Stochastic Hamiltonian systems and reduction

Transcription

1 Stochastic Hamiltonian systems and reduction Joan Andreu Lázaro Universidad de Zaragoza Juan Pablo Ortega CNRS, Besançon Geometric Mechanics: Continuous and discrete, nite and in nite dimensional Ban, August

2 Motivations Modeling of physical systems subjected to random perturbations. Translate some fruitful concepts/techniques of classical mechanics (symmetries, conserved quantities, reduction,... ) to the stochastic context. Details available in two papers: Lázaro-Camí, J.-A. and Ortega, J.-P [2007] Stochastic Hamiltonian dynamical systems. Lázaro-Camí, J.-A. and Ortega, J.-P [2007] Reduction and reconstruction of symmetric stochastic di erential equations. 2

3 Stochastic integrals on R ; F; ff t g t0 ; P a standard ltered probability space. Let Y : R +! R be a process whose paths are left-continuous and have right limits and X : R +! R a continuous semimartingale. We de ne the Itô integral as Z t 0 Y dx := jx n 1 lim n Y t n i probability i=0 X t n i+1 X t n i where 0 t n 0 ::: t n j n = t is a time partition such that max 0ijn 1 t n i+1 t n i! 0 as n! 1. If Y and X are two semimartingales, we de ne their quadratic variation as R R [X; Y ] := XY X 0 Y 0 XdY Y dx and the Stratonovich integral Z Z Y X := Y dx [X; Y ] : 3

4 Stratonovich SDE on manifolds M and N two nite dimensional manifolds X : R +! N a N-valued semimartingale (i.e., f X is a real semimartingale 8f 2 C 1 (N)) Stratonovich operator: family fs (x; y)g x2n;y2m such that S (x; y) : T x N! T y M is a linear map that depends smoothly on its two entries. Denote S (x; y) : T y M! T x N the adjoint of S (x; y). For any N-valued semimartingale X there exits a linear map 7! R h; Xi that associates a real semimartingale to any T N-valued process : R +! T N over X (i.e., T N ( t (!)) = X t (!) for any (t;!) 2 R + ) 1. 1 Émery, M. [1989] Stochastic Calculus on Manifolds. Universitext. Springer-Verlag. 4

5 Stratonovich SDE on manifolds A M-valued semimartingale is solution of the Stratonovich SDE = S (X; ) X de ned by X and S if, for any 2 (M), the following equality holds Z Z h; i = hs (X; ) () ; Xi (?) Theorem 1 (Existence and Uniqueness of solutions) Given a semimartingale X in N, a F 0 measurable random variable 0, and a Stratonovich operator S from N to M, there are a predictable stopping time and a solution of (?) with initial condition 0 de ned on the set f(t;!) 2 R + j t 2 [0; (!))g that has the following maximality and uniqueness property: if 0 is another stopping time such that 0 < and 0 is another solution de ned on f(t;!) 2 R + j t 2 [0; 0 (!))g, then 0 and coincide in this set. 2 2 Émery, M. [1989] Stochastic Calculus on Manifolds. Universitext. Springer-Verlag. 5

6 Hamiltonian SDE (M; f; g) a Poisson manifold and X : R +! V semimartingale with values on the vector space V. h : M! V a smooth function, f 1 ; : : : ; r g basis of V so that h = P r i=1 h i i, h i 2 C 1 (M). The Hamilton equations with stochastic component X, and Hamiltonian function h are the Stratonovich stochastic di erential equation h = H(X; h )X; de ned by the Stratonovich operator H(v; z) : T v V! T z M given by rx H(v; z)(u) := h j ; uix hj (z), where X hj = f; h j g : j=1 6

7 Hamiltonian SDE Evolution of observables f 2 C 1 (M) in Stratonovich form f( h ) f( h 0) = rx Z j=1 0 ff; h j g( h )X j : and in Itô form f h f h 0 = rx j= Z 0 rx h ff; h j g dx j Z j;i=1 0 fff; h j g ; h i g h d X j ; X i for any stopping time :! [0; 1). 7

