Helmholtz Theorem for Stochastic Hamiltonian Systems

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1 Advnces in Dynmicl Systems nd Applictions ISSN , Volume 1, Number 2, pp ) Helmholtz Theorem for Stochstic Hmiltonin Systems Frédéric Pierret Observtoire de Pris SYRTE UMR CNRS 863 Pris, Frnce Abstrct We derive the Helmholtz theorem for stochstic Hmiltonin systems. Precisely, we give theorem chrcterizing Strtonovich stochstic differentil equtions, dmitting Hmiltonin formultion. Moreover, in the ffirmtive cse, we give the ssocited Hmiltonin. This result shows coherence with the pproch of [13] where the uthors nswered to this problem through the preservtion of the symplectic form but without providing the Hmiltonin ssocited. AMS Subject Clssifictions: 6H1, 37K5, 65P1. Keywords: Stochstic differentil equtions, stochstic clculus of vritions, stochstic Hmilton s equtions, stochstic inverse problem of clculus of vritions. 1 Introduction A clssicl problem in Anlysis is the well-known Helmholtz s inverse problem of the clculus of vritions: find necessry nd sufficient condition under which system of) differentil equtions) cn be written s n Euler Lgrnge or Hmiltonin eqution nd, in the ffirmtive cse, find ll the possible Lgrngin or Hmiltonin formultions. This condition is usully clled Helmholtz condition. The Lgrngin Helmholtz problem hs been studied nd solved by J. Dougls [7], A. Myer [12] nd A. Hirsch [8, 9]. The Hmiltonin Helmholtz problem hs been studied nd solved up to our knowledge by R. Sntilli in his book [16]. Generliztion of this problem in the discrete clculus of vritions frmework hs been done in [4] nd [1] in the discrete Lgrngin cse. For the Hmiltonin cse it hs been done for the discrete clculus of vritions in [1] using the frmework of [11] nd in [6] using discrete embedding procedure derived in [5]. In the cse of time-scle Received August 29, 215; Accepted August 31, 215 Communicted by Peter Kloeden

2 22 Frédéric Pierret clculus i.e., mixing between continuous nd discrete sub-intervls of time, it hs been done in [15]. In this pper we generlize the Helmholtz theorem for Hmiltonin systems in the cse of Strtonovich stochstic clculus. We recover the clssicl cse when there is no stochstic counterprt. The pper is orgnized s follows: In Section 2, we give some generlities nd nottions bout the Strtonovich stochstic clculus. In Section 3, we remind definitions nd results bout clssicl nd stochstic Hmiltonin systems. In Section 4, we give brief survey of the clssicl Helmholtz Hmiltonin problem nd then we prove the min result of this pper, the stochstic Hmiltonin Helmholtz theorem. Finlly, in Section 5, we conclude nd give some prospects. 2 Generlities nd Nottions In this section we remind the bsic definitions nd nottions of stochstic processes following closely the presenttion done in the book of B. Øksendl [14] for which we refer for more detils nd proofs. In ll the pper, we consider Ω, F, P ) to be probbility spce. Definition 2.1. Let Bt) be n-dimensionl Brownin motion. Then we define F t = F n) t to be the σ-lgebr generted by the rndom vrible {B i s)} 1 i n, s t. Definition 2.2 See [14, Definition 3.3.2, p. 35]). Let W H, b) be the clss of functions such tht f : [, + ) Ω R t, ω) ft, ω) = f t ω) 1. t, ω) ft, ω) is B F-mesurble, where B denotes the Borel σ-lgebr on [, + ). 2. There exist filtrtion H on Ω, F) defining n incresing fmily of σ-lgebrs H t, t such tht ) B t is mrtingle with respect to H t b) f t is H t -dpted ) 3. P fs, ω) 2 ds = 1. We put W H = b> W H, b).

