The dynamics of Schrödinger bridges

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1 Stochastic processes and statistical machine learning February, 15, 2018

2 Plan of the talk The Schrödinger problem and relations with Monge-Kantorovich problem Newton s law for entropic interpolation The entropy along the entropic interpolations The talk is based on G. Conforti. A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost. Probability Theory and Related Fields(to appear)

3 Part I: The of Schrödinger problem and relations with the Monge-Kantorovich problem

4 Schrödinger s thought experiment An old story from Schrödinger back in Imaginez que vous observez un système de particules en diffusion, qui soient en équilibre thermodynamique. Admettons qu à un instant donné 0 vous les ayez trouvées en répartition à peu près uniforme et quà 1 vous ayez trouvé un écart spontané et considérable par rapport à cette uniformité. On vous demande de quelle manière cet écart sest produit. Quelle en est la manière la plus probable? A more recent story from C.Villani s textbook.. Take a perfect gas in which particles do not interact, and ask him to move from a certain prescribed density field at time t = 0, to another prescribed density field at time t = 1. Since the gas is red lazy, he will find a way to do so that it needs a minimal amount of work (least action path).

5 Schrödinger s thought experiment An old story from Schrödinger back in Imaginez que vous observez un système de particules en diffusion, qui soient en équilibre thermodynamique. Admettons qu à un instant donné 0 vous les ayez trouvées en répartition à peu près uniforme et quà 1 vous ayez trouvé un écart spontané et considérable par rapport à cette uniformité. On vous demande de quelle manière cet écart sest produit. Quelle en est la manière la plus probable? A more recent story from C.Villani s textbook.. Take a perfect gas in which particles do not interact, and ask him to move from a certain prescribed density field at time t = 0, to another prescribed density field at time t = 1. Since the gas is red lazy, he will find a way to do so that it needs a minimal amount of work (least action path).

6 Schrödinger s thought experiment An old story from Schrödinger back in Imaginez que vous observez un système de particules en diffusion, qui soient en équilibre thermodynamique. Admettons qu à un instant donné 0 vous les ayez trouvées en répartition à peu près uniforme et quà 1 vous ayez trouvé un écart spontané et considérable par rapport à cette uniformité. On vous demande de quelle manière cet écart sest produit. Quelle en est la manière la plus probable? A more recent story from C.Villani s textbook.. Take a perfect gas in which particles do not interact, and ask him to move from a certain prescribed density field at time t = 0, to another prescribed density field at time t = 1. Since the gas is lazy, he will find a way to do so that it needs a minimal amount of work (least action path).

7 Schrödinger bridge; Problem formulation To model Schrödinger s experiment we need An ambient space A Riemannian manifold M The equilibrium dynamics for the particles stationary Brownian motion P The non-interacting particles X 1,..., X N stationary Brownian motions independent The particle-configuration empirical measure µ N µ N (A) = 1 N Card({i : Xi A}) The initial and final prescribed configurations ν 0, ν 1.

8 Schrödinger bridge; Problem formulation To model Schrödinger s experiment we need An ambient space A Riemannian manifold M The equilibrium dynamics for the particles stationary Brownian motion P The non-interacting particles X 1,..., X N stationary Brownian motions independent The particle-configuration empirical measure µ N µ N (A) = 1 N Card({i : Xi A}) The initial and final prescribed configurations ν 0, ν 1.

9 Schrödinger bridge; Problem formulation To model Schrödinger s experiment we need An ambient space A Riemannian manifold M The equilibrium dynamics for the particles stationary Brownian motion P The non-interacting particles X 1,..., X N stationary Brownian motions independent The particle-configuration empirical measure µ N µ N (A) = 1 N Card({i : Xi A}) The initial and final prescribed configurations ν 0, ν 1.

10 Schrödinger bridge; Problem formulation To model Schrödinger s experiment we need An ambient space A Riemannian manifold M The equilibrium dynamics for the particles stationary Brownian motion P The non-interacting particles X 1,..., X N stationary Brownian motions independent The particle-configuration empirical measure µ N µ N (A) = 1 N Card({i : Xi A}) The initial and final prescribed configurations ν 0, ν 1.

11 Schrödinger bridge; Problem formulation To model Schrödinger s experiment we need An ambient space A Riemannian manifold M The equilibrium dynamics for the particles stationary Brownian motion P The non-interacting particles X 1,..., X N stationary Brownian motions independent The particle-configuration empirical measure µ N µ N (A) = 1 N Card({i : Xi A}) The initial and final prescribed configurations ν 0, ν 1.