8 Example: Bismut s Hamiltonian di usions Let h : M! R r+1 be the Hamiltonian function m 7! (h 0 (m) ; : : : ; h r (m)) and X : R +! R r+1 given by: (t;!) 7! t; B 1 t (!) ; : : : ; B r t (!) ; where B j ; j = 1; :::; r, are r-independent Brownian motions. That is, B i ; B j = t ij t. In this setup, for any f 2 C 1 (M) Z t rx Z t f h t f h 0 = 0 + ff; h 0 g rx j=1 Z t 0 ff; h j g h 1 s ds + 2 i=1 h s db j s : 0 fff; h i g ; h i g h s ds In particular, the classical energy h 0 needs not to be conserved. Taking averages d dt E h 1 rx h 0 t = E h ffh 0 ; h i g ; h i g s t=s 2 i=1 2 Bismut, J.-M.[1981] Mécanique Aléatoire. Lecture Notes in Mathematics, volume 866. Springer- Verlag. 8

9 Some properties of Hamiltonian SDE Energy is not automatically preserved. Resemblance with double bracket dissipation 3. Liouville s Theorem: Let : M! [0; 1] be such that (z) is the maximal stopping time associated to the solution of the stochastic Hamilton s equations starting at z a.s., z 2 M. Let F be the ow such that for any z 2 M, F (z) : [0; (z)]! M is the above-mentioned solution. The map z 2 M 7! F t (z; ) 2 M is a local di eomorphism of M, for each t 0 and almost all 2. (t; ) 2 [0; (z)], F t (z; )! =! Then, for any z 2 M and any Additionally, the volume form! ^ : n) : : ^! is also preserved. Preservation of symplectic leaves. 3 A. Bloch, P.S. Krishnaprasad, J. E. Marsden, T. S. Ratiu. The Euler-Poincare Equations and ouble Bracket Dissipation. Commun. Math. Phys. 175, 1-42 (1996) 9

10 Conserved quantities and Stability De nition 2 A function f 2 C 1 (M) is said to be a strongly (respectively, weakly) conserved quantity of the stochastic system (?) if for any solution we have that f ( ) = f ( 0 ) (respectively, E [f ( )] = E [f ( 0 )], for any stopping time ). For instance, if f 2 C 1 (M) is such that S (x; y) (df (y)) = 0, then f is a strongly conserved quantity of (?). In a Hamiltonian system M; f; g ; h = P r i=0 h i i such that ff; h 0 g = ::: = ff; h r g = 0 =) f is a strong conserved quantity. If the driving noise X : R +! V is a Brownian motion, then the converse is also true. If we have rx ff; h 0 g + fff; h i g ; h i g = 0 i=1 instead, then f is a weakly conserved quantity. 10

11 Conserved quantities and Stability De nition 3 Suppose that the point z 0 2 M is an equilibrium of (?), that is, the constant process t (!) := z 0 is a solution (?). Denote by x the unique solution of (?) such that x 0 (!) = x for all! 2. Then we say that the equilibrium z 0 is 1. Almost surely (Lyapunov) stable when for any open neighborhood U of z 0 there exists another neighborhood V U of z 0 such that for any x 2 V we have x U a.s.. 2. Stable in probability if for any " > 0 lim P x!z 0 sup d ( x t ; z 0 ) > " t0 = 0 where d : M M! R + is any distance function that generates the manifold topology of M. 11

12 Conserved quantities and Stability Theorem 4 (Stochastic Dirichlet s Criterion) Suppose that z 0 2 M is an equilibrium point of (?) and there exists a function f 2 C 1 (M) such that df (z 0 ) = 0 and that the quadratic form d 2 f (z 0 ) is (positive or negative) de nite. If f is a strongly (respectively, weakly) conserved quantity for the system (?) then the equilibrium z 0 is almost surely stable (respectively, stable in probability). Other tools to check stability in probability: Lyapunov functions. 4 4 Hasminskii, R. Z. [1980] Stochastic Stability of Di erential Equations. Translated from the Russian by D. Louvish. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7. Sijtho and Noordho, Alphen aan den Rijn-Germantown, Md. 12

13 Example: Damped harmonic oscillator Physical interpretation: Brownian motion models random instantaneous bursts of momentum that are added to the particle by collision with lighter particles =) Collisions slow down of the particle. Mathematically, the expected value q e := E [q] satis es q e = _q e : This description is accurate but does not provide any information about the mechanism that links the presence of the Brownian perturbation to the emergence of damping. should be 0 when the Brownian motion is switched o. 13