3 Helmholtz Theorem for Stochstic Hmiltonin Systems 23 Definition 2.3. Let W d n H, b) to be the set of d n mtrices v = [v ijt, ω)] where ech entry v ij t, ω) W H, b). We lso put in the sme mnner s the previous definition, W d n H = W d n H, b). b> We now cn introduce the set of Itô processes see [14, Definition 4.1.1, p. 44]). Definition dimensionl Itô processes). Let Bt) be 1-dimensionl Brownin motion on Ω, F, P ). A 1-dimensionl Itô process or Itô integrl is stochstic process Xt) on Ω, F, P ) of the form Xt) = X) + us, ω)ds + where v W H, u is H t -dpted nd is such tht P In the differentil form, X t is written s ) us, ω) ds < for ll t = 1. vs, ω)dbs), 2.1) dxt) = uds + vdb t 2.2) We now turn to higher dimensions see [14, Section 4.2, p. 48]). Definition 2.5 d-dimensionl Itô processes). Let Bt) be n-dimensionl Brownin motion on Ω, F, P ). A d-dimensionl Itô process is stochstic process Xt) on Ω, F, P ) of the form Xt) = X) + us, ω)ds + vs, ω)dbs), 2.3) where u is d-dimensionl vector nd v is d n mtrix such tht ech entry u i t, ω) nd v i,j t, ω)) defined 1-dimensionl Itô process, for i = 1,..., d s in the previous definition. Generlly, when ut, ω) = µt, Xt) nd vt, ω) = σt, Xt)) where µ : R d+1 R d, σ : R d+1 R d n, dxt) = µt, Xt))dt + σt, Xt))dBt) 2.4) is clled n Itô stochstic differentil eqution We now define the Strtonovich integrl from the Itô integrl see [14, Prgrph A comprison of Itô nd Strtonovich integrls, p. 35 nd Section 6.1, p. 85]).

4 24 Frédéric Pierret Definition 2.6 Strtonovich integrl). Let Bt) be n-dimensionl Brownin motion on Ω, F, P ) nd let Xt) d-dimensionl Itô process such tht Xt) = X) + {µs, Xs)) + µs, Xs))} ds + σs, Xs))dBs), 2.5) where µ, µ : R d+1 R d, σ : R d+1 R d n, nd where for ll 1 i d we hve µ i t, x) = 1 2 n d j=1 k=1 σ ij x k σ kj. The term µs, Xs))ds + σs, Xs))dBs) defines the Strtonovich integrl see [14, p.24]) nd it is written s σs, Xs)) dbs) = µs, Xs))ds + Then, the corresponding Strtonovich process is written s Xt) = X) + or in differentil form s µs, Xs))ds + σs, Xs))dBs). 2.6) σs, Xs)) dbs), 2.7) dxt) = µs, Xs))ds + σs, Xs)) dbs). 2.8) which is clled Strtonovich stochstic differentil eqution. We define the set S[, b], R d ) s the set of d-dimensionl Strtonovich processes defined in the Definition 2.6 over the intervl of time [, b]. We formlly introduce the set S [, b], R d ) s the set of Strtonovich stochstic process in their differentil form i.e., element x S [, b], R d ) correspond to element Xt) S[, b], R d ) such tht x = dxt). We lso define S [, b], R d ) = {X S[, b], R d ), X, ω) = Xb, ω) = for ll ω}. From [14, Theorem 4.1.5], we hve the following integrtion by prts formul for Strtonovich processes. Theorem 2.7 Strtonovich integrtion by prts formul). Let X, Y S[, b], R d ). We hve Xt) dy t) = Xb) Y b) X) Y ) Y t) dy t). 2.9)