12 Schrödinger bridge problem: dynamic formulation We denote the law µ N by P N Sanov Theorem 1 N log PN ( µ N = Q ) H (Q P) Thus, the most likely evolution is found solving Schrödinger Problem (SP) inf Q H path(q P) Q P(C([0, 1], M)), (X 0 ) # Q = ν 0, (X 1 ) # Q = ν 1 H path is the relative entropy for laws on the path space C([0, 1], M) The Schrödinger bridge (SB) is the optimal solution of (SP)

13 Schrödinger bridge problem: dynamic formulation We denote the law µ N by P N Sanov Theorem 1 N log PN ( µ N = Q ) H (Q P) Thus, the most likely evolution is found solving Schrödinger Problem (SP) inf Q H path(q P) Q P(C([0, 1], M)), (X 0 ) # Q = ν 0, (X 1 ) # Q = ν 1 H path is the relative entropy for laws on the path space C([0, 1], M) The Schrödinger bridge (SB) is the optimal solution of (SP)

14 Some notation and two classical results Marginal flow of SB entropic interpolation (µ t ) The optimal value of SP is the entropic cost T H (ν 0, ν 1 ) Theorem (fg-decompostion) Under some mild regularity assumptions on M, ν 0, ν 1 there exist non-negative functions f t, g t such t [0, 1], µ t = f t g t f t,g t solve the equation t f t = 1 2 f t, t g t = 1 2 g t Theorem (Vague statement) In the small noise regime (SP) Γ-converges to the Monge-Kantorovich problem.

15 Some notation and two classical results Marginal flow of SB entropic interpolation (µ t ) The optimal value of SP is the entropic cost T H (ν 0, ν 1 ) Theorem (fg-decompostion) Under some mild regularity assumptions on M, ν 0, ν 1 there exist non-negative functions f t, g t such t [0, 1], µ t = f t g t f t,g t solve the equation t f t = 1 2 f t, t g t = 1 2 g t Theorem (Vague statement) In the small noise regime (SP) Γ-converges to the Monge-Kantorovich problem.

16 What about Machine Learning? The use of Sinkhorn s algorithm to compute (approximate) solutions of OT has led to a dramatic reduction in the computational cost, O(d 2 ) vs. O(d 3 log d). Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in neural information processing systems, pages , 2013 The regularized problem solved using Sinkhorn is a discrete Schrödinger problem! The reason why we can use Sinkhorn s algorithm is the fg decomposition Theorem

17 What about Machine Learning? The use of Sinkhorn s algorithm to compute (approximate) solutions of OT has led to a dramatic reduction in the computational cost, O(d 2 ) vs. O(d 3 log d). Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in neural information processing systems, pages , 2013 The regularized problem solved using Sinkhorn is a discrete Schrödinger problem! The reason why we can use Sinkhorn s algorithm is the fg decomposition Theorem

18 Some references(incomplete list) E. Schrödinger. La théorie relativiste de l électron et l interprétation de la mécanique quantique. Ann. Inst Henri Poincaré, (2): , 1932 C. Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete and Continuous Dynamical Systems, 34(4): , 2014 H. Föllmer. Random fields and diffusion processes. In École d Été de Probabilités de Saint-Flour XV XVII, , pages Springer, 1988 C. Léonard. From the Schrödinger problem to the Monge Kantorovich problem. Journal of Functional Analysis, 262(4): , 2012 T. Mikami. Monges problem with a quadratic cost by the zero-noise limit of h-path processes. Probability Theory and Related Fields, 129(2): , 2004

19 Motivating question What is the shape of the particle cloud at t = 1 2? Entropy minimization particles try to arrange according to the equilibrium configuration m. Prescribed initial and final configurations particles are forced into a configuration far from equilibrium at t = 0, 1. Does µ 1/2 look like m? How big is H marg (µ 1/2 m)? The key to answer the question is to view the entropic interpolation (µ t ) as a curve in a Riemannian manifold.

20 Motivating question What is the shape of the particle cloud at t = 1 2? Entropy minimization particles try to arrange according to the equilibrium configuration m. Prescribed initial and final configurations particles are forced into a configuration far from equilibrium at t = 0, 1. Does µ 1/2 look like m? How big is H marg (µ 1/2 m) := S(µ 1/2 )? The key to answer the question is to view the entropic interpolation (µ t ) as a curve in a Riemannian manifold.

