Invariance properties of Brownian motion via Lie's symmetries

In v ariance prop erties of Bro wnian motion via Lie’s symmetries Susanna Deh` o ∗ F rancesco C. De V ecc hi † P aola Morando ‡ Stefania Ugolini § Abstract The in v ariance prop erties of Bro wnian motion are inv estigated and revisited within a recent Lie symmetry approac h to sto c hastic differential equations. Some notable prop erties of the process can b e reco vered by a related integration by parts formula dev elop ed in the same research area. Keyw ords: Lie symmetry analysis of SDEs; Inv ariance prop erties of Bro wnian motion; Integration b y parts formula and Stein identities AMS Sub ject Classification: 60H10; 58D19; 60H07 In tro duction The study of symmetries in differen tial equations, pioneered by Sophus Lie, constitutes a fundamen tal geometric approach to understand the inv ariance prop erties gov erning a dynamical system ([ 20 ],[ 21 ]). By identifying the groups of transformations that preserve the equation’s structure, this framework pro- vides systematic to ols for order reduction, the construction of exact solutions, and the simplification of complex problems ([ 16 ]). Although symmetry analysis stands as a classical pillar for determinis- tic ordinary and partial differential equations (ODEs and PDEs), supp orted b y extensiv e literature, its extension to Stochastic Differen tial Equations (SDE) represen ts a relatively recent field of research ([ 6 ],[ 22 ],[ 23 ],[ 17 ],[ 9 ],[ 8 ],[ 10 ], [ 14 ],[ 1 ], [ 11 ],[ 12 ],[ 2 ],[ 7 ]). The reduction techniques that hav e prov en useful in the deterministic setting can also b e form ulated in the sto c hastic case ([ 18 ],[ 12 ],[ 8 ]). How ever, prop erly extending the classical Lie symmetry approach to SDEs requires careful attention: the generality and ric hness of sto chastic processes demand precise definitions and a clear distinction b etw een differen t types of symmetries and transformations. In this pap er, we focus on Brownian SDEs and their inv ariance prop erties, emphasizing, in particu- lar, Bro wnian motion itself, whic h, as a w ell-known and extensively studied case, provides a clear and illustrativ e example to highlight the full p oten tial of our approach and its unifying p ersp ectiv e. W e show that these inv ariance prop erties can b e inv estigated b oth through the recent approach to Lie symmetries of SDEs (dev elop ed in Milan, [ 9 ], [ 8 ], [ 10 ], [ 14 ], [ 1 ], [ 11 ],[ 12 ], [ 2 ],[ 7 ]), and via a nov el related in tegration b y parts formula ([ 13 ],[ 15 ]). The cornerstone of the theory of symmetries for SDEs is provided by sto c hastic transformations. While classical transformations of differen tial equations consist primarily of time changes t 7→ f ( t ) and space- time diffeomorphisms ( x, t ) 7→ Φ( x, t ), the framework of Bro wnian-driven SDEs admits at least tw o additional op erations in timately link ed to the noise term: random rotations of the driving Brownian motion and c hanges of measure via Girsanov’s theorem ([ 9 ],[ 1 ]). Let us analyze how sto c hastic transformations op erate on a solution ( X , W ) of a giv en SDE in a filtered probability space: dX t = µ ( X t , t ) dt + σ ( X t , t ) dW t . (1) First, we observe that a sto c hastic transformation T acts simultaneously on the solution process and on the SDE itself, ensuring structural consistency: the transformed process remains a solution to the transformed equation, p ossibly within a transformed filtered probabilit y space. Consequently , we denote ∗ Dipartimento di Matematica, Universit` a degli Studi di Milano, susanna.deho@unimi.it † Dipartimento di Matematica “F elice Casorati”, Univ ersit` a degli Studi di P avia, francescocarlo.devecchi@unipv.it ‡ DISAA, Univ ersit` a degli Studi di Milano, paola.morando@unimi.it § Dipartimento di Matematica, Universit` a degli Studi di Milano, stefania.ugolini@unimi.it 1 b y P T the action of the transformation T on the solution pro cess ( X , W ), and by E T its action on the equation’s co efficien ts ( µ, σ ). Mirroring the standard distinction b etw een strong and weak solutions to SDEs, we distinguish b et w een str ong sto chastic tr ansformations , whic h mo dify neither the driving Bro wnian motion nor the filtered probability space (fixed a priori), and we ak sto chastic tr ansformations . The latter, as it is w ell-known, ma y c hange the driving Brownian motion, the filtration, or the underlying probabilit y measure. F or weak transformations, admissibilit y is a crucial requirement: the transformed driving process must remain a Brownian motion with resp ect to the (possibly transformed) filtered probabilit y space. Let us start by considering a diffeomorphism T = Φ ∈ C 2 , 1 . By applying Itˆ o’s lemma to Φ( X t , t ), one immediately obtains: d Φ( X t , t ) = L (Φ)(Φ − 1 (Φ( X t , t )) , t ) dt + (( D Φ) σ )(Φ − 1 (Φ( X t , t )) , t ) dW t . Consequen tly , the transformed pro cess P T ( X, W ) := (Φ( X t , t ) , W t ) solv es the equation with transformed co efficien ts E T ( µ ) := L (Φ) ◦ (Φ − 1 , id ) and E T ( σ ) := ( D Φ) σ ◦ (Φ − 1 , id ). Since the Brownian motion and the probability space remain unchanged, diffeomorphisms are classified as strong sto c hastic transforma- tions. Differen tly , the following op erations are weak transformations. Considering a time change t ′ = f ( t ) = R t 0 f ′ ( s ) ds , if W t is an adapted Bro wnian motion, it is a standard result (see, e.g. [ 19 ]) that ˜ W t := R f − 1 ( t ) 0 p f ′ ( s ) dW s is a Brownian motion with resp ect to the time-c hanged filtration F f − 1 ( t ) . Using Itˆ o’s lemma and the time-change formula for SDEs, the pro cess X f − 1 ( t ) satisfies: dX f − 1 ( t ) = µ f ′ ◦ (Id , f − 1 ) dt + σ √ f ′ ◦ (Id , f − 1 ) d ˜ W t . Th us, the action on the pro cess is P T ( X, W ) := ( X f − 1 ( t ) , ˜ W t ), and the action on the SDE is E T ( µ, σ ) := ( µ f ′ ◦ (Id , f − 1 ) , σ √ f ′ ◦ (Id , f − 1 )). Similarly , let T = B ( x, t ) b e an orthogonal matrix-v alued function. One can prov e that the pro cess ¯ W t := R t 0 B ( X s , s ) dW s remains a Bro wnian motion by L´ evy’s characterization. Substituting dW t = B − 1 d ¯ W t in to ( 1 ) yields: dX t = µdt + σ B − 1 d ¯ W t . In our framework, this means P T ( X, W ) := ( X t , ¯ W t ) and the coefficients transform as E T ( µ, σ ) := ( µ, σ B − 1 ). Although the probabilit y space is fixed, the mo dification of the driving noise classifies this as a w eak transformation. Finally , let us in tro duce a Girsanov transformation defined by T = h ( x, t ). Under suitable assump- tions (see [ 1 ]), the pro cess W ′ t = W t − R t 0 h ( X s , s ) ds is a Brownian motion under the new measure Q . Substituting dW t = dW ′ t − hdt into the original equation leads to: dX t = ( µ + σ h ) dt + σdW ′ t . Here, the action on the pro cess is P T ( X, W ) := ( X t , W ′ t ), while the co efficients transform as E T ( µ, σ ) := ( µ + σ h, σ ). This is a w eak transformation as it mo difies both the underlying measure and the driving pro cess. W e can unify these examples into a general definition: a sto c hastic transformation T is a quadruplet T = (Φ , f , B , h ) encompassing these four op erations on the SDE. The global action is obtained by com- p osing these individual effects (see Theorem 1.2 ). Within the set of sto c hastic transformations, a sub class of particular in terest consists of those trans- formations that preserve the structure of the original SDE. W e name symmetries these transformations, and they are deeply linked to in v ariance properties of the underlying diffusion, such as self-similarit y (see Example 3.2.1 ) or conditioning (see Example 1.2 ). In the sto castic framework, different notions of in v ariance yield different definitions of symmetry: w e define str ong symmetries as sto c hastic transfor- mations that preserve the set of strong solutions; we ak symmetries as transformations that preserve the set of weak solutions; and G -we ak symmetries as transformations that preserv e the law of the solution pro cess (see, [ 2 ],[ 15 ]). By imp osing inv ariance conditions, w e can c haracterize the symmetries of a giv en SDE through a set of finite determining e quations , which provide necessary and sufficien t conditions for a transformation T to qualify as a symmetry (see Theorem 2.2 ). 