5 Helmholtz Theorem for Stochstic Hmiltonin Systems 25 We now introduce the definition of stochstic field. Definition 2.8. A stochstic field XZt)) ssocited to stochstic process Zt) S[, b], R d ) is the dt of its deterministic prt nd its purely stochstic prt s follows: XZt)) = {µt, Zt)), σt, Zt))}. In wht follow, when there is no misleding, we omit for nottions convenience the explicit dependence in ω nd we put the dependency in time t s subscript when there is no possible confusion with the coordintes. 3 Summry bout Hmiltonin Systems 3.1 Clssicl Hmiltonin Systems For simplicity we consider time-independent Hmiltonin. The time-dependent cse cn be done in the sme wy. Definition 3.1 Clssicl Hmiltonin). A clssicl Hmiltonin is function H : R d R d R such tht for q, p) C 1 [, b], R d ) C 1 [, b], R d ) we hve the time evolution of q, p) given by the clssicl Hmilton s equtions Hq, p) q = p 3.1) Hq, p) ṗ =. q H A vector nottion is obtined posing z = q, p nd H = q, H where p T denotes the trnsposition. The Hmilton s equtions re then written s dz = J H 3.2) dt ) Id where J = with I I d d the identity mtrix on R d denotes the symplectic mtrix. An importnt property of Hmiltonin systems is tht there solutions correspond to criticl points of given functionl i.e., follow from vritionl principle. Theorem 3.2. The points q, p) C 1 [, b], R d ) C 1 [, b], R d ) stisfying Hmilton s equtions re criticl points of the functionl L H : C 1 [, b], R d ) C 1 [, b], R d ) R q, p) L H q, p) = where L H : R d R d R d R d R is the Lgrngin defined by L H x, y, v, w) = y v Hx, y). L H qt), pt), qt), ṗt)) 3.3)

6 26 Frédéric Pierret 3.2 Stochstic Hmiltonin Systems Stochstic Hmiltonin systems re formlly defined s follows. Definition 3.3. A stochstic differentil eqution is clled stochstic Hmiltonin system if one cn find H = {H D, H S } with H D : R 2d R nd H S : R 2d R n such tht dq = H D dt + H S db t dp = H D dt H 3.4) S db t. We recover the clssicl lgebric structure of Hmiltonin systems. The min properties supporting this definition re the following one, lredy proved in [2]: Liouville s property: Let Q, P ) R 2d, we consider the stochstic differentil eqution { dq = fp, Q)dt + σp, Q) dbt, 3.5) dp = gp, Q)dt + γp, Q) db t. The phse flow of 3.5) preserves the symplectic structure if nd only if it is stochstic Hmiltonin system. Hmilton s principle: Solutions of stochstic Hmiltonin system correspond to criticl points of stochstic functionl defined by L H Q, P ) = 4 Stochstic Helmholtz Problem L HD Qt), P t))dt + L HS Qt), P t)) db t. 3.6) In this section, we solve the inverse problem of the stochstic clculus of vritions in the Hmiltonin cse. We first recll the usul wy to derive the Helmholtz conditions following the presenttion mde by R. Sntilli [16]. We hve two min derivtions. One is relted to the chrcteriztion of Hmiltonin systems vi the symplectic two-differentil form nd the fct tht by dulity the ssocited one-differentil form to Hmiltonin vector field is closed. Such conditions re clled integrbility conditions. The second one use the chrcteriztion of Hmiltonin systems vi the selfdjointness of the Fréchet derivtive ssocited to the differentil opertor ssocited to the eqution. These conditions re usully clled Helmholtz conditions. Of course, we hve coincidence of the two procedures in the clssicl cse. In the stochstic cse the first chrcteriztion hs been studied by [13]. As consequence, we follow the second wy to obtin the stochstic nlogue of the Helmholtz conditions nd we show tht there is coherence between the two pproches.