21 Part II: Newton s law for the entropic interpolation

22 The Otto metric Formally, it is the Riemannian metric on P 2 (M) whose associated geodesic distance is the Wasserstein distance W 2. The tangent space at µ P 2 (M) is identified with the gradient vector fields T µ = { ϕ, ϕ C c } L2 (µ) We define the Riemannian metric on it Riemannian metric on T µ ϕ, ψ Tµ := M ϕ, ψ dµ. The velocity of an absolutely continuous curve (µ t ) is given by Continuity equation t µ t + (v t µ t ) = 0, v t T µt

23 The Otto metric Formally, it is the Riemannian metric on P 2 (M) whose associated geodesic distance is the Wasserstein distance W 2. The tangent space at µ P 2 (M) is identified with the gradient vector fields T µ = { ϕ, ϕ C c } L2 (µ) We define the Riemannian metric on it Riemannian metric on T µ ϕ, ψ Tµ := M ϕ, ψ dµ. The velocity of an absolutely continuous curve (µ t ) is given by Continuity equation t µ t + (v t µ t ) = 0, v t T µt

24 Construction of the covariant derivative The Benamou-Brenier formula tells indeed that the geodesic distance for this Riemannian metric is the Wasserstein distance. Displacement interpolations are geodesics W 2 2(ν 0, ν 1 ) = inf (µ,v) µ 0 =ν 0,µ 1 =ν v t 2 T µt dt, In a Riemannian manifold, the acceleration of a curve is the covariant derivative of its velocity Acceleration of a curve W 2 v t v t = t v t ( v t 2)

25 Construction of the covariant derivative The Benamou-Brenier formula tells indeed that the geodesic distance for this Riemannian metric is the Wasserstein distance. Displacement interpolations are geodesics W 2 2(ν 0, ν 1 ) = inf (µ,v) µ 0 =ν 0,µ 1 =ν v t 2 T µt dt, In a Riemannian manifold, the acceleration of a curve is the covariant derivative of its velocity Acceleration of a curve W 2 v t v t = t v t ( v t 2)

26 The acceleration of a SB Particles move from configuration ν 0 to configuration ν 1 minimizing relative entropy The natural way of doing it would be to follow the gradient flow Gradient flow v t = 1 2 W 2 S(µ t ), µ 0 = ν 0. If particles go along the gradient flow µ 1 ν 1 IDEA: Modify the gradient flow equation as little as possible in order to be able to impose the terminal condition

27 The acceleration of a SB Particles move from configuration ν 0 to configuration ν 1 minimizing relative entropy The natural way of doing it would be to follow the gradient flow Gradient flow v t = 1 2 W 2 S(µ t ), µ 0 = ν 0. If particles go along the gradient flow µ 1 ν 1 IDEA: Modify the gradient flow equation as little as possible in order to be able to impose the terminal condition

28 The acceleration of a SB Particles move from configuration ν 0 to configuration ν 1 minimizing relative entropy The natural way of doing it would be to follow the gradient flow Gradient flow v t = 1 2 W 2 S(µ t ), µ 0 = ν 0. If particles go along the gradient flow µ 1 ν 1 IDEA: Modify the gradient flow equation as little as possible in order to be able to impose the terminal condition

29 The acceleration of a SB Particles move from configuration ν 0 to configuration ν 1 minimizing relative entropy The natural way of doing it would be to follow the gradient flow Gradient flow v t = 1 2 W 2 S(µ t ), µ 0 = ν 0. If particles go along the gradient flow µ 1 ν 1 IDEA: Modify the gradient flow equation as little as possible in order to be able to impose the terminal condition

30 Tweaking a gradient flow Gradient flow in R d ẋt = S(xt) Second order equation for the gradient flow ẍ t = 2 S(x t ) ẋ t = 2 S(x t ) S(x t ) = 1 2 ( S(x t ) 2) Back to the OT setting (S = Relative entropy) The Fisher information I is the norm squared of the gradient of the entropy I(µ) = W 2 S( ) 2 T µ Thus, we have a candidate equation...

31 Tweaking a gradient flow Gradient flow in R d ẋt = S(xt) Second order equation for the gradient flow ẍ t = 2 S(x t ) ẋ t = 2 S(x t ) S(x t ) = 1 2 ( S(x t ) 2) Back to the OT setting (S = Relative entropy) The Fisher information I is the norm squared of the gradient of the entropy I(µ) = W 2 S( ) 2 T µ Thus, we have a candidate equation...