2 Using these (finite) determining equations, it is straightforw ard to verify whether a sp ecific candidate T is a symmetry for a giv en SDE. F or instance, in the case of Bro wnian motion, one can easily chec k that the reflection transformation satisfies the determining equations (see Example 2.1 ), consistent with the well-kno wn fact that if W t is a Bro wnian motion, then − W t is again a Brownian motion. Ho wev er, the inv erse problem is significantly more c hallenging. Since these equations are highly nonlin- ear in the comp onents Φ , f , B , h , deriving an unknown symmetry T for a fixed SDE is a difficult task. F or this reason, aiming to systematically determine symmetries for general SDEs, we follow the classical deterministic approac h (see [ 20 ]) and lo ok for symmetries in a linearized setting. Indeed, the (p ossible) transition to a Lie algebra structure would provide significan tly more p o werful computational to ols. But in order to mov e to a Lie algebra structure, the set of sto c hastic transformations should constitute a Lie group, that is, a group with a differentiable structure (see [ 20 ] for a complete discussion). Regarding the group structure, w e just need to define the comp osition b et ween transformations and the inv ersion op eration. Of course, this could b e done in differen t wa ys, but we wan t the group structure to hav e a meaningful probabilistic coun terpart. Th us, giv en t wo stochastic transformations T 1 and T 2 , we define T 1 ◦ T 2 as the unique sto c hastic transformation under whic h the transformed solution process ( P T 1 ◦ T 2 ( X )) and the transformed coefficients ( E T 1 ◦ T 2 ( µ, σ )) coincide with the iteration of individual transformations (resp ectiv ely , P T 1 ( P T 2 ( X )) and E T 1 ( E T 2 ( µ, σ ))). Once established the algebraic structure, w e ha ve now to endow the group of sto c hastic transforma- tions with a differentiable structure to c haracterize it as a Lie group. A go od w ay to ac hieve this goal is to establish a one-to-one corresp ondence b etw een the group of sto c hastic transformations and the group of diffeomorphisms of suitable principal bundles. By endo wing the latter with an appropriate algebraic structure, this correspondence b ecomes a group isomorphism. Since the group of diffeomorphisms is a w ell-known Lie group, the group of sto chastic transformations inherits the Lie group structure. In particular, w e can consider one-parameter groups of s tochastic transformations. The elemen ts of the corresp onding Lie algebra are obtained in the standard manner by differen tiating the elements of the one-parameter group with respect to the flo w parameter and ev aluating the result at zero. In tuitively , one can view the Lie group elements as in tegral curv es and the corresp onding Lie algebra elements as the tangen t vector fields to these curves; formally , the Lie algebra is isomorphic to the tangent space of the Lie group at the iden tity (see [ 20 ]). Consequently , w e define an infinitesimal sto chastic tr ansformation as an element of this Lie algebra. Con versely , given an infinitesimal sto chastic transformation, it is p ossible to recov er the corresp onding finite transformation b y reconstructing the flo w (see Theorem 3.6 ).W e can no w define infinitesimal symmetries as those infinitesimal sto c hastic transformations that corresp ond to finite symmetries within the Lie group structure. These symmetries can b e now characterized as the solution to infinitesimal determining e quations , which th us pro vide necessary and sufficient conditions for an infinitesimal transformation to b e an infinitesimal symmetry . This approach ov ercomes the inheren t difficult y of solving finite determining equations directly . Instead, one solv es the infinitesimal determining equations—whic h are significan tly easier to handle due to their linearized setting—to find infinitesimal symmetries. Subsequently , the corresp onding finite symmetries are recov ered by reconstructing the flo w. The study of symmetries is of particular interest in determining the in v ariance prop erties of diffusion pro cesses, but this is not the only application of this research field. Indeed, in [ 13 ], the authors demon- strated that this geometric approac h to studying symmetries leads to the formulation of an integration b y parts formula in volving the infinitesimal symmetries of a given SDE, inspired b y Bismut’s approac h to Malliavin calculus ([ 4 ]). Since originally the in tegration b y parts theorem was derived without including rotations of the Bro wnian motion in the set of admissible sto c hastic transformations, in [ 15 ] the result is generalized to include rotations as w ell. The theorem states that if ( Y , m, C, H ) is an infinitesimal symmetry for a giv en SDE ( 1 )—where Y = P i Y i ∂ x i is a vector field in the Lie algebra corresp onding to the diffeomorphism Φ, m is a time function in the Lie algebra corresp onding to the time c hange f , C is an antisymmetric matrix corresp onding to the rotation matrix B , and H is a vector corresponding to the Girsanov drift h —then, under suitable regularity assumptions (see Section 4 ), the following holds for all sufficien tly smo oth functions F : − m ( t ) E P [ L ( F ( X t ))] + E P  F ( X t ) Z t 0 H ( X s , s ) dW s  + E P [ Y ( F ( X t ))] − E P [ Y ( F ( X 0 ))] = 0 . (2) 3 F ormula ( 2 ) exhibits a clear integration-b y-parts structure. In general, it inv olves zero-order terms ( F ( X t )), first-order terms ( Y ( F ( X t )), which entails the first deriv atives of F as Y is a vector field), and second-order terms ( L ( F ( X t )), where L is the infinitesimal generator of X , thus requiring, in principle, b oth first and second deriv atives). Naturally , certain symmetries may hav e null comp onen ts, causing sp ecific terms in ( 2 ) to v anish; consequently , we may obtain formulas of order zero, one, or tw o. One of the strongest features of the new Lie symmetry approach to SDEs is the inclusion of Girsano v transformations within the set of admissible transformations. This c hoice leads to an infinite-dimensional family of symmetries, dep ending on arbitrary functions of time (see, for instance, the Bro wnian motion example in Section 3.4 ). Consequen tly , when applied to sp ecific examples, formula ( 2 ) incorp orates these arbitrary time-dep enden t functions. A particularly comp elling asp ect of the integration by parts formula ( 2 ) is that by sp ecializing it through suitable choices of these functions, we can reco ver well-kno wn formulas from probabilit y theory (see Section 5 ). Among these, Stein’s iden tities are p erhaps the most significan t. These identities are fundamen tal tools that characterize probability distributions and serv e as the cornerstone of Stein’s metho d for distributional approximation in statistics. How ever, in the existing literature, such iden tities are typically deriv ed for sp ecific distributions on a case-by-case basis, and a fully unified approach is still largely missing. F or instance, the classical Stein identit y for a Gaussian random v ariable Z ∼ N ( µ, σ 2 ) is giv en by E [( Z − µ ) F ( Z )] = σ 2 E [ F ′ ( Z )]. In the specific case of a Brownian motion at time t , where W t ∼ N (0 , t ), this identit y implies: E [ W t F ( W t )] = t E [ F ′ ( W t )] . (3) F rom our persp ective, the emergence of such relations is not surprising. Indeed, although classically deriv ed using different metho ds, it is imp ortan t to observe that identit y ( 3 ) naturally descends from the symmetries of Brownian motion, and in particular from its inv ariance under Girsano v transformations. This p erspective is closely related to Bism ut’s form ulation of Mallia vin calculus. Sp ecifically , one can consider a p erturbation of the Brownian motion by a constant drift h λ = λ . Since W t − λt is a Brownian motion under the measure Q λ defined by the Radon-Nik o dym deriv ative e λW t − 1 2 λ 2 t , we ha ve the equalit y of exp ectations: E P [ F ( W t )] = E Q λ [ F ( W t − λt )] = E P h e λW t − 1 2 λ 2 t F ( W t − λt ) i . By differentiating b oth sides with resp ect to λ and ev aluating at λ = 0, one recov ers precisely the target iden tity ( 3 ). In our framew ork, form ula ( 2 ), b eing muc h more general, captures this relationship directly . W e obtain the iden tity as a particular case b y specializing the arbitrary time-dependent function appearing in the form ula to the specific c hoice corresponding to the Girsano v symmetry used in the argumen t abov e. Indeed, consider a one-dimensional Brownian motion dX t = dW t . In this case, formula ( 2 ) ensures that for any time-dep enden t function β ( t ) and any b ounded function F ∈ C 2 b , the follo wing holds: E P  F ( W t ) Z t 0 − β ′ ( s ) dW s  + E P [ β ( t ) F ′ ( W t )] − E P [ β (0) F ′ ( W 0 )] = 0 . (4) By setting β ( t ) = t , iden tity ( 4 ) reduces to: E P [ W t F ( W t )] = t E P [ F ′ ( W t )] . As exp ected, this result coincides exactly with the Bro wnian Stein identit y ( 3 ). Stein’s iden tity is not the only classical result reco verable from ( 2 ). By v arying the choices of parameters, w e also obtain other well-kno wn inv ariance prop erties: Isserlis’ theorem, L ´ evy’s sto c hastic area, and m uch more (see Section 5 ). Indeed, classical iden tities of Gaussian analysis emerge here as an unified manifestations of the underlying symmetry groups asso ciated with the Wiener process. The flexibilit y in c ho osing temporal parameters suggests that the integration by parts theorem acts as a generating mec hanism for an infinite family of conserv ation la ws for the sp ecific diffusion at hand. While, for simplicity , Section 5 of this pap er fo cuses exclusively on Bro wnian motion, we stress that the integration b y parts formula can be deriv ed for generic diffusion pro cesses, yielding equally compelling results. 