7 Helmholtz Theorem for Stochstic Hmiltonin Systems Helmholtz Conditions for Hmiltonin Systems: A Brief Survey Symplectic Sclr Product In this section we work on R 2d, d 1, d N. The symplectic sclr product, J is defined for ll X, Y R 2d by X, Y J = X, J Y, 4.1) where, denotes the usul sclr product. We lso consider the L 2 symplectic sclr product induced by, J defined for f, g C 1 [, b], R 2d ) by f, g L 2,J = Adjoint of Differentil Opertor ft), gt) J dt. 4.2) In the following, we consider first order differentil equtions of the form ) ) d q Xq q, p) =. 4.3) dt p X p q, p) The ssocited differentil opertor is written s ) O,b q X q, p) = Xq q, p) ṗ X p q, p) A nturl notion of djoint for differentil opertor is then defined.. 4.4) Definition 4.1. Let A : C 1 [, b], R 2n ) C 1 [, b], R 2n ). We define the djoint A J of A with respect to, L 2,J by A f, g L 2,J = A J g, f L 2,J. 4.5) An opertor A will be clled self-djoin if A = A J with respect to the L 2 symplectic sclr product Hmiltonin Helmholtz Conditions The Helmholtz conditions in the Hmiltonin cse re given by the following theorem. Theorem 4.2 See [16, Theorem , p ]). Let Xq, p) be vector field defined by Xq, p = X q q, p), X p q, p)). The differentil eqution 4.3) is Hmiltonin if nd only the ssocited differentil opertor O,b X given by 4.4) hs self-djoint Fréchet derivtive with respect to the symplectic sclr product. In this cse the Hmiltonin is given by Hq, p) = 1 [p X q λq, λp) q X p λq, λp)] dλ. 4.6)

8 28 Frédéric Pierret The conditions for the self-djointness of the differentil opertor cn be mde explicitly. They coincide with the integrbility conditions chrcterizing the exctness of the one-form ssocited to the vector field by dulity. Theorem 4.3 See [16, Theorem.2.7.3, p. 88]). Let Xq, p = X q q, p), X p q, p)) be vector field. The differentil opertor O,b X given by 4.4) hs self-djoint Fréchet derivtive with respect to the symplectic sclr product if nd only if X q q + Xp =, p X q p nd X p q re symmetric. 4.7) Of course, the first condition corresponds to the fct tht Hmiltonin systems re divergence free i.e., we hve divx =. 4.2 Stochstic Helmholtz s Conditions We derive the min result of the pper giving the chrcteriztion of stochstic differentil equtions which re stochstic Hmiltonin systems Stochstic Symplectic Sclr Product Following the clssicl cse, we introduce stochstic nlogue of the symplectic sclr product nd notion of self-djointness for stochstic differentil equtions. Consider two stochstic process X t, Y t S[, b], R 2d ) written s dx t = At, X t )dt + Bt, X t ) db t nd dy t = Ct, Y t )dt + Dt, Y t ) db t. Definition 4.4 L 2 -Strtonovich sclr product). We define the L 2 -Strtonovich sclr product of X t nd dy t s X t, dy t L 2,Strto = The symplectic version is obtined s in the clssicl cse. X t dy t. 4.8) Definition 4.5 L 2 -Strtonovich symplectic sclr product). We define the L 2 symplectic sclr product of X t nd dy t such s X t, dy t L 2,J,Strto = X t, J dy t L 2,Strto. 4.9) Adjoint of Stochstic Differentil Opertor We consider the stochstic differentil equtions of the form ) ) dq XQ,D dt + X = Q,S db t. 4.1) dp X P,D dt + X P,S db t

