) Lie algebra {H i } o.n. basis of G, right invariant, (we suppose finite dimensional), Hi H i = Brownian motion on G with generator = Laplace-Beltrami operator: dg ( (t) = T e R g (t) H i dw i (t) ) ( = T e R g (t) H i dw i (t) ) i where T a R g(t) : T a G T ag(t) G is the differential of the right translation R g(t) (x) := xg(t), x G at the point x = a G. i 1 / 2

11 Brownian motion on Lie groups General diffusion processes: ( dg(t) = T e R g(t) H i dw i (t) + u(t)dt ) i T e R g(t) u(t) = D g(t) dt where, by definition, D is the (mean) derivative defined as: for ξ(t) = t T s dg(s), where T : T g() G T g( ) G is the stochastic parallel transport associated to, define and Dξ(t) dt 1 [ ] = lim ɛ ɛ E ξ(t + ɛ) ξ(t) P t D g(t) dt := T t Dξ(t) dt 11 / 2

12 Girsanov Theorem and entropy From Girsanov s Theorem the law Q of g on C([, 1]; G) is absolutely continuous w.r.t. the law P of g (same initial distributions) with density given by dq dp = exp{ Therefore the entropy is < T g R g 1 i 1 T g R g 1 H(Q; P) = 1 1 T g R 2 g 1 D g(t), H i dw i (t) > dt D g(t) 2 dt} dt D g(t) 2 dtdq dt 12 / 2

13 Csiszär Theorem Denote by α the invariant measure on G, P x law of the Brownian motion g on G starting at x, P t (x, α(dy)) its transition semigroup. P µ = P x µ(dx) Assume µ and σ prob. measures, to be abs. cont. w.r.t. α. M(µ, σ) := {π prob. measures on G G : π(g () ) = µ( ), π(g (1) ) = σ( ), π abs. cont. w.r.t. µ P 1 } For π M(µ, σ), define P π (dω) = G G prob. measure on C([, 1]; G). G π(dx, dy)p(dω, x; 1, y) 13 / 2

14 Csiszär Theorem By Csiszär s Theorem, if π M(µ, σ) : H(π, µ P 1 ) <, then 1 Q on C([, 1]; G] attaining the with Q = P π, π attaining inf Q: Q((g(),g(1) ) M(µ,σ) H(Q; P µ ) inf π H(π; µ P 1 ) Moreover Q is the law of a Markov process. 14 / 2

15 Csiszär Theorem Combining with Girsanov s result, we have H(Q; P µ ) = 1 1 T g R 2 g 1 := A[g] D g(t) 2 dtdq dt 15 / 2

16 Stochastic Euler-Poincaré reduction Stochastic Euler-Poincaré reduction theorem Theorem. The G-valued semi-martingale ( dg(t) = T e R g(t) H i dw i (t) + u(t)dt ) is a critical point of A if and only if the time-dependent vector field u( ) satisfies the equation, i d dt u(t) = ad u(t) u(t) K (u(t)), where for u G, adu : G G is the adjoint of ad u : G G with respect to the metric < >, and the operator K : G G is defined as K (u) = 1 2 (de Rham-Hodge Laplacian) i ( Hi Hi u + R(u, H i )H i ) = 1 2 (u) 16 / 2

17 Stochastic Euler-Poincaré reduction (Left) variations: L A[ξ( )] = d dε ε=a[ξ ε,v ( )] with ξ ε,v (t) = e ε,v (t)ξ(t) { d dt e ε,v(t) = εt e R eε,v (t) v(t), e ε,v () = e (1) for v( ) C 1 ([, T ]; G), v() = v(t ) = We differentiate in the direction of v ( d dɛ ε= e ε,v = v) 17 / 2

18 Stochastic Euler-Poincaré reduction "Proof" of the Theorem: For g ε (t) = e ε,v (t)g(t), by Itô s formula, dg ε (t) = i ( + T e R gε(t) T e R gε(t)ad e 1 ε (t) H i dwt i Ad e 1 ε (t) ( ) ) u(t) + Teε(t)R e 1 (t)ėε(t) dt, ε (2) (no contraction term) We have T eε(t)r e 1 (t)ėε(t) d = ε v(t), ε dε ε= Ad e 1 ε and d dε ε= Ad e 1 = ad ε (t)h i v(t) H i (t) u(t) = ad v(t)u(t) 18 / 2

19 Stochastic Euler-Poincaré reduction da[g ε ( )] dε ε= = E = 1 1 < u(t), v(t) ad (u(t)) v(t) + 1 ( adv(t) H 2 i H i + Hi (ad v(t) H i )) > dt = i 1 < d dε (T D g ε (t) g ε(t)r g 1 ) ε (t) dt ε=, u(t) > dt < u(t) adu(t) u(t) K (u(t)), v(t) > dt (3) Remark : When H i = we recover the deterministic Euler-Poncaré reduction theorem. 19 / 2

20 Hamilton-Jacobi-Bellman equation The change of variables u(t) = ϕ(t) gives the Hamilton-Jacobi-Bellman equation ϕ t = 1 2 ϕ LB(ϕ) Extra remarks: In the assumptions of Csiszär s theorem ( π M(µ, σ) : H(π, µ P 1 ) < ), if π is abs. cont. w.r.t. µ P 1, then the minimiser is of the form π(dx, dy) = ψ(x)φ(y)α(dx)p 1 (x, α(dy)) and dµ dα = ψp dν 1φ, dα = φp 1 ψ a.e. which allows to transform initial and final conditions µ, σ in φ, ψ, initial and final conditions for the equations satisfied by the drift and the one of its time reversed. 2 / 2

21 References M. Arnaudon, X. Chen, A. B. C., Stochastic Euler-Poincaré reduction, J. Math. Physics (214) M. Arnaudon, A.B.C., C. Léonard, J.C. Zambrini, An entropic interpolation problem for incompressible viscid fluids, arxiv: (infinite dimensional entropic approach ) X. Chen, A. B. C., T. S. Ratiu, Constrained and stochastic variational principles for dissipative equations with advected quantities arxiv: (extension of stochastic geometric mechanics results ) A.B.C., Liming Wu, J.C. Zambrini, Bernstein processes associated with a Markov process, Trends Math., Birkhäuser Boston (2) C. Léonard, A survey of the Schrödinger problem and some of its connections with optimal transport, Discrete and Cont. Dyn. Systems (214) (general results on the entropy approach ) 2 / 2