4 1 Finite sto c hastic transformations Throughout this pap er, we denote by SDE µ,σ the following equation: dX t = µ ( X t , t ) dt + σ ( X t , t ) dW t , (5) where W is an d -dimensional Brownian motion, µ ( x, t ) ∈ R n , and σ ( x, t ) ∈ R n × d . W e denote by D F the deriv ativ e of a function F , whose interpretation dep ends on the dimension: for F : R → R , it represents the first deriv ative; for F : R n → R , it denotes the gradient; and for F : R n → R d , it corresponds to the Jacobian matrix. Similarly , we denote by D 2 F the second deriv ative (representing the standard second deriv ativ e or the Hessian matrix, depending on the dimension). A sto c hastic transformation T acts simultaneously on a given SDE and on its solution process ( X , W ). F ollowing the notation in [ 9 , 11 , 13 ], we denote by E T the action on the SDE co efficien ts and by P T the action on the solution pro cess. This structure ensures consistency: if ( X , W ) is a (w eak) solution to S D E µ,σ , then the transformed pro cess P T ( X, W ) := ( P T ( X ) , P T ( W )) is a (w eak) solution to the transformed equation E T ( S D E µ,σ ) := S D E E T ( µ ) ,E T ( σ ) . W e distinguish b et ween str ong and we ak sto c hastic transformations. A strong transformation pre- serv es the Brownian motion (i.e., P T ( W ) = W ) and the underlying probability space. Conv ersely , a w eak transformation mo difies the Bro wnian component ( P T ( W )  = W ) or the filtered probability space; in this latter case, well-posedness requires verifying that P T ( W ) remains a Bro wnian motion, p ossibly under a new filtration or probability measure. Definition 1.1. L et M , M ′ b e op en subsets of R n . Consider S D E µ,σ ( 5 ) , driven by an m -dimensional Br ownian motion W . A (finite) sto chastic tr ansformation fr om M to M ′ is a quadruple T = (Φ , f , B , h ) of smo oth functions, whose c omp onents act as fol lows: • Sp atial diffe omorphism ( Φ ): The function Φ : M × R + → M ′ defines a time-dep endent sp atial c o or dinate change T 1 . It le aves the pr ob ability sp ac e unchange d. Its action is: P T 1 ( X ) t = Φ( X t , t ) , P T 1 ( W ) t = W t , E T 1 ( µ ) = L t (Φ) ◦ (Φ − 1 , id) , E T 2 ( σ ) = ( D (Φ) σ ) ◦ (Φ − 1 , id) , wher e L t = ∂ t + n P i =1 µ i ∂ i + n P i,j =1 1 2 ( σ σ T ) ij ∂ ij is the infinitesimal gener ator of S D E µ,σ . • Time change ( f ): The strictly p ositive function f ′ : R + → R + defines an absolutely c ontinuous time change f ( t ) := R t 0 f ′ ( s ) ds . This tr ansformation, denote d by T 2 , changes the filtr ation fr om F t to F f − 1 ( t ) . Its action is: P T 2 ( X ) t = X f − 1 ( t ) , P T 2 ( W ) t = Z f − 1 ( t ) 0 p f ′ ( s ) dW s , E T 2 ( µ ) = µ f ′ ◦ (Id , f − 1 ) , E T 2 ( σ ) = σ √ f ′ ◦ (Id , f − 1 ) . • R andom r otation ( B ): The function B : M × R + → S O ( d ) defines a r otation of the Br ownian motion T 3 . It pr eserves the pr ob ability sp ac e. Its action is: P T 3 ( X ) = X , P T 3 ( W ) t = Z t 0 B ( X s , s ) dW s , E T 3 ( µ ) = µ, E T 3 ( σ ) = σ B − 1 . • Drift change/Girsanov ( h ): The function h : M × R + → R d defines a tr ansformation T 4 that changes the me asur e fr om P to Q via the R adon-Niko dym derivative: d Q d P    F t = exp  Z t 0 h ( X s , s ) · dW s − 1 2 Z t 0 | h ( X s , s ) | 2 ds  . 5 Assuming r e gularity c onditions to ensur e non-explosivity (se e [ 11 ]), the action of T 4 is: P T 4 ( X ) = X , P T 4 ( W ) t = W t − Z t 0 h ( X s , s ) dt, E T 4 ( µ ) = µ + σ h, E T 4 ( σ ) = σ . F or e ach i = 1 , . . . , 4 , the tr ansformation T i is wel l-define d. Sp e cific al ly, the tr ansforme d driving Br ow- nian motion, P T i ( W ) , r emains a Br ownian motion with r esp e ct to the filter e d pr ob ability sp ac e induc e d by T i . F or the non-trivial c ases, we r efer to [ 19 ] for time change and to [ 3 ] for r otation and change of me asur e. The general sto c hastic transformation is obtained by comp osing these elementary actions. Theorem 1.2 (Double action of the transformation) . L et ( X, W ) b e a (we ak) solution to S D E µ,σ and let T = (Φ , f , B , h ) b e a sto chastic tr ansformation. Then, the tr ansforme d pr o c ess P T ( X, W ) is a solution to E T ( S D E µ,σ ) . The tr ansforme d pr o c ess P T ( X, W ) := ( P T ( X ) , P T ( W )) is define d by: P T ( X ) t = Φ( X f − 1 ( t ) , f − 1 ( t )) , P T ( W ) t = Z f − 1 ( t ) 0 p f ′ ( s ) B ( X s , s )( dW s − h ( X s , s ) ds ) , wher e P T ( W ) is a Br ownian motion with r esp e ct to the filtr ation F ′ t = F f − 1 ( t ) and the pr ob ability me asur e Q define d by the density: d Q d P    F t = exp  Z t 0 h ( X s , s ) · dW s − 1 2 Z t 0 | h ( X s , s ) | 2 ds  . The tr ansforme d c o efficients of E T ( S D E µ,σ ) := S D E E T ( µ ) ,E T ( σ ) ar e given by: E T ( µ ) =  1 f ′  L t (Φ) + D (Φ) σ h   ◦ (Φ − 1 , f − 1 ) , E T ( σ ) =  1 √ f ′ D (Φ) σB − 1  ◦ (Φ − 1 , f − 1 ) . Pr o of. The result follows b y iterating the actions of the four comp onents described in Definition 1.1 . See [ 11 ] for details. ■ Remark 1.3. The distinction b etwe en str ong and we ak tr ansformations mirr ors that of str ong and we ak solutions. A strong sto chastic tr ansformation acts on a fixe d pr ob ability sp ac e and filtr ation, le aving the Br ownian motion unchange d. This r e quir es h = 0 (no me asur e change), f = id (no change in the filtr ation), and B = I d (no change in the driving Br ownian motion). Thus, a tr ansformation is str ong if and only if T = (Φ , id , I d , 0) . Remark 1.4. Any elementary tr ansformation c an b e viewe d as a sub class of the gener al tuple T : a diffe o- morphism c orr esp onds to (Φ , id , I d , 0) ), a time change to (Id , f , I d , 0) , a r andom r otation to (Id , id , B , 0) , and a me asur e change to (Id , id , I d , h ) . 1.1 F rom Bro wnian motion to Geometric Bro wnian motion via strong trans- formation Consider a one-dimensional Bro wnian motion dX t = dW t , with drift µ 0 := 0 and diffusion co efficien t σ 0 := 1. Applying the strong transformation T = (Φ , id , 1 , 0) defined by Φ( x, t ) = z 0 exp  ( µ − 1 2 σ 2 ) t + σ x  6 yields the Geometric Brownian Motion (GBM). Indeed, by Theorem 1.2 , the transformed process Z t := P T ( X ) t = Φ( X t , t ) solves the SDE determined b y the transformed co efficien ts ¯ µ := E T ( µ 0 ) and ¯ σ := E T ( σ 0 ). Computing the action of the generator L t = ∂ t + 1 2 ∂ xx and the deriv ativ e D = ∂ x on Φ, w e obtain: L t (Φ) = ∂ t Φ + 1 2 ∂ xx Φ = ( µ − 1 2 σ 2 )Φ + 1 2 σ 2 Φ = µ Φ , D (Φ) = σ Φ . The transformed co efficien ts are obtained by comp osing with Φ − 1 (see Definition 1.1 ): ¯ µ ( z , t ) := E T ( µ 0 )( z , t ) = L t (Φ) ◦ Φ − 1 = µz , ¯ σ ( z , t ) := E T ( σ 0 )( z , t ) = D (Φ) ◦ Φ − 1 = σ z . Th us, T yields the standard geometric Brownian motion equation: d Z t = µZ t dt + σ Z t dW t , Z 0 = z 0 . (6) Since T is strong, the underlying filtered probabilit y space remains unchanged. The generalization to the n -dimensional case is straightforw ard and follo ws the same logic comp onen t-wise. 1.2 F rom Bro wnian motion to Bro wnian Bridge via w eak transformation Consider an n -dimensional Brownian motion dX t = dW t , with X = ( X 1 , · · · , X n ) T , W = ( W 1 , · · · , W n ) T , drift µ 0 := 0 and diffusion co efficien t σ 0 := I n . Let us apply the weak stochastic transformation T = (Φ , f , I n , h ) defined by: f ( t ) = tT 2 1 + tT , Φ( x, t ) = ( T − f ( t )) x, h ( x, t ) = f ′ ( t ) T − f ( t ) x. According to Theorem 1.2 , the transformed pro cess P T ( X , W ) solv es the SDE with co efficien ts E T ( µ, σ ). The transformed solution pro cess is given by: Z t := P T ( X ) t = Φ( X f − 1 ( t ) , f − 1 ( t )) = ( T − t ) X f − 1 ( t ) . The transformed driving noise is: P T ( W ) t = Z f − 1 ( t ) 0 p f ′ ( s ) ( dW s − h ( X s , s ) ds ) = ˜ W t − Z t 0 Z s T − s ds, (7) where ˜ W t := R f − 1 ( t ) 0 p f ′ ( s ) dW s is a Brownian motion under the transformed filtration (according to Definition 1.1 ). W e no w compute the transformed co efficien ts E T ( µ 0 , σ 0 ). The new drift is: µ ′ := E T ( µ 0 ) = h 1 f ′  L t (Φ)+ D Φ · h i ◦ (Φ − 1 , f − 1 ) = h 1 f ′  − f ′ ( t ) x +( T − f ( t )) f ′ ( t ) T − f ( t ) x i ◦ (Φ − 1 , f − 1 ) = 0 . The new diffusion co efficient is: σ ′ := E T ( σ 0 ) = h 1 √ f ′ ( D Φ) I n i ◦ (Φ − 1 , f − 1 ) = h T − f ( t ) T − f ( t ) I n i ◦ (Φ − 1 , f − 1 ) = I n . Consequen tly , the transformed pro cess satisfies the SDE: dZ t = µ ′ dt + σ ′ dP T ( W ) t = dP T ( W ) t . Substituting the expression for P T ( W ) t from ( 7 ), w e obtain: dZ t = − Z t T − t dt + d ˜ W t . Th us, the transformation T applied to a standard Bro wnian motion yields a Br ownian Bridge . 