9 Helmholtz Theorem for Stochstic Hmiltonin Systems 29 The stochstic differentil equtions 4.1) is ssocited with the stochstic field XQ, P ) = {X D Q, P ), X S Q, P )}. 4.11) Its ssocited stochstic differentil opertor is written s ) dq XQ,D dt X O X Q, P ) = Q,S db t dp X P,D dt X P,S db t. 4.12) A notion of symplectic djoint for stochstic differentil opertors cn be defined. Definition 4.6. Let A : S[, b], R 2d ) S [, b], R 2d ). We define the djoint A J : S[, b], R 2d ) S [, b], R 2d ) of A with respect to, L 2,J,Strto by A X t, Y t L 2,J,Strto = A J Y t, X t L 2,J. 4.13) As in the clssicl cse, min role will be plyed by self-djoint stochstic differentil opertors. Definition 4.7. A stochstic differentil opertor A is sid to be self-djoint with respect to the symplectic sclr product, L 2,J,Strto if A = A J Stochstic Helmholtz Conditions The min result of this section is the stochstic nlogue of the Hmiltonin Helmholtz conditions for stochstic differentil equtions. Proposition 4.8. Let U, V S [, b], R d ). The Fréchet derivtive DOQ, P ) of the opertor 4.12) is given by DOQ, P )U, V ) = [ XQ,D du [ XP,D dv U + X Q,D U + X P,D ] dt ] dt [ [ XP,S U + X Q,S U + X P,S ] db t ] db t nd its djoint DO JQ, P ) with respect to the symplectic sclr product, L 2,J,Strto is given by DOJQ, P )U, V ) = [ XP,D du U + [ XP,D dv U XQ,D XQ,D ) ] [ T dt ) ] T dt XP,S [ XP,S U + U ] ) ] T db t. db t

10 21 Frédéric Pierret Remrk 4.9. The nottion T corresponds to trnsposition s follows: As X Q,S nd X P,S re in R d n then ech prtil derivtive with respect to Q or P re in R d n d. For exmple, ) X Q,S XQ,S = ) i,j k nd then = XQ,S ) k,j 1 i d, 1 j n, 1 k d ) with respect to the sme indices. i 1 i d, 1 j n, 1 k d Proof. Let U, V S [, b], R d ) nd F, G S[, b], R d ). The Fréchet derivtive DOQ, P ) follows from simple computtions nd is given by DOQ, P )U, V ), F, G) L 2,J,Strto = [ XQ,D du G U + X Q,D dv F + [ XP,D U + X P,D ] {[ G dt ] F dt + {[ XP,S U + X Q,S U + X P,S Using the Strtonovich integrtion by prts formul, we obtin ] } db t ] db t } F G ). DOQ, P )U, V ), F, G) L 2,J,Strto = [ XQ,D U dg G) U + {[ ) } T G] db t U [ XP,D ) V df + F U + {[ XP,S ) ] } T + F db t U {[ XP,D {[ XP,S XQ,D ) ] T G) ) ] } G db t F ) ] dt ) ] } ) F db t. dt As consequence, the symplectic stochstic djoint of DOQ, P ) is given by DOJQ, P )F, G) = [ XP,D df F + [ XP,D dg F which concludes the proof. XQ,D XQ,D ) [ T G] dt G] dt XP,S [ XP,S F + F G] G] db t db t

11 Helmholtz Theorem for Stochstic Hmiltonin Systems 211 It follows directly, n explicit chrcteriztion of stochstic vector fields stisfying the stochstic Hmiltonin Helmholtz conditions. By coherence with the clssicl cse, we cll them stochstic integrbility conditions. Proposition 4.1 Stochstic integrbility conditions). The opertor O X defined by 4.12) hs self-djoint Fréchet derivtive t Q, P ) S[, b], R) S[, b], R) if nd only if the conditions X Q,D + XP,D =, 4.14) X Q,D = XQ,D nd X P,D = XP,D, 4.15) X Q,S + XP,S =, 4.16) X Q,S = nd X P,S = XP,S, 4.17) re stisfied over [, b]. Remrk Theses conditions re the sme tht the uthors in [13] obtined. It mens, s in the clssicl cse, there is coherence between the conservtion of the symplectic form ssocited with the stochstic field XQ, P ) nd the self-djointness of the stochstic differentil opertor O X. Theorem 4.12 Stochstic Hmiltonin Helmholtz theorem). Let XQ, P ) = {X D Q, P ), X S Q, P )} be stochstic field. The stochstic differentil equtions 4.1) is stochstic Hmiltonin eqution if nd only if the opertor O X defined by 4.12) hs self-djoint Fréchet derivtive with respect to the symplectic sclr product, L 2,J,Strto. Moreover, in this cse, the Hmiltonin H = {H D, H S } is given by H D Q, P ) = H S Q, P ) = 1 1 [P X Q,D λq, λp ) Q X P,D λq, λp )] dλ, 4.18) [P X Q,S λq, λp ) Q X P,S λq, λp )] dλ. 4.19) Proof. If X is stochstic Hmiltonin field, then there exist function H D C 2 R d R d, R) such tht X Q,D = H D nd X P,D = H D,