32 Tweaking a gradient flow Gradient flow in R d ẋt = S(xt) Second order equation for the gradient flow ẍ t = 2 S(x t ) ẋ t = 2 S(x t ) S(x t ) = 1 2 ( S(x t ) 2) Back to the OT setting (S = Relative entropy) The Fisher information I is the norm squared of the gradient of the entropy I(µ) = W 2 S( ) 2 T µ Thus, we have a candidate equation...

33 Tweaking a gradient flow Gradient flow in R d ẋt = S(xt) Second order equation for the gradient flow ẍ t = 2 S(x t ) ẋ t = 2 S(x t ) S(x t ) = 1 2 ( S(x t ) 2) Back to the OT setting (S = Relative entropy) The Fisher information I is the norm squared of the gradient of the entropy I(µ) = W 2 S( ) 2 T µ Thus, we have a candidate equation...

34 Second order equation for entropic interpolation Theorem (C. 17) Let (µ t ) be the entropic interpolation between ν 0 and ν 1 and (v t ) its velocity field. Under suitable regularity assumptions (µ t ) solves the equation W 2 v t v t = 1 8 W 2 I(µ t ) The equation answers in a precise way What kind of 2nd order equation the bridge of a diffusion satisfies? and thus gives grounding to the intuition that the Brownian bridge is the stochastic version of a geodesic. To prove a rigorous statement, we took advantage of Gigli s rigorous version of the Otto calculus

35 Second order equation for entropic interpolation Theorem (C. 17) Let (µ t ) be the entropic interpolation between ν 0 and ν 1 and (v t ) its velocity field. Under suitable regularity assumptions (µ t ) solves the equation W 2 v t v t = 1 8 W 2 I(µ t ) The equation answers in a precise way What kind of 2nd order equation the bridge of a diffusion satisfies? and thus gives grounding to the intuition that the Brownian bridge is the stochastic version of a geodesic. To prove a rigorous statement, we took advantage of Gigli s rigorous version of the Otto calculus

36 Proof sketch It relies on the representation µ t = f t g t. Lemma (Representation of the velocity field) The velocity field of (µ t ) is 1 2 (log g t log f t ). log f t (resp log g t ) solve the forward(backward) HJB equation HJB t log f t = 1 2 log f t log f t 2, t log g t = 1 2 log g t 1 2 log g t 2 Gradient of the Fisher information We have W 2 I(µ) = 2 log µ log µ 2

37 Proof sketch Recall that W 2 v t v t = t v t ( v t 2). We have, using HJB t v t = 1 2 t log f t t log g t HJB = 1 4 ( log f t + log g t ) 1 4 [ log f t 2 + log g t 2 ] µ t =f t g t = 1 4 log µ t 1 4 [ log f t 2 + log g t 2 ] polarization = 1 4 log µ t 1 8 log f t + log g t log g t log f t 2 = 1 8 [2 log µ t log µ t 2 ] 1 2 v t 2 = 1 8 W 2 I(µ t ) 1 2 v t 2 which (formally) concludes the proof.

38 Proof sketch Recall that W 2 v t v t = t v t ( v t 2). We have, using HJB t v t = 1 2 t log f t t log g t HJB = 1 4 ( log f t + log g t ) 1 4 [ log f t 2 + log g t 2 ] µ t =f t g t = 1 4 log µ t 1 4 [ log f t 2 + log g t 2 ] polarization = 1 4 log µ t 1 8 log f t + log g t log g t log f t 2 = 1 8 [2 log µ t log µ t 2 ] 1 2 v t 2 = 1 8 W 2 I(µ t ) 1 2 v t 2 which (formally) concludes the proof.

39 Proof sketch Recall that W 2 v t v t = t v t ( v t 2). We have, using HJB t v t = 1 2 t log f t t log g t HJB = 1 4 ( log f t + log g t ) 1 4 [ log f t 2 + log g t 2 ] µ t =f t g t = 1 4 log µ t 1 4 [ log f t 2 + log g t 2 ] polarization = 1 4 log µ t 1 8 log f t + log g t log g t log f t 2 = 1 8 [2 log µ t log µ t 2 ] 1 2 v t 2 = 1 8 W 2 I(µ t ) 1 2 v t 2 which (formally) concludes the proof.