7 It is worth noting that since E T ( µ 0 ) = µ 0 and E T ( σ 0 ) = σ 0 , the structural form of the SDE is preserv ed: Z t solv es a Brownian motion SDE driv en by P T ( W ). How ever, P T ( W ) is a Brownian motion only under the probability measure Q defined by the Dol´ eans-Dade exp onen tial asso ciated with h . This reflects the fact that a Brownian Bridge can b e viewed as a Brownian motion conditioned on reaching 0 at time T (see [ 5 ]). W e refer to such transformations—those preserving the form of the SDE co efficien ts—as symmetries . These are intrinsically link ed to the in v ariance prop erties of the diffusion pro cess and will be discussed extensiv ely in the follo wing section. 2 In v ariance prop erties and finite symmetries Example 1.2 sho ws that sp ecific transformations can preserv e the structure of an SDE, reflecting intrinsic prop erties of the underlying diffusion such as conditioning or self-similarit y (see Example 3.2.1 ). W e define symmetries as sto c hastic transformations that leav e the original SDE inv arian t. F ollowing the framew ork in [ 10 , 11 , 13 ], an SDE is (strongly/w eakly) in v arian t if the transformation T preserv es the set of its (strong/w eak) solutions. A broader notion, introduced in [ 15 ], requires only the preserv ation of the solution’s la w, linking symmetry to the inv ariance of an entire family of SDEs rather than of a single equation. Definition 2.1. Consider the sto chastic differ ential e quation S D E µ,σ dX t = µ ( X t , t ) dt + σ ( X t , t ) dW t on the filter e d pr ob ability sp ac e (Ω , F , ( F t ) t , P ) . • A str ong sto chastic tr ansformation T = (Φ , id , I d , 0) is a strong (finite) symmetry if, for every str ong solution X with driving Br ownian motion W , the tr ansforme d pr o c ess P T ( X ) r emains a solution to S D E µ,σ in the same sp ac e driven by P T ( W ) = W . Being c omp ose d of diffe omorphisms, T mo difies neither the pr ob ability sp ac e nor the Br ownian motion. • A we ak sto chastic tr ansformation T = (Φ , f , B , h ) is a weak (finite) symmetry if, for every we ak solution ( X , W ) , the tr ansforme d p air P T ( X, W ) r emains a we ak solution to S D E µ,σ in (Ω , F , ( F ′ t ) t , Q ) , wher e Q and F ′ t ar e the me asur e and the filtr ation induc e d by T , r esp e ctively. • A we ak sto chastic tr ansformation T = (Φ , f , B , h ) is a G -we ak symmetry if it pr eserves the solu- tion set of the martingale pr oblem asso ciate d with ( µ, σ σ T ) . Equivalently, for every we ak solution ( X, W ) , the tr ansforme d pr o c ess P T ( X ) has the same law as X . Imp osing these inv ariance conditions characterizes symmetries via the co efficien ts of the SDE. Theorem 2.2. L et S D E µ,σ b e define d as ab ove. • T = (Φ , id , I d , 0) is a str ong symmetry if and only if µ = L t (Φ) ◦ (Φ − 1 , id) , σ =  D Φ · σ  ◦ (Φ − 1 , id) . • T = (Φ , f , B , h ) is a we ak symmetry if and only if µ =  1 f ′  L t (Φ) + D Φ σ h   ◦ (Φ − 1 , f − 1 ) , σ =  1 √ f ′ D Φ σ B − 1  ◦ (Φ − 1 , f − 1 ) . • T = (Φ , f , B , h ) is a G -we ak symmetry if and only if µ =  1 f ′  L t (Φ) + D Φ σ h   ◦ (Φ − 1 , f − 1 ) , σ σ T =  1 f ′ D Φ σ σ T D Φ T  ◦ (Φ − 1 , f − 1 ) . Pr o of. Proofs for the first t w o statements are in [ 11 ], and the third in [ 15 ]. The logic relies on Theorem 1.2 , whic h states that P T ( X, W ) solv es the transformed equation S D E E T ( µ ) , E T ( σ ) . 8 • F or strong and weak symmetries, in v ariance requires the transformed co efficien ts to match the original ones: µ = E T ( µ ) and σ = E T ( σ ). Substituting the explicit forms of E T yields the stated equations. • F or G -weak symmetries, preserving the la w implies preserving the infinitesimal generator L . Th us, w e require µ = E T ( µ ) and σ σ T = E T ( σ ) E T ( σ ) T . ■ The relations in Theorem 2.2 are called finite determining e quations . They form a highly nonlinear system. T o ov ercome this complexity , we turn to infinitesimal transformations. This approach linearizes the problem: instead of solving nonlinear functional equations for T , we determine the generator of a one-parameter group of transformations. This motiv ates the algebraic structure introduced in Section 3 . Remark 2.3. The symmetry notions in Definition 2.1 fol low the hier ar chy Str ong ⊂ We ak ⊂ G -we ak. Str ong symmetries ar e we ak symmetries which, in p articular, pr eserve the driving Br ownian motion and the filter e d pr ob ability sp ac e. We ak symmetries ar e G -we ak as they pr eserve the law. The c onverses gener al ly fail: a we ak symmetry is str ong only if f = id , B = I d , and h = 0 , while a G -we ak symmetry is not ne c essarily we ak unless E T ( σ ) = σ . This is b e c ause the determining e quations for G -we ak symmetries do not c onstr ain the r otation matrix ( B ), making any r otation a trivial G -we ak symmetry: if T = (Id , id , B , 0) is a r otation, then fr om Definition 1.1 E T ( σ ) = σ B − 1 ; sinc e B is a r otation matrix, by c onstruction B B T = I d , so it is imme diate to verify that E T ( σ ) E T ( σ ) T = σ σ T . Equivalenc e b etwe en we ak and G -we ak symmetries holds only under structur al c onditions, such as the invertibility of σ σ T . In this c ase, de c omp osing and using The or em 2.2 σ = σ σ T σ ( σ T σ ) − 1 =  1 f ′ D (Φ) σσ T D (Φ) T  ◦ (Φ − 1 , f − 1 ) · σ ( σ T σ ) − 1 , we find that to satisfy the we ak symmetry c ondition, B is uniquely determine d by B − 1 ◦ Φ − 1 =  1 √ f ′ σ T D (Φ) T  ◦ (Φ − 1 , f − 1 ) · σ ( σ T σ ) − 1 , which yields an ortho gonal matrix. Se e [ 15 ] for the c onne ction to the “squar e r o ot of a matrix field” pr oblem. Remark 2.4. If X satisfies dX t = µdt + σ dW t and B ∈ S O ( d ) , then X is also a we ak solution to dX t = µdt + ( σ B − 1 ) dW ′ t , driven by W ′ t = R t 0 B dW s . The inclusion of r otations implies that the law of a we ak solution is no longer identifie d by the single S D E µ,σ , but by the whole e quivalenc e class G ⊣⊓}⌉ ( S DE µ,σ ) := { S D E µ,σ B − 1 | B ∈ S O ( d ) } . Conse quently, G -we ak symmetries ar e tr ansformations pr eserving the solution set of G ⊣⊓}⌉ ( S D E µ,σ ) . Se e [ 15 , 1 ] for the c onne ction with gauge symmetries. 2.1 Reflection of Brownian motion It is well known that if W t is an n -dimensional Brownian motion, then Y t := − W t is also a Brownian motion on the same filtered probability space (see [ 3 ]). W e aim to recov er this reflection as a symmetry of the SDE dX t = dW t , where µ = 0 and σ = I n . First, consider the transformation T = (Φ , id , I n , 0) with Φ( x ) = − x . This map inv olves only a diffeomorphism of the state space, implying Φ( X t ) = − W t . According to Theorem 1.2 , the co efficien ts of the transformed SDE are: E T ( µ ) = L t (Φ) ◦ Φ − 1 = 1 2 D 2 Φ ◦ Φ − 1 = 0 = µ, E T ( σ ) = ( D Φ · σ ) ◦ Φ − 1 = − I n . While E T ( σ )  = σ , we observ e that E T ( σ ) E T ( σ ) T = ( − I n )( − I n ) T = I n = σ σ T . Thus, T is a G -weak symmetry for Definition 2.1 . In this context, T preserv es the law of the solution process but fails to preserv e the SDE structure, as it interprets the reflected pro cess via dY t = − dW t . Since σ σ T = I n is inv ertible, Remark 2.3 ensures the equiv alence b et ween w eak and G -weak sym- metries. T o recov er the weak symmetry T 1 = (Φ , B , 1 , 0), following the Remark 2.3 w e determine the orthogonal matrix B suc h that: B − 1 ◦ Φ − 1 = ( σ T D Φ T ) ◦ Φ − 1 · σ ( σ T σ ) − 1 = − I n , 9 yielding B = − I n . The resulting transformation T 1 = (Φ , id , − I n , 0) maps X to − W and preserv es the SDE co efficien ts: E T 1 ( µ ) = µ, E T 1 ( σ ) = ( D Φ · σ B − 1 ) ◦ Φ − 1 = ( − I n )( I n )( − I n ) = I n = σ. Th us, T 1 is a weak symmetry according to Definition 2.1 . It in terprets the reflected pro cess as dY t = d ( − W t ). Remark 2.5. Both T and T 1 ar e discrete symmetries, sinc e they c annot b e gener ate d by the flow of an infinitesimal gener ator. Conse quently, they ar e r e c over e d only as finite tr ansformations (se e [ 20 ]). 3 Algebraic structure of sto c hastic transformations W e aim to endow the set of sto c hastic transformations with a group structure. Naturally , one could define comp osition and inv ersion in several different wa ys; how ev er, the resulting algebraic structure should satisfy t wo key requiremen ts: 1. It m ust rest on a meaningful probabilistic foundation; 2. It must allo w for the formulation of an asso ciated Lie algebra, thereby pro viding more p o werful computational to ols in a suitably linearized setting. 