12 212 Frédéric Pierret nd there exist function H S C 2 R d R d, R n ) such tht X Q,S = H S nd X P,S = H S. The stochstic integrbility conditions re stisfied by the Schwrz lemm. Reciproclly, we suppose tht X stisfies the stochstic integrbility conditions. First, we compute H D. We denote ) = λq, λp ) nd we hve H 1 D Q, P ) = XQ,D ) λp X Q,D λq XP,D ) ) dλ. Using the stochstic integrbility conditions for the deterministic prt, we obtin H 1 D Q, P ) = λp X P,D ) X Q,D λq X ) P,D ) dλ. Remrking tht X P,D ) λ = X P,D ) we deduce H D Q, P ) = nd finlly, integrting with respect to λ 1 P + X P,D ) λ λx P,D )) dλ Q, In the sme wy we obtin H D Q, P ) = X P,DQ, P ). H D Q, P ) = X Q,DQ, P ). Second, we compute H S. We hve H 1 ) S Q, P ) = ) XP,S ) λp X Q,S λq dλ. Using the stochstic integrbility conditions for the purely stochstic prt, we obtin using the sme trick s previously nd This concludes the proof. H S Q, P ) = X P,SQ, P ) 4.2) H S Q, P ) = X Q,SQ, P ). 4.21)

13 Helmholtz Theorem for Stochstic Hmiltonin Systems 213 Remrk The Strtonovich clculus seems to be more pproprite for such computtions nd lso for its result bout the vritionl structure developed by J-M Bismut in [2] compred with the Itô clculus. However, one cn define notion of Itô Hmiltonin systems following the self-djointness chrcteriztion. Indeed, the definition of Strtonovich symplectic sclr product cn be lso defined with the Itô integrl. However, one will hve counterprt in the integrtion by prts in the proof of Proposition 4.8. It leds to n dditionl set of conditions on the deterministic prt of the stochstic field due to the extr term ppering with the Itô formul which cn be very restrictive. Also, in the Strtonovich clculus, the Hmiltonin defined is first integrl over the solutions of the stochstic differentil equtions considered i.e., constnt process, s in the clssicl cse. Wheres the definition obtined with the Itô clculus s explined previously leds to second extr set of conditions in order to hve the Hmiltonin defined s first integrl. It leds to rel need of interprettion of Hmiltonin systems for stochstic clculus. 5 Conclusion nd Prospects We proved result on Strtonovich stochstic differentil equtions which llows us to find the existence of Hmiltonin structure ssocited nd in the ffirmtive cse to give the Hmiltonin. Our result covers the clssicl cse when there is no purely stochstic counterprt. An importnt extension of this result concern the stochstic time-scle clculus nd more precisely the stochstic clculus on time scles developed in [17] nd [3]. First, the work is to define notion of Strtonovich clculus on time-scle s it hs been done for Itô clculus in [17] nd [3]. Second, the work is to define nturl notion of stochstic Hmiltonin on time scles nd then to give the stochstic time-scle version of Theorem This extension is work in progress nd will be the subject of future pper. References [1] I.D. Albu nd D. Opris. Helmholtz type condition for mechnicl integrtors. Novi Sd Journl of Mthemtics, 293):11 21, [2] J.M. Bismut. Mécnique létoire, volume 929 of Lecture notes in mthemtics. Springer-Verlg, [3] M. Bohner, O.M. Stnzhytskyi, nd A.O. Brtochkin. Stochstic dynmic equtions on generl time scles. Electronic Journl of Differentil Equtions, 21357):1 15, 213.

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