40 Proof sketch Recall that W 2 v t v t = t v t ( v t 2). We have, using HJB t v t = 1 2 t log f t t log g t HJB = 1 4 ( log f t + log g t ) 1 4 [ log f t 2 + log g t 2 ] µ t =f t g t = 1 4 log µ t 1 4 [ log f t 2 + log g t 2 ] polarization = 1 4 log µ t 1 8 log f t + log g t log g t log f t 2 = 1 8 [2 log µ t log µ t 2 ] 1 2 v t 2 = 1 8 W 2 I(µ t ) 1 2 v t 2 which (formally) concludes the proof.

41 Proof sketch Recall that W 2 v t v t = t v t ( v t 2). We have, using HJB t v t = 1 2 t log f t t log g t HJB = 1 4 ( log f t + log g t ) 1 4 [ log f t 2 + log g t 2 ] µ t =f t g t = 1 4 log µ t 1 4 [ log f t 2 + log g t 2 ] polarization = 1 4 log µ t 1 8 log f t + log g t log g t log f t 2 = 1 8 [2 log µ t log µ t 2 ] 1 2 v t 2 = 1 8 W 2 I(µ t ) 1 2 v t 2 which (formally) concludes the proof.

42 Proof sketch Recall that W 2 v t v t = t v t ( v t 2). We have, using HJB t v t = 1 2 t log f t t log g t HJB = 1 4 ( log f t + log g t ) 1 4 [ log f t 2 + log g t 2 ] µ t =f t g t = 1 4 log µ t 1 4 [ log f t 2 + log g t 2 ] polarization = 1 4 log µ t 1 8 log f t + log g t log g t log f t 2 = 1 8 [2 log µ t log µ t 2 ] 1 2 v t 2 = 1 8 W 2 I(µ t ) 1 2 v t 2 which (formally) concludes the proof.

43 Reference list (incomplete) Felix Otto. The geometry of dissipative evolution equations: the porous medium equation. Communications in Partial Differential Equations, 26(1-2): , 2001 N. Gigli. Second Order Analysis on (P2(M),W2). Memoirs of the American Mathematical Society Max-K von Renesse. An Optimal Transport view of Schrödinger s equation. Canadian mathematical bulletin, 55(4): , 2012 E. Nelson. Dynamical theories of Brownian motion, volume 2. Princeton university press Princeton, 1967

44 Part III: The entropy along entropic interpolations

45 First and second derivative of the entropy We want to study how does the particle configuration evolves How much does µ 1/2 look like m? The relative entropy can be decomposed into S(µ t ) = log f t dµ t + log g t dµ t := h (t) + h (t) M M h (t) is the forward entropy, h (t) the backward entropy. First derivative -forward entropy We have 1 t h (t) = vt W 2 S 2 T µt Second derivative-forward entropy 1 tt h (t) = W 2 2 ξ t W 2 S, ξ t T µt with ξ t := 1 2 W 2 S v t

46 First and second derivative of the entropy We want to study how does the particle configuration evolves How much does µ 1/2 look like m? The relative entropy can be decomposed into S(µ t ) = log f t dµ t + log g t dµ t := h (t) + h (t) M M h (t) is the forward entropy, h (t) the backward entropy. First derivative -forward entropy We have 1 t h (t) = vt W 2 S 2 T µt Second derivative-forward entropy 1 tt h (t) = W 2 2 ξ t W 2 S, ξ t T µt with ξ t := 1 2 W 2 S v t

47 Curvature enters the game Assume now that the Ricci curvature is bounded below, i.e. Ric x (w, w) λ w 2 uniformly in x, w T x M. A fundamental result of OT is that S is a λ-convex functional. Differential inequality-forward entropy 1 tt h (t) = W 2 2 ξ t W 2 S, ξ t T µt λ 2 ξ t 2 T µt = λ W 2 2 S v t = λ t h (t) T µt

48 The entropy along entropic interpolations Theorem (C. 17) Let M be a compact manifold with Ricci curvature bounded below x M, w T x M, Ric x (w, w) λ w 2 Then, for all ν 0, ν 1 and t [0, 1] the entropic interpolation (µ t ) satisfies: S(µ t ) 1 exp( λ(1 t)) S(ν 0 ) + 1 exp( λt) 1 exp( λ) 1 exp( λ) S(ν 1) cosh( λ 2 ) cosh( λ(t 1 2 )) sinh( λ 2 ) T H (ν 0, ν 1 ).