3.1 Probabilistic foundation of the algebraic structure The algebraic structure of stochastic transformations must be grounded in a well-defined probabilistic in terpretation. Sp ecifically , we require that the comp osition of transformations corresponds to the com- p osition of the induced pro cesses on solutions and co efficien ts. That is, for any T 1 , T 2 , the comp osed transformation T 2 ◦ T 1 is the unique transformation satisfying: P T 2 ◦ T 1 ( X, W ) = P T 2 ( P T 1 ( X, W )) and E T 2 ◦ T 1 ( µ, σ ) = E T 2 ( E T 1 ( µ, σ )) . Let T 1 = (Φ 1 , f 1 , B 1 , h 1 ) and T 2 = (Φ 2 , f 2 , B 2 , h 2 ). Assuming T 2 ◦ T 1 = ( ˜ Φ , ˜ f , ˜ B , ˜ h ), we compare the transformed solution pro cess X and the driving noise W . First, for the pro cess X , we hav e: P T 2 ◦ T 1 ( X ) = ˜ Φ( X ˜ f − 1 ( t ) , ˜ f − 1 ( t )) . (8) Con versely , applying the transformations sequen tially: P T 2 ( P T 1 ( X )) t = (Φ 2 ◦ Φ 1 )( X ( f 2 ◦ f 1 ) − 1 ( t ) , ( f 2 ◦ f 1 ) − 1 ( t )) . (9) Comparing ( 8 ) and ( 9 ) identifies the spatial and temporal comp onen ts: ˜ Φ = Φ 2 ◦ Φ 1 , ˜ f = f 2 ◦ f 1 . Similarly , for the Bro wnian motion, the composed transformation yields P T 2 ◦ T 1 ( W ) = Z ˜ f − 1 ( t ) 0 q ˜ f ′ ( s ) ˜ B ( X s )( dW s − ˜ h ( X s , s ) ds ) , (10) while the sequen tial application gives P T 2 ( P T 1 ( W )) = Z ( f 2 ◦ f 1 ) − 1 ( t ) 0 q [( f ′ 2 ◦ f 1 ) f ′ 1 ]( s ) [( B 2 ◦ Φ 1 ) B 1 ]( X s , s ) · dW s −  h 1 ( X s , s ) + q f ′ 1 ( s ) B − 1 1 ( X s , s )( h 2 ◦ Φ 1 )( X s , s )  ds ! . (11) Matc hing the terms in ( 10 ) and ( 11 ) determines ˜ B = ( B 2 ◦ Φ 1 ) B 1 , ˜ h = h 1 + p f ′ 1 B − 1 1 · ( h 2 ◦ (Φ 1 , id)) . 10 It is easy to verify that with these choices for the comp onen ts of ˜ T , w e also ha ve that E T 2 ◦ T 1 ( µ, σ ) ≡ E T 2 ( E T 1 ( µ, σ )) . T o define a group structure, it remains to establish the neutral element and the inv erse. The neutral elemen t T 0 = (Φ 0 , f 0 , B 0 , h 0 ) must satisfy T ◦ T 0 = T = T 0 ◦ T , whic h yields Φ 0 = Id , f 0 = id B 0 = I n , h 0 = 0 . (12) The inv erse T − 1 = ( ¯ Φ , ¯ f , ¯ B , ¯ h ) is defined by T ◦ T − 1 = T − 1 ◦ T = T 0 . Solving the comp osition equations for the barred comp onents results in: T − 1 =  Φ − 1 , f − 1 , ( B ◦ (Φ − 1 , id)) − 1 , − 1 √ f ′ ( B · h ◦ (Φ − 1 , id))  . (13) W e summarize these results in the following definition. Definition 3.1. The set of sto chastic tr ansformations forms a gr oup with the fol lowing op er ations: • Comp osition: Given T 1 = (Φ 1 , f 1 , B 1 , h 1 ) and T 2 = (Φ 2 , f 2 , B 2 , h 2 ) , T 2 ◦ T 1 =  Φ 2 ◦ Φ 1 , f 2 ◦ f 1 , ( B 2 ◦ (Φ 1 , id)) · B 1 , h 1 + p f ′ 1 B − 1 1 · ( h 2 ◦ (Φ 1 , id))  . • Inverse: Given T = (Φ , f , B , h ) , T − 1 =  Φ − 1 , f − 1 , ( B ◦ Φ − 1 ) − 1 , − 1 √ f ′ ( B · h ◦ (Φ − 1 , id))  . • Neutr al Element: T 0 = (Id , id , I n , 0) . 3.1.1 Geometric Brownian motion (again) W e previously obtained the GBM via a strong transformation (Example 1.1 ). Here, we sho w it can also b e constructed from a standard Brownian motion via a we ak transformation in volving a change of probabilit y measure. In this example, w e will also illustrate how the group structure described in Definition 3.1 is well-defined in terms of pro cess comp osition and transformed co efficien ts. Let the target GBM on (Ω , F , ( F t ) t , Q ) be d Y t = µY t dt + σ Y t d ¯ W t , (14) and the source standard Brownian motion on (Ω , F , ( F t ) t , P ) be dX t = dW t , (i.e., µ 0 = 0 , σ 0 = 1) . (15) Consider the w eak transformation T = (Φ , I n , 1 , h ) defined by Φ( x ) = e σ x , h = µ σ − σ 2 . Applying Theorem 1.2 , the transformed pro cess ( P T ( X ) , P T ( W )) is given by Y t := Φ( X t ) = e σ X t , ¯ W t := W t − Z t 0 h ds = W t −  µ σ − σ 2  t. The change of measure defining Q is d Q d P    F t = exp  Z t 0 h dW s − 1 2 Z t 0 h 2 ds  = exp   µ σ − σ 2  W t − 1 2  µ σ − σ 2  2 t  . The transformed co efficien ts E T ( µ 0 , σ 0 ) are computed as follows: µ ′ ( x ) := E T ( µ 0 ) =  1 2 D 2 Φ + D Φ · h  ◦ Φ − 1 ( x ) =  1 2 σ 2 e σ ( · ) + σe σ ( · )  µ σ − σ 2  ◦ Φ − 1 ( x ) = µx, σ ′ ( x ) := E T ( σ 0 ) = ( D Φ) ◦ Φ − 1 ( x ) = ( σ e σ ( · ) ) ◦ Φ − 1 ( x ) = σ x. Th us, P T ( X, W ) solv es ( 14 ), confirming that T maps the standard Bro wnian motion to the GBM. Alternativ ely , T can b e decomp osed as T = T 2 ◦ T 1 (see Definition 3.1 ), where: 11 • T 1 = (Id , I , 1 , h ): A pure Girsanov transformation (drift change). • T 2 = (Φ , I , 1 , 0): A pure change of v ariable (Itˆ o transformation). By Theorem 1.2 , T 1 maps ( X, W ) to ( Z, ¯ W ) where Z t = X t and ¯ W t is as defined abov e. The in termediate SDE on (Ω , F , Q ) has co efficien ts: µ 1 = E T 1 ( µ 0 ) = h = µ σ − σ 2 , σ 1 = E T 1 ( σ 0 ) = σ 0 = 1 . Applying T 2 to the result of T 1 , we transform Z t in to Y t = Φ( Z t ) = e σ X t , while the Brownian motion ¯ W remains unchanged (since h 2 = 0). The final co efficients are: µ 2 = E T 2 ( µ 1 ) =  µ 1 D Φ + 1 2 D 2 Φ  ◦ Φ − 1 =  µ σ − σ 2  σ Φ + 1 2 σ 2 Φ  ◦ Φ − 1 = µx, σ 2 = E T 2 ( σ 1 ) = ( D Φ) ◦ Φ − 1 = σ x. This confirms that E T 2 ◦ T 1 = E T 2 ◦ E T 1 and P T 2 ◦ T 1 = P T 2 ◦ P T 1 , yielding the same GBM solution ( 14 ). The pro cess is summarized in the follo wing diagram: dX t = dW t in (Ω , F , ( F t ) t , P ) d Z t = ( µ σ − σ 2 ) dt + d ¯ W t in (Ω , F , ( F t ) t , Q ) d Y t = µY t dt + σ Y t d ¯ W t in (Ω , F , ( F t ) t , Q ) T = T 2 ◦ T 1 T 1 T 2 3.2 Lie algebra structure and infinitesimal transformations The group of sto chastic transformations lacks an ob vious in trinsic differen tiable structure. T o define a Lie algebra, we utilize a geometric approac h: w e identify the stochastic transformations with a closed subgroup of the diffeomorphism group of a sp ecific principal bundle. This allows the sto c hastic group to inherit a natural Lie group structure from the geometry of the bundle. Consider the trivial princip al bund le P = ˜ M × G . Here, the base manifold ˜ M := M × R + represen ts the spacetime domain (with co ordinates ( x, t )), while the structur e gr oup is defined as G = SO( d ) × R + × R n . Conceptually , given a sto chastic transformation T = (Φ , f , B , h ), the fiber G enco des the parameters ( B , f ′ , h ) that act on the driving Brownian motion, while the base ˜ M corresp ond to the comp onen ts (Φ , f ) that act on the solution pro cess X . The bundle is equipp ed with the standard pro jection π M : (( x, t ) , g ) 7→ ( x, t ) and the right action R g 1 (( x, t ) , g ) = (( x, t ) , g ∗ g 1 ). Here ∗ denotes the group op eration in G , which is not fixed a priori but must b e determined to ensure compatibilit y with the composition of sto c hastic transformations. Definition 3.2. Given two (trivial) princip al bund les ˜ M × G and ˜ M ′ × G , an isomorphism b etwe en princip al bund les is a diffe omorphism F : ˜ M × G → ˜ M ′ × G that pr eserves the princip al bund le structur e of ˜ M × G and ˜ M ′ × G . In other wor ds, ther e exists a diffe omorphism Ψ : ˜ M → ˜ M ′ such that π ˜ M ′ ◦ F = Ψ ◦ π ˜ M F ◦ R g = R g ◦ F ∀ g ∈ G. Remark 3.3. A bund le isomorphism F is uniquely determine d by its value at the identity se ction (( x, t ) , e ) , wher e e = (0 , 1 , I d ) is the neutr al element in G . This me ans that if F (( x, t ) , e ) = (Ψ( x, t ) , ˜ g ( x, t )) , then F (( x, t ) , h ) = (Ψ( x, t ) , ˜ g ( x, t ) ∗ h ) for al l h in G . W e define a map from the group of sto c hastic transformations to the group of bundle isomorphisms of ˜ M × G : T = (Φ , f , B , h ) 7− → F T , (16) where, thanks to Remark 3.3 , F T is defined via its action on the iden tity section: F T (( x, t ) , e ) :=  (Φ( x, t ) , f ( t )) | {z } Ψ( x,t ) , ( B ( x, t ) , f ′ ( x, t ) , h ( x, t ))  . 