14 Example: Damped harmonic oscillator Consider the following stochastic Hamiltonian system: M = R 2 with its canonical symplectic two form. X : R +! R given by X t (!) = (t + B t (!)) where 2 R and B t is a Brownian motion. h 2 C 1 (M) is the energy of a harmonic oscillator, h = 1 2m p kq2. Writing q e (t) = E h q t it holds that mq e (t) = 2 4 k k _q e (t) k 4m + 1 q e (t) : If = 0 the damping vanishes =) we get the di erential equation of a free harmonic oscillator. h is a strongly conserved quantity =) (0; 0) is an a.s. Lyapunov stable critical point by Dirichlet s criterion. 14

15 Critical action principle: Introduction S (M) and S (R) the M and the real valued semimartingales. A sequence fx n g n2n of processes X n : R +! R converge uniformly on compacts in probability to a process X if, for any t 2 R and any " > 0, P sup st jx n s Xj > "! n!1 0: Directional derivative: Let F : S (M)! S (R), and 2 S (M). F is di erentiable at in the direction of a one parameter group of di eomorphisms ' s : ( "; ")M! M, if for any sequence fs n g n2n R, such that s n! 0; the family X n = 1 n!1 s n F ' sn ( ) F ( ) converges uniformly on compacts in probability (ucp) to a process that we will denote s=0 F (' s ( )) and that is referred to as the directional derivative of by d ds F at in the direction of ' s. 15

16 Critical action principle: Introduction Admissible vector elds: A one parameter group of di eomorphisms ' : ( "; ")M! M is admissible for a set D M if ' s (y) = y 8y 2 D, 8s 2 ( "; "). The corresponding vector eld satis es that Y j D = 0 and is also called admissible for D. Stochastic action associated: Let (M;! = manifold and S : S(M)! S(R) given by Z Z Dbh E S ( ) = h; i ( ) ; X ; d) be an exact symplectic where b h ( ) : R +! V V is b h ( ) (t;!) := (Xt (!); h( t (!))) and h : M! V is the Hamiltonian. 16

17 Critical action principle Theorem 5 (M;! = d) exact symplectic manifold, X : R +! V a semimartingale, and h : M! V a Hamiltonian function. Let m 0 2 M be a point in M and : R +! M a continuous semimartingale de ned on [0; ) such that 0 = m 0. Suppose that there exist a measurable set U containing m 0 such that U < a.s., where U = inf ft > 0 j t (!) =2 Ug is the rst exit time of with respect to U. If the semimartingale satis es the stochastic Hamilton equations (with initial condition 0 = m 0 ) on the interval [0; U ] then for any local one parameter group of di eomorphisms ' : ( "; ") M! M admissible for fm 0 g we have d 1 f U <1g ds S (' s ( )) s=0 U = 0 a.s.: 17

18 Critical action principle Important: This critical action principle does not admit a converse within the set of hypotheses in which it is formulated due to the lack of di erentiability. There are counterexamples! That is, unlike it happens in classical mechanics, d ds s=0 S (' s ( )) = 0 does not imply that the Hamilton t equations hold for any s 2 [0; t]. To get a converse (as in the variational principle presented by Nawaf Bou-Rabee) we need to require the action to be critical with respect to a broader class of variations () future work). Noether s theorem If the action is invariant by ' s, that is, S (' s ( )) = S ( ) ; then the function i Y is a strongly conserved quantity of the stochastic Hamiltonian system associated to h : M! V. 18

19 Symmetries A di eomorphism : M! M is a symmetry of (?) if, for any x 2 N and any y 2 M, T x N that is, S (x; (y)) = T y S (x; y). S(x;(y))! T (y) M S(x;y) & ] % Ty T y M Proposition 6 Let : M! M be a symmetry. If ( ) is also a solution. is a solution of (?), then Let G a Lie Group and : G M! M a di erential map. The system (?) is G-invariant if g : M! M de ned g (m) = (g; m) is a symmetry for any g 2 G. 19