49 About the entropy bound If M has a Ricci curvature bound, then the particle configuration at t = 1 2 is very close to the equilibrium measure m, and we have a way to quantify this. In the small noise regime, the entropy estimate becomes the well known convexity of the entropy along entropic interpolations. There is a version of the Theorem when P is the Langevin dynamics

50 About the entropic transportation cost The entropic cost T H (µ, m) measures how difficult it is to steer Brownian particles which start out of equilibrium into the equilibrium configuration m in one unit of time. We expect that the more µ looks like m, the smaller is T H How to bound T H? And with what? Theorem (C. 17) Assume Ric λ. Then for all µ we have T H (µ, m) 1 1 exp( λ) S(µ) We call this an entropy-entropy inequality.

51 About the entropic transportation cost The entropic cost T H (µ, m) measures how difficult it is to steer Brownian particles which start out of equilibrium into the equilibrium configuration m in one unit of time. We expect that the more µ looks like m, the smaller is T H How to bound T H? And with what? Theorem (C. 17) Assume Ric λ. Then for all µ we have T H (µ, m) 1 1 exp( λ) S(µ) We call this an entropy-entropy inequality.

52 About the entropic transportation cost The entropic cost T H (µ, m) measures how difficult it is to steer Brownian particles which start out of equilibrium into the equilibrium configuration m in one unit of time. We expect that the more µ looks like m, the smaller is T H How to bound T H? And with what? Theorem (C. 17) Assume Ric λ. Then for all µ we have T H (µ, m) 1 1 exp( λ) S(µ) We call this an entropy-entropy inequality.

53 About the entropy-entropy inequality The bound is useful since T H (µ, m) is hard to compute and S(µ) is easy to compute (it is just an integral). In the small noise regime the entropy-entropy inequality becomes Talagrand s transportation entropy inequality W 2 2(µ, m) 2λS(µ) The inequality implies concentration of measure properties for m (work in progress) It allows to bound a joint entropy, T H (µ, m) with a marginal entropy (S(µ))!

54 About the entropy-entropy inequality The bound is useful since T H (µ, m) is hard to compute and S(µ) is easy to compute (it is just an integral). In the small noise regime the entropy-entropy inequality becomes Talagrand s transportation entropy inequality W 2 2(µ, m) 2λS(µ) The inequality implies concentration of measure properties for m (work in progress) It allows to bound a joint entropy, T H (µ, m) with a marginal entropy (S(µ))!

55 Dual form of the entropy-entropy inequality Under the curvature condtion, it is known that the heat semigroup (P t ) t 0 is hypercontractive. For all p, q 1 s.t. q 1 p 1 = exp(2λt) we have f s.t. fdm = 0, P t f L q (m) f L p (m) It is known that hypercontractivity is equivalent to the Logarithmic Sobolev inequality. Theorem (C. 18) The following are equivalent i) The entropy entropy inequality with constant 1/(1 exp( λ)). ii) For all p (0, 1), q< 1 s.t. q 1 p 1 = exp(2λt), and for all f s.t. fdm = 0, P t f L q (m) f L p (m)

56 References list(incomplete) Felix Otto and Cédric Villani. Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality. Journal of Functional Analysis, 173(2): , 2000 G. Conforti and M. Von Renesse. Couplings,gradient estimates and logarithmic Sobolev inequality for Langevin bridges. to appear in Probability Theory and Related Fields Christian Léonard. On the convexity of the entropy along entropic interpolations. arxiv preprint arxiv: , 2013 N. Gozlan, C. Roberto, P.M. Samson, and P. Tetali. Kantorovich duality for general transport costs and applications, to appear in j. funct. anal., preprint (2014)

57 Some thoughts for the future Is the entropy bound equivalent to curvature even if we do not look at the small noise regime? How close is the entropic interpolation to the displacement interpolation? How to construct a Schrödinger bridge for a system of weakly interacting particles system? Is there a Netwon s law?

58 Some thoughts for the future Is the entropy bound equivalent to curvature even if we do not look at the small noise regime? How close is the entropic interpolation to the displacement interpolation? How to construct a Schrödinger bridge for a system of weakly interacting particles system? Is there a Netwon s law?

59 Thank you very much!

Some geometric consequences of the Schrödinger problem

Some geometric consequences of the Schrödinger problem Christian Léonard To cite this version: Christian Léonard. Some geometric consequences of the Schrödinger problem. Second International Conference