12 This map is well-defined and bijective: every T defines a unique F T , and conv ersely , any isomorphism F defines a unique T by extracting the base diffeomorphism (Φ , f ) and the group comp onen ts ( B , h ) from F (( x, t ) , e ). Since the group of bundle isomorphisms is a closed subgroup of Diff ( ˜ M × G ) (that is, the group of Diffeomorphisms of ˜ M × G ), it is a Lie group by Cartan’s theorem. T o transfer this structure to the group of sto c hastic transformations, ( 16 ) must b e a group homomorphism. Sp ecifically , we require F T 2 ◦ T 1 = F T 2 ◦ F T 1 . This condition determines the group la w ∗ on G . Recalling the composition la w for sto c hastic transformations (Definition 3.1 ), we hav e F T 2 ◦ T 1 (( x, t ) , e ) =  (Φ 2 ◦ Φ 1 , f 2 ◦ f 1 ) , ( B 2 ◦ Φ 1 ) B 1 , ( f ′ 2 ◦ Φ 1 ) f ′ 1 , p f ′ 1 B − 1 1 ( h 2 ◦ Φ 1 ) + h 1  . (17) Con versely , ev aluating the comp osition of isomorphisms F T 2 ( F T 1 (( x, t ) , e )) yields: F T 2  (Φ 1 , f 1 ) , ( B 1 , f ′ 1 , h 1 )  =  (Φ 2 ◦ Φ 1 , f 2 ◦ f 1 ) , ( B 2 ◦ Φ 1 , f ′ 2 ◦ f 1 , h 2 ◦ Φ 1 ) ∗ ( B 1 , f ′ 1 , h 1 )  . (18) Comparing the group comp onents of ( 17 ) and ( 18 ), the op eration ∗ m ust satisfy: ( B 2 , f ′ 2 , h 2 ) ∗ ( B 1 , f ′ 1 , h 1 ) = ( B 2 B 1 , f ′ 2 f ′ 1 , p f ′ 1 B − 1 1 h 2 + h 1 ) . This structure identifies G as a semidirect pro duct. Let K = S O ( d ) × R + with the comp onen t-wise pro duct. W e define the homomorphism ψ : K → Aut( R d ) by ψ ( B ,f ′ ) ( h ) = √ f ′ B − 1 h , where Aut( R d ) denotes the group of automorphism of R d . Then: G ∼ = K ⋉ ψ R d with pro duct law ⋉ ψ defined in a standard wa y as ( k 2 , h 2 ) ⋉ ψ ( k 1 , h 1 ) = ( k 2 k 1 , ψ k 1 ( h 2 ) + h 1 ) =  k 2 k 1 , p f ′ 1 B − 1 1 h 2 + h 1  . With this group la w, the map T 7→ F T is a group isomorphism, preserving inv erses ( F T − 1 = F − 1 T ) and the identit y ( F T 0 = I d ). Theorem 3.4. The gr oup of sto chastic tr ansformations is isomorphic to the gr oup of bund le isomor- phisms of ˜ M × G , wher e ˜ M = M × R + and G = ( S O ( d ) × R + ) ⋉ ψ R d . Conse quently, the gr oup of sto chastic tr ansformations inherits a Lie gr oup structur e. W e can no w define the Lie algebra V d ( M ) associated with this group. Elements of the Lie algebra corresp ond to tangen t v ectors at the identit y of a one-parameter subgroup { T λ } λ , computed as deriv atives at λ = 0: Y ( x, t ) = ∂ λ Φ λ ( x, t )   λ =0 , m ( t ) = ∂ λ f λ ( t )   λ =0 , C ( x, t ) = ∂ λ B λ ( x, t )   λ =0 , H ( x, t ) = ∂ λ h λ ( x, t )   λ =0 . (19) Definition 3.5. An infinitesimal sto c hastic transformation is an element V = ( Y , m, C, H ) ∈ V d ( M ) , wher e Y = n P i =1 Y i ( x, t ) ∂ x i is a ve ctor field on M , m = m ( t ) is a function on R + , C : M × R + → so ( d ) is an antysimmetric matrix, and H : M × R + → R d . If V = ( Y , 0 , 0 , 0) , it is c al le d a strong infinitesimal tr ansformation. Finally , given an elemen t of the Lie algebra, w e can recov er the corresp onding finite transformation. Theorem 3.6 (Reconstruction of the flow) . L et V = ( Y , C , τ , H ) b e an infinitesimal sto chastic tr ansfor- mation. The c orr esp onding one-p ar ameter gr oup T λ is determine d by the unique solution to the system:          ∂ λ Φ λ ( x, t ) = Y (Φ λ ( x, t ) , t ) , Φ 0 = Id R n , ∂ λ f λ ( t ) = m ( f λ ( t )) , f 0 = id R , ∂ λ B λ ( x, t ) = C (Φ λ ( x, t ) , t ) B λ ( x, t ) , B 0 = I d , ∂ λ h λ ( x, t ) = p f ′ λ ( t ) B − 1 λ ( x, t ) H (Φ λ ( x, t ) , t ) , h 0 = 0 . (20) Pr o of. See [ 11 ]. ■ Remark 3.7. F ol lowing the usual one-p ar ameter gr oup notations, we wil l denote by T − λ the inverse of the tr ansformation T λ , that is, T − λ = T − 1 λ . In the same way, Φ − λ ≡ Φ − 1 λ , f − λ ≡ f − 1 λ , B − λ ≡ B − 1 λ and h − λ = h − 1 λ . 13 3.2.1 Time inv ariance of Bro wnian motion Consider a n − dimensional Bro wnian motion dX t = dW t , with X = ( X 1 , ..., X n ) T , W = ( W 1 , ..., W n ) T , drift µ 0 := 0 and diffusion co efficien t σ 0 := I n . Let us apply to this SDE the infinitesimal sto c hastic transformation V = ( Y , m, C , H ), defined by: Y = n X i =1 x i ∂ x i , m ( t ) = 2 t, C = 0 n × n , H = 0 n . W e aim to reco ver the corresp onding one-parameter group of finite transformations. By the flo w recon- struction theorem (Theorem 3.6 ), the comp onents of T λ satisfy: ∂ λ Φ λ ( x, t ) = Φ λ ( x, t ) , ∂ λ f λ ( t ) = 2 f λ ( t ) , ∂ λ B λ = 0 , ∂ λ h λ = 0 . The last t wo equations imply that B λ and h λ are independent of λ . The first tw o equations can be solv ed b y separation of v ariables: d Φ λ Φ λ = dλ, d f λ f λ = 2 dλ. Imp osing the initial conditions Φ 0 = Id , f 0 = id , B 0 = I n , h 0 = 0 n , we obtain the group T λ = (Φ λ , f λ , B λ , h λ ): Φ λ ( x ) = e λ x, f λ ( t ) = e 2 λ t, B λ = I n , h λ = 0 n with λ ranging in R . Setting a = e λ (with a > 0), the transformation T := T ln( a ) tak es the form: Φ( x ) = ax, f ( t ) = a 2 t, B = I n , h = 0 n . T o v erify the action of T on the original Bro wnian motion SDE, w e apply Theorem 1.2 . Since f ′ ( t ) = ∂ t f ( t ) and f − 1 ( t ) = t a 2 , the transformed pro cess is given by the pair: P T ( X ) t = Φ( X f − 1 ( t ) ) = aX t/a 2 , P T ( W ) t = Z f − 1 ( t ) 0 p f ′ ( s ) dW s = Z t/a 2 0 a dW s = aW t/a 2 . On the other hand, the transformed co efficien ts are E T ( µ 0 ) = h 1 f ′  L t (Φ) + D (Φ) σ h  i ◦ (Φ − 1 , f − 1 ) = h 1 a 2  0 + 0  i = 0 n = µ 0 , E T ( σ 0 ) =  1 √ f ′ D (Φ) σB − 1  ◦ (Φ − 1 , f − 1 ) =  1 a aI n · I n  = I n = σ 0 . Since h = 0 n , the probability measure remains unchanged ( d Q /d P = 1). How ev er, the filtration is time-scaled (Theorem 1.2 ): F ′ t = F f − 1 ( t ) = F t/a 2 . Since E T ( µ 0 ) = µ 0 and E T ( σ 0 ) = σ 0 , T is a w eak symmetry according to Definition 2.1 . The transformed pro cess P T ( X, W ) satisfies the original Brownian motion SDE with resp ect to the transformed filtration ( F t/a 2 ) t . This recov ers the well-kno wn self-similarit y (or time inv ariance) prop ert y of Bro wnian motion (see Prop osition 3.2 in [ 3 ]): if W t is a F t − Bro wnian motion, then aW t a 2 is a F t a 2 − Bro wnian motion. 3.3 Infinitesimal symmetries Definition 3.8. A sto chastic infinitesimal tr ansformation V gener ating a one-p ar ameter gr oup T λ is c al le d an infinitesimal symmetry—r esp e ctively, str ong, we ak, or G -we ak—for S D E µ,σ if T λ is a finite symmetry of the c orr esp onding typ e for S D E µ,σ . The next theorem pro vides a useful c haracterization of infinitesimal symmetries, yielding the in- finitesimal determining e quations . These constitute the infinitesimal coun terparts of the equations in Theorem 2.2 , but are far simpler to solve and therefore form an effective to ol for the explicit computa- tion of symmetries. 14 Theorem 3.9. L et V = ( Y , 0 , 0 , 0) b e a str ong infinitesimal tr ansformation. Then V is a str ong infinites- imal symmetry if and only if V gener ates a one-p ar ameter gr oup of tr ansformations and the fol lowing c onditions hold: Y ( µ ) − L t ( Y ) = 0 , [ Y , σ ] = 0 , wher e [ Y , σ ] iα = Y ( σ iα ) − n X k =1 ∂ k ( Y i ) σ k α = n X k =1  Y k ∂ k ( σ i α ) − ∂ k ( Y i ) σ k α  . L et V = ( Y , m, C, H ) b e a we ak infinitesimal tr ansformation. Then V is a we ak infinitesimal sym- metry of S D E µ,σ if and only if V gener ates a one-p ar ameter gr oup of tr ansformations and the fol lowing c onditions hold: Y ( µ ) + m ( t ) ∂ t µ − L t ( Y ) − σ H + m ′ ( t ) µ = 0 , [ Y , σ ] + 1 2 m ′ ( t ) σ + σ C = 0 . (21) L et V = ( Y , m, C , H ) b e a we ak infinitesimal tr ansformation. Then V is a G -we ak infinitesimal symmetry of S D E µ,σ if and only if V gener ates a one-p ar ameter gr oup of tr ansformations and the fol lowing determining e quations hold: Y ( µ ) + m ( t ) ∂ t µ − L t ( Y ) − σ H + m ′ ( t ) µ = 0 , [ Y , σ σ T ] + τ σ σ T = 0 , (22) wher e [ Y , σ σ T ] = Y ( σσ T ) − D ( Y ) σ σ T − σσ T D ( Y ) T . Pr o of. See [ 15 ]. The main idea consists of differentiating the corresp onding finite determining equations with resp ect to the flow parameter. ■ No w that w e hav e established a Lie algebra structure, our strategy to study the in v ariance prop erties of a given diffusion is the following: instead of solving the finite determining equations given in Theorem 2.2 , whic h are generally difficult to handle, w e first solve the infinitesimal determining equations pro vided in Theorem 3.9 , and then apply Theorem 3.6 to reco ver the corresp onding finite symmetries. 3.4 Symmetries of Brownian Motion W e aim to reco ver the symmetries of an n -dimensional Brownian motion describ ed b y the SDE d X t = d W t , where X = ( X 1 , . . . , X n ) T and W = ( W 1 , . . . , W n ) T , with drift µ = 0 and diffusion co efficient σ = I n . Our approac h pro ceeds in tw o steps: first, we solv e the infinitesimal determining equations to iden tify the infinitesimal symmetries; second, we reconstruct the asso ciated flow to deriv e the corresponding group of finite symmetries. According to Theorem 3.9 , a w eak infinitesimal symmetry is defined b y the quadruple V = ( Y , m, C, H ), where: Y = n X i =1 Y i ( x , t ) ∂ x i , m = m ( t ) , C = [ c ij ( x , t )] i,j =1 ,...,n , H =    H 1 ( x , t ) . . . H n ( x , t )    . The infinitesimal generator must satisfy the determining equations ( 21 ). Given µ = 0 and σ = I n , the first determining equation, L t ( Y i ) − Y ( µ i ) − m ( t ) ∂ t ( µ i ) + ( σ H ) i − m ′ ( t ) µ i = 0, reduces to: H i ( x , t ) = − ∂ t Y i ( x , t ) − 1 2 ∆ Y i ( x , t ) . (23) Since for every k, i, α we ha ve ∂ k σ iα = 0, the second determining equation, [ Y , σ ] + 1 2 m ′ ( t ) σ + σ C = 0, simplifies to − D ( Y ) + 1 2 m ′ ( t ) I n + C = 0. In component form, this is expressed as: − ∂ Y i ∂ x j + 1 2 m ′ ( t ) δ ij + c ij = 0 . (24) 15 Com bining ( 23 ) and ( 24 ), w e obtain the complete system of determining equations:              H i ( x , t ) = − ∂ t Y i ( x , t ) − 1 2 ∆ Y i ( x , t ) , ∀ i = 1 , . . . , n, ∂ x i Y i ( x , t ) = 1 2 m ′ ( t ) , ∀ i = 1 , . . . , n, ∂ x j Y i ( x , t ) = c ij ( x , t ) , ∀ i  = j. (25) This system depends on tw o sets of free time-dep enden t parameters, m ( t ) and C ( t ), which generate tw o indep enden t families of infinitesimal symmetries. First F amily ( V α ) By setting C = 0 and m = α ( t ), where α ( t ) is an arbitrary smo oth function, we obtain the infinitesimal symmetry: V α = n X i =1 1 2 α ′ ( t ) x i ∂ x i , α ( t ) , 0 , − 1 2 α ′′ ( t ) x ! . (26) Second F amily ( V β ) If we set m = 0, the determining system ( 25 ) implies that the spatial part of the symmetry is go verned by the an tisymmetric matrix C ( t ) = ( c ij ( t )) ∈ so ( n ). Solving the equations 1 , w e find: V β =    n X i,j =1 C β ( t ) i,j x j ∂ x i  , 0 , C β ( t ) , − C ′ β ( t ) x   , (27) where C β ( t ) is a smo oth antisymmetric matrix parametrized by the v ector of functions β ( t ). In the one-dimensional case ( n = 1), rotations are trivial ( C = 0), and the spatial part Y b ecomes an arbitrary function of time β ( t ). Thus, V β reduces to: V β = ( β ( t ) ∂ x , 0 , 0 , − β ′ ( t )) . (28) In the t wo-dimensional case ( n = 2), V β explicitly accounts for rotations: V β =  β ( t )( x 2 ∂ x 1 − x 1 ∂ x 2 ) , 0 ,  0 β ( t ) − β ( t ) 0  ,  − x 2 β ′ ( t ) x 1 β ′ ( t )  . (29) W e no w pro ceed to reconstruct the flo w and reco v er the corresponding finite symmetries. While the study of T λ for V α is conducted in arbitrary dimension n , for V β w e fo cus on n = 1 and n = 2 to illustrate the distinction b et w een translational and rotational inv ariance. The results for n ≥ 3 generalize directly from the t wo-dimensional case. The family V α and time in v ariance Consider an n -dimensional Brownian motion d X t = d W t , where X = ( X 1 , . . . , X n ) T and W = ( W 1 , . . . , W n ) T . Consider the family of infinitesimal symmetries V α computed in ( 26 ) V α = n X i =1 1 2 α ′ ( t ) x i ∂ x i , α ( t ) , 0 , − 1 2 α ′′ ( t ) x ! . This family is related to the time-in v ariance prop erties of Brownian motion. It is w ell known (see [ 19 ]) that if W t is a Brownian motion on a filtered probability space (Ω , F , ( F t ) , P ) and f ( t ) = R t 0 f ′ ( s ) ds is a time change, then the process ¯ W t := R t 0 p f ′ ( f − 1 ( s )) d W s is a (Ω , F , ( F f − 1 ( t ) ) , P )-Brownian motion. The family V α generalizes this by considering, instead of ¯ W , the process ˜ W t := p f ′ ( f − 1 ( t )) W f − 1 ( t ) . While 1 Recall that since C is by construction antisymmetric then c ij = − c j i ∀ i  = j and c ii = 0 ∀ i . 16 ˜ W is not a P -Bro wnian motion, Itˆ o’s calculus and Girsano v’s theorem show it is a (Ω , F , ( F f − 1 ( t ) ) , Q )- Bro wnian motion, where d Q /d P is the Dol´ eans-Dade exponential of the drift h := f ′′ ( f − 1 ( t )) 2[ f ′ ( f − 1 ( t ))] 3 / 2 , since: d ˜ W t = p f ′ ( f − 1 ( t )) d W f − 1 ( t ) | {z } d ¯ W t − f ′′ ( f − 1 ( t )) 2[ f ′ ( f − 1 ( t ))] 3 / 2 W f − 1 ( t ) dt | {z } h dt . According to the flow reconstruction (Theorem 3.6 ), V α generates a one-parameter group of symmetries T λ = (Φ λ , f λ , B λ , h λ ) satisfying: ∂ λ f λ ( t ) = α ( f λ ( t )) , ∂ λ Φ λ = 1 2 α ′ ( f λ ( t ))Φ λ , ∂ λ B λ = 0 , ∂ λ h λ = − 1 2 q f ′ λ α ′′ ( f λ ( t ))Φ λ . In tegrating with resp ect to λ with initial conditions Φ 0 ( x , t ) = x , f 0 ( t ) = t , B 0 ( x , t ) = I n and h 0 ( x , t ) = 0 , and applying the chain rule α ′ ( f λ ( t )) = ∂ λ α ( f λ ( t )) α ( f λ ( t )) , we obtain: Φ λ ( x , t ) = x p | α ( f λ ( t )) | , f ′ λ ( t ) = | α ( f λ ( t )) | , B λ = I n h λ ( x , t ) = − 1 2 x Z α ′′ ( f λ ( t )) | α ( f λ ( t )) | dλ = − 1 2 x sgn( α ( f λ ( t ))) α ′ ( f λ ( t )) . By Theorem 1.2 , the transformed pair ( P T λ ( X ) , P T λ ( W )) solves the same SDE as ( X , W ). Sp ecifically , the transformed pro cesses are: P T λ ( X t ) = Φ λ ( X f − λ ( t ) , f − λ ( t )) = p | α ( t ) | X f − λ ( t ) , P T λ ( W t ) = Z f − λ ( t ) 0 q f ′ λ ( t )  d W t − h ( X s , s ) ds  = Z t 0 p | α ( s ) |  d W f − λ ( s ) + 1 2 X f − λ ( s ) sgn( α ( s )) α ′ ( s ) | α ( s ) | ds  . Consisten t with the original equation d X t = d W t , Itˆ o’s calculus and Girsanov’s theorem confirm that P T λ ( X , W ) solves the SDE under the measure Q λ , since: d  p | α ( t ) | X f − λ ( t )  = p | α ( t ) | d W f − λ ( t ) | {z } d ¯ W t −      − 1 2 p | α ( t ) | X f − λ ( t ) sgn( α ( t )) α ′ ( t ) | {z } ˜ h dt      . One-dimensional family V β and Girsano v in v ariance Consider a one-dimensional Brownian motion dX t = dW t and the family of infinitesimal symmetries V β computed in ( 28 ): V β = ( β ( t ) ∂ x , 0 , 0 , − β ′ ( t )) . The v ector field V β enco des the in v ariance prop erties of Brownian motion described by Girsanov’s theo- rem. Sp ecifically , if W is a P -Brownian motion, then W t − R t 0 h ( X s , s ) ds is a Q -Brownian motion under the measure c hange defined by the Dol ´ eans-Dade exp onen tial of h . By reconstructing the flo w (Theorem 3.6 ), V β generates a one-parameter group of finite symmetries T λ = (Φ λ , f λ , B λ , h λ ) satisfying: d Φ λ dλ = β ( t ) , d f λ dλ = 0 , dB λ dλ = 0 , dh λ dλ = − β ′ ( f λ ( t )) , with iden tity initial conditions at λ = 0. The resulting finite symmetry T λ consists of the spatial diffeomorphism Φ λ ( x, t ) = x + λβ ( t ) and a Girsanov drift transformation h λ ( x, t ) = − λβ ′ ( t ). Since V β is a symmetry , the transformed process satisfies the original SDE under the new measure Q λ . According to Theorem 1.2 , the transformed processes are: P T λ ( X t ) = Φ λ ( X t , t ) = X t + λβ ( t ) , P T λ ( W t ) = W t − Z t 0 h λ ( X s , s ) ds = W t + λ Z t 0 β ′ ( s ) ds = W t + λβ ( t ) . Consisten t with Itˆ o calculus and the SDE dX t = dW t , it follo ws that d ( X t + λβ ( t )) = d ( W t + λβ ( t )), whic h is a Q λ -Bro wnian motion by Girsanov’s theorem. 17 Tw o-dimensional family V β and rotational in v ariance Consider a tw o-dimensional Brownian motion d X t = d W t , where X = ( X 1 , X 2 ) T and W = ( W 1 , W 2 ) T . Consider the t wo-dimensional family of infinitesimal symmetries V β computed in ( 29 ) V β =  β ( t )( x 2 ∂ x 1 − x 1 ∂ x 2 ) , 0 ,  0 β ( t ) − β ( t ) 0  ,  − x 2 β ′ ( t ) x 1 β ′ ( t )  . This family generalizes the well-kno wn rotational inv ariance of Brownian motion. By L´ evy’s character- ization, if W t is a P -Brownian motion and B ( t ) is a rotation matrix, then ¯ W t = R t 0 B ( s ) d W s is still a P -Bro wnian motion. V β extends this property b y considering the directly rotated pro cess ˜ W t := B ( t ) W t . Although ˜ W is not a Brownian motion under P , Girsano v’s theorem ensures it is one under a new mea- sure Q with Radon-Nikodym deriv ative given by the Dol´ eans-Dade exp onen tial of h ( x , t ) = − B ′ ( t ) x , where B ′ ( t )is the matrix whose entries are the time deriv ativ es of the en tries of B ( t ) . F rom the reconstruction of the flow (Theorem 3.6 ), V β generates a one-parameter group of finite symmetries T λ = (Φ λ , f λ , B λ , h λ ) defined b y: Φ λ ( X t , t ) =  cos( λβ ( t )) X 1 t + sin( λβ ( t )) X 2 t − sin( λβ ( t )) X 1 t + cos( λβ ( t )) X 2 t  , f λ ( t ) = t, B λ ( t ) =  cos( λβ ( t )) sin( λβ ( t )) − sin( λβ ( t )) cos( λβ ( t ))  , h λ ( X t , t ) =  − λβ ′ ( t ) X 2 t λβ ′ ( t ) X 1 t  . Since V β is a symmetry , the transformed pro cess satisfies the original SDE under the measure Q λ . According to Theorem 1.2 , the transformed pro cesses are: P T λ ( X t ) = Φ λ ( X t , t ) , P T λ ( W t ) = Z t 0 B λ ( s ) ( d W s − h λ ( X s , s ) ds ) . Recalling the original equation d X t = d W t , it is straigh tforward to v erify that T λ acts as a symmetry . Applying Itˆ o’s formula reveals that the transformed pro cess acquires a drift term, which can b e remov ed via Girsanov’s theorem to recov er the original Brownian dynamics under a new measure Q λ . Explicitly , w e ha ve: d  cos( λβ ( t )) X 1 t + sin( λβ ( t )) X 2 t − sin( λβ ( t )) X 1 t + cos( λβ ( t )) X 2 t  =  cos( λβ ( t )) sin( λβ ( t )) − sin( λβ ( t )) cos( λβ ( t ))   dW 1 t dW 2 t  | {z } d ¯ W t −  − λβ ′ ( t ) sin( λβ ( t )) X 1 t + λβ ′ ( t ) cos( λβ ( t )) X 2 t − λβ ′ ( t ) cos( λβ ( t )) X 1 t − λβ ′ ( t ) sin( λβ ( t )) X 2 t  | {z } ˜ h dt. 4 Lie symmetries and in tegration b y parts form ulas In [ 13 ], the study of inv ariance prop erties for diffusions in R n via infinitesimal transformations led to the deriv ation of explicit, closed-form integration b y parts formulas. This approach, inspired by the Bism ut metho d in Mallia vin calculus, w as initially developed for symmetries that did not account for rotations. In [ 15 ], we extended this framework to include rotational transformations, proving the rotational inv ari- ance of the resulting integration by parts formula—a prop ert y deeply linked to the isotropy of Brownian