20 Symmetries and Conserved Quantities We consider free and proper actions : G M! M. In a Hamiltonian system (M; f; g) with energy function h = P r i=0 h i i, if s denotes the local one-parameter group of di eomorphisms associated to X f = f; fg 2 X (M), ff; h 0 g = ::: = ff; h r g = 0 () s are symmetries ( ) f is a strongly conserved quantity) 20

21 Reduction Theorem 7 (Reduction theorem) Let X : R +! N be a N-valued semimartingale and let S : T N M! T M be a Stratonovich operator that is invariant with respect to a proper and free action of the Lie group G on the manifold M. Denote by : M! M=G the canonical projection. The Stratonovich operator S M=G : T N M=G! T (M=G) given by S M=G (x; (y)) = T y (S(x; y)) ; x 2 N; y 2 M is such that, if is a solution semimartingale of the stochastic system associated to S with initial condition 0 and stochastic component X, then so is M=G := ( ) with respect to S M=G with initial condition ( 0 ) and stochastic component X. 21

22 Reduction For Hamiltonian systems more geometrical tools are available: momentum maps, Poisson and symplectic reduction... For example: As in mechanics, if (M; f; g) is Poisson and the G-action is canonical, then M=G is again Poisson and the reduced system is Hamiltonian with Hamilton function determined by h M=G = h: If the G-action on (M; f; g) has a momentum map associated J : M! g then its level sets are left invariant by the stochastic Hamiltonian system associated to h and X. Moreover, its components are conserved quantities. For any 2 g, the function h : M := J 1 () =G! V uniquely determined by the equality h = h i, i : J 1 (),! M, induces a stochastic Hamiltonian system on the symplectic reduced space (M ;! ). 22

23 Reconstruction Step 1: Find a solution M=G for the reduced stochastic di erential equation associated to the reduced Stratonovich operator S M=G on the dimensionally smaller space M=G. Step 2: Take an auxiliary principal connection A 2 1 (M; g) and a semimartingale d : R +! M such that d 0 = 0, (d) = M=G, and R ha; di = 0 2 g. 5 Step 3: Solve on G the stochastic di erential equation g t = T e L gt (Y t ) with initial condition g 0 = e a.s., where Y = R hs (X; d) (A) ; Xi. The solution of the original stochastic di erential equation (?) with initial condition 0 is then = g (d). 5 Shigekawa, I. [1982] On Stochastic Horizontal Lifts. Z. Wahrscheinlichkeitstheorie verw. Gebiete 9,

24 Example: Pertubation of the free rigid body Deterministic description in body coordinates: the con guration space is G = SO (3) and the energy function h 0 : G g! R given by (g; ) 7! h; ()i where : g! g is the inverse of the inertia tensor. The reduced Euler equations are _ t = t ( t ) and the reconstructed equations on the group _A t = A t Ad A t () ( ) where = J R is the conserved spatial angular momentum map. 24

25 Example: Pertubation of the free rigid body Under random bombardments (modeled by a spatial Brownian motion) M. Liao and L. Wang 6 claim that the spatial angular momentum is no longer preserved and replace dt in ( ) by dt + P 3 i=1 i Bt. i So, they propose the following SDE on SO (3) 3X A t = A t Ad A t () dt + A t Ad A t i Bt i i=1 But, can we introduce the perturbation preserving the symmetries of the problem? Consider the following stochastic Hamiltonian system: Take h 0 : G g! R the classical energy function and h 1 = J R 2 g the momentum map, both invariant. Consider X : R +! R g the driven stochastic process de ned as (!; t) 7! t; B 1 t (!) ; B 2 t (!) ; B 3 t (!). 6 Liao, M. and Wang, L. [2005] Motion of a rigid body under random perturbation. Elect. Comm. n Probab. 10,

26 Example: Pertubation of the free rigid body The reduced Lie-Poisson equations are t = t ( t ) dt + 3X i=1 t i B i t and the reconstructed equations on SO (3) A t = A t Ad A t () dt + 3X i=1 A t i B i t where = J R, which is now conserved. Same drift as in Liao and Wang s model. Di erent perturbation. The spatial angular momentum is preserved. 26

27 So... Thank you very much for your attention! (We are looking forward to having your feedback! If you have SDE + manifolds let us know...) 27