motion. While the formula’s structure remains inv ariant under rotations, the regularity assumptions required for its deriv ation undergo subtle changes. F urthermore, [ 15 ] introduced the notion of G -weak symmetry as the m ost general class of transformations for which this theorem holds, as it relies fundamentally on the in v ariance of the la w of the solution pro cess under the action of the symmetry group. T o ensure the v alidit y of the theorem, we assume the following integrabilit y condition: Hyp othesis A. L et X t b e a solution to S D E µ,σ with deterministic initial c onditions. Each c omp o- nent of the fol lowing ve ctors or matric es: C H ( X t , t ) , H ( X t , t ) , Y ( H )( X t , t ) , L ( Y )( X t , t ) , Σ( Y )( X t , t ) , L ( Y ( Y i ))( X t , t ) , Σ( Y ( Y i ))( X t , t ) is in L 2 (Ω) for i = 1 , . . . , n and for al l t ∈ [0 , T ] . Her e by Σ we me an the ve ctor-value d differ ential op er ator define d as Σ = σ T · ∇ , that is, Σ α = n X j =1 σ j α ∂ x j , α = 1 , .., d. 18 Theorem 4.1 (Integration b y parts formula) . L et ( X , W ) b e a solution to S D E µ,σ and let V = ( Y , m, C, H ) b e an infinitesimal sto chastic symmetry for the system. Under Hyp othesis A, the fol low- ing identity holds for every t ∈ [0 , T ] and for any b ounde d functional F with b ounde d first and se c ond derivatives: − m ( t ) E P [ L ( F ( X t ))] + E P  F ( X t ) Z t 0 H ( X s , s ) dW s  + E P [ Y ( F ( X t ))] − E P [ Y ( F ( X 0 ))] = 0 . (30) Pr o of. Let T λ = (Φ λ , f λ , B λ , h λ ) b e the one-parameter group generated by V . W e provide here a sketc h of the argumen t; the complete pro of is a v ailable in [ 15 ]. • Since V is an infinitesimal symmetry , the generated flo w T λ is a finite sto chastic symmetry that preserv es the law of the solution process. Thus, for an y t ∈ [0 , T ]: E P [ F ( X t )] = E Q λ [ F ( P T λ ( X t ))] , where Q λ is the probability measure obtained via Girsano v’s theorem under the transformation T λ . • Applying the Itˆ o form ula and utilizing the martingale prop ert y of the stochastic integral with resp ect to the Q λ -Bro wnian motion P T λ ( W ), we obtain: E P [ F ( X t )] = E Q λ  Z t 0 L ( F (Φ λ ( X f − λ ( s ) ))) ds  . • Changing v ariables s = f λ ( u ) and applying the Radon-Nikodym deriv ative d Q λ d P yields: E P [ F ( X t )] = E P " d Q λ d P     F t Z f − λ ( t ) 0 L ( F (Φ λ ( X s , s ))) f ′ λ ( s ) ds # . • The result follows b y differentiating both sides with resp ect to the flow parameter λ and ev aluating at λ = 0. Since the left-hand side is independent of λ , its deriv ativ e v anishes. Applying the Leibniz rule on the right-hand side and mo ving the deriv ative inside the expectation (justified b y the integrabilit y conditions in Hypothesis A) concludes the proof. ■ Remark 4.2. While the analytic al c onditions in Hyp othesis A may app e ar stringent, they ar e often verifiable in pr actic e using Lyapunov the ory to ensur e the r e quir e d inte gr ability. In the sp e cific c ase of Br ownian motion in R n , these c onditions ar e natur al ly satisfie d, as the moments of Gaussian r andom variables ar e finite for al l or ders. A detaile d discussion on these r e gularity r e quir ements is pr ovide d in [ 13 , 15 ]. 5 In tegration b y parts form ulas and Stein iden tities W e no w apply our in tegration b y parts theorem to the symmetries of Brownian motion derived in Section 3.4 . As w e shall see, this result generalizes sev eral well-kno wn identities in probability theory , most no- tably the Stein identities. F or the sake of conciseness, we fo cus on one-dimensional and tw o-dimensional Bro wnian motion—the latter b eing the first instance in volving rotations—noting that the theorem ex- tends naturally to general diffusions in R n . W e omit the verification of the regularity assumptions, whic h can b e found in [ 15 ]. Consider a one-dimensional Bro wnian motion dX t = dW t in R and the family of symmetries V β computed in ( 28 ): V β = ( β ( t ) ∂ x , 0 , 0 , − β ′ ( t )) . Theorem 4.1 ensures that for an y time-dep enden t function β ( t ) and an y bounded functional F ∈ C 2 b ( R n ), we hav e: E P  F ( W t ) Z t 0 − β ′ ( s ) dW s  + E P [ β ( t ) F ′ ( W t )] − E P [ β (0) F ′ ( W 0 )] = 0 . (31) 19 Setting β ( t ) = t , identit y ( 31 ) reduces to: E P [ W t F ( W t )] = t E P [ F ′ ( W t )] . (32) Equation ( 32 ) is precisely the Stein iden tity for a Brownian motion at fixed time t . Recall that the well- kno wn Stein’s Lemma c haracterizes a random v ariable Z ∼ N ( µ, σ 2 ) b y the iden tity E [( Z − µ ) F ( Z )] = σ 2 E [ F ′ ( Z )]; since W t ∼ N (0 , t ), the corresp ondence is exact. The result in Theorem 4.1 is closely related to the Bismut approac h to Malliavin calculus. In one dimension, the family V β reflects the inv ariance of Bro wnian motion under a Girsano v-type change of measure. Sp ecifically , considering a constan t drift h λ = λ , w e kno w that if ( W t ) t is a P -Brownian motion, then ( W t − λt ) t is a Q λ -Bro wnian motion with Radon-Nik o dym deriv ative: d Q λ d P     F t = e λW t − 1 2 λ 2 t . Th us, for any sufficiently smo oth F : E P [ F ( W t )] = E Q λ [ F ( W t − λt )] = E P h e λW t − 1 2 λ 2 t F ( W t − λt ) i . If w e now differentiate b oth sides with resp ect to λ and ev aluate the result at λ = 0, we obtain exactly the Stein iden tity for 1-dimensional Brownian motion ( 32 ). Stein’s identit y is not the unique classical result recov erable from ( 31 ) through a suitable c hoice of β . F or instance, by setting β ( t ) = t and c ho osing F ( x ) ≡ 1 or F ( x ) = x , identit y ( 31 ) yields, resp ectiv ely: E P [ W t ] = 0 , E P [ W 2 t ] = t. Alternativ ely , if w e c ho ose a constan t β ( t ) ≡ 1 in ( 31 ), we find that for any sufficiently smo oth G = F ′ : E P [ G ( W t )] = E P [ G ( W 0 )] , reflecting the constancy of the exp ectation of Brownian motion, a prop ert y deeply tied to its martingale nature. F urthermore, by selecting the indicator function β ( u ) = min ( u, s ) for a fixed s < t and choosing F ( x ) = x , equation ( 31 ) specializes into the well-kno wn cov ariance prop ert y (see [ 3 ]): E P [ W t W s ] = min( t, s ) . These examples demonstrate that our integration by parts theorem effectiv ely encodes the fundamental in v ariance properties of the underlying diffusion. Consider now a tw o-dimensional Brownian motion d W t = ( dW 1 t , dW 2 t ) T and the family of symmetries V β asso ciated with rotational inv ariance, as derived in ( 29 ): V β =  β ( t ) y ∂ x − β ( t ) x∂ y , 0 ,  0 β ( t ) − β ( t ) 0  ,  − y β ′ ( t ) xβ ′ ( t )  . Theorem 4.1 ensures that for every arbitrary time-dep endent function β and for every functional F ∈ C 2 b ( R 2 ), it holds: E P  F ( W 1 t , W 2 t ) Z t 0 ( − W 2 s dW 1 s + W 1 s dW 2 s ) β ′ ( s )  + E P  β ( t )  W 2 t ∂ x F − W 1 t ∂ y F  = 0 . (33) By choosing β ( t ) = t , w e obtain: E P  F ( W 1 t , W 2 t ) Z t 0 ( W 1 s dW 2 s − W 2 s dW 1 s )  = t E P  W 1 t ∂ y F ( W 1 t , W 2 t ) − W 2 t ∂ x F ( W 1 t , W 2 t )  . (34) The sto chastic integral on the left-hand side is precisely the L´ evy sto c hastic area. This identit y pro vides a characterization of the L ´ evy area through the lens of rotational symmetries; for instance, c ho osing 20 F ≡ 1 immediately recov ers its zero-mean prop erty . Alternativ ely , if w e c ho ose β ≡ 1, iden tity ( 33 ) specializes into: E P [ W 2 t ∂ x F ( W 1 t , W 2 t )] = E P [ W 1 t ∂ y F ( W 1 t , W 2 t )] , whic h is a direct consequence of Isserlis’ theorem applied to the Gaussian v ector ( W 1 t , W 2 t ). F or the sake of completeness, we examine the application of the theorem to the family V α , which is asso ciated with time-inv ariance. F o cusing on the one-dimensional case dX t = dW t , consider the symme- try vector field computed in ( 26 ) : V α =  1 2 α ′ ( t ) x∂ x , α ( t ) , 0 , − 1 2 α ′′ ( t ) x  . Theorem 4.1 giv es: − α ( t ) E P  1 2 F ′′ ( W t )  + E P  F ( W t ) Z t 0 − 1 2 α ′′ ( s ) W s dW s  + E P  1 2 α ′ ( t ) W t F ′ ( W t )  = 0 . (35) Cho osing α ( t ) = t yields: t E P [ F ′′ ( W t )] = E P [ W t F ′ ( W t )] , (36) while choosing α ( t ) = t 2 yields: t 2 E P [ F ′′ ( W t )] = − 2 E P  F ( W t ) Z t 0 W s dW s  + 2 t E P [ W t F ′ ( W t )] . Substituting the sto c hastic integral R t 0 W s dW s = 1 2 ( W 2 t − t ) and recalling ( 36 ), we obtain: t 2 E P [ F ′′ ( W t )] = E P [ F ( W t )( W 2 t − t )] . (37) Both ( 36 ) and ( 37 ) represent classical iden tities. In particular, for t = 1, we recov er the standard Gaus- sian integration by parts formulas for Z ∼ N (0 , 1): E [ Z F ′ ] = E [ F ′′ ] and E [ F ( Z 2 − 1)] = E [ F ′′ ]. In summary , the classical identities of Gaussian analysis emerge here as unified manifestations of the underlying symmetry groups of the Wiener pro cess. 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