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Change of Variables in Infinite-Dimensional Space

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Change of variables in infinite-dimensional space Denis Bell Denis Bell Change of variables in infinite-dimensional space Change of variables formula in R Let φ be a continuously differentiable function on [a; b] and f an integrable function. Then Z φ(b) Z b f (y)dy = f (φ(x))φ0(x)dx: φ(a) a Denis Bell Change of variables in infinite-dimensional space Higher dimensional version Let φ : B 7! φ(B) be a diffeomorphism between open subsets of Rn. Then the change of variables theorem reads Z Z f (y)dy = (f ◦ φ)(x)Jφ(x)dx: φ(B) B where @φj Jφ = det @xi is the Jacobian of the transformation φ. Denis Bell Change of variables in infinite-dimensional space Set f ≡ 1 in the change of variables formula. Then Z Z dy = Jφ(x)dx φ(B) B i.e. Z λ(φ(B)) = Jφ(x)dx: B where λ denotes the Lebesgue measure (volume). Denis Bell Change of variables in infinite-dimensional space Version for Gaussian measure in Rn − 1 jjxjj2 Take f (x) = e 2 in the change of variables formula and write φ(x) = x + K(x). Then we obtain Z Z − 1 jjyjj2 <K(x);x>− 1 jK(x)j2 − 1 jjxjj2 e 2 dy = e 2 Jφ(x)e 2 dx: φ(B) B − 1 jxj2 So, denoting by γ the Gaussian measure dγ = e 2 dx, we have Z <K(x);x>− 1 jjKx)jj2 γ(φ(B)) = e 2 Jφ(x)dγ: B Denis Bell Change of variables in infinite-dimensional space Infinite-dimesnsions There is no analogue of the Lebesgue measure in an infinite-dimensional vector space. However, there is an analogue of Gaussian measure. Denis Bell Change of variables in infinite-dimensional space Brownian motion x w(t) T Denis Bell Change of variables in infinite-dimensional space Figure : a2 + b2 = c2 The Wiener process The Wiener processes is the mathematical model for Brownian motion. It is a stochastic process w : [0; T ) × Ω 7! R (a random function) with the properties (i) w0 = 0: (ii) The function t 7! wt is (a.s.) continuous. n (iii) For a measurable set B ⊂ R and times 0 < t1 < ··· < tn ≤ T P(w(t1);:::; w(tn) 2 B) = 1 × n=2p (2π) t1(t2 − t1) ::: (tn − tn−1) 2 2 2 Z 1 x1 (x2−x1) (xn−xn−1) − 2 t + (t −t ) +::: (t −tn ) e 1 2 1 n 1 dx1 ::: dxn B Denis Bell Change of variables in infinite-dimensional space Underlying geometric structure: the Cameron-Martin space Consider the space H of continuous, piecewise C 1 functions R T 0 2 h : [0; T ] 7! R with h0 = 0 and energy 0 ht dt < 1: Equip H with the inner product Z T 0 0 < h; k >≡ ht kt dt; h; k 2 H: 0 Then P(w(t1);:::; w(tn) 2 B) = Z 1 − 1 jjh(x)jj2 n p e 2 dx (2π) 2 t1(t2 − t1) ::: (tn − tn−1) B where x = (x1 ::: xn) and h(x) is the polygonal path between the points (0; 0); (t1; x1);::: (tn; xn); (T ; xn). Denis Bell Change of variables in infinite-dimensional space The Wiener measure Denote by C0 the Banach space of continuous paths fσ : [0; T ] 7! R; σ0 = 0g: We define the Wiener measure γ on C0 to be the law of Brownian motion w. i.e. γ(S) = P(w 2 S) on Borel sets S ⊂ C0. Denis Bell Change of variables in infinite-dimensional space Change of variables for Wiener measure: the Girsanov theorem Theorem Let h : [0; T ] 7! R be a continuous adapted path (for any t, ht = f (ws =s ≤ t)). Define φ : C0 7! C0 by Z t φ(w)t = wt + hs ds: 0 Then for any Borel set B ⊂ C0, Z R T h dw− 1 R T h2ds γ(φ(B)) = e 0 s 2 0 s dγ: B Compare with finite-dimensional formula. Note that there is no requirement that the map w 7! h (and hence φ) be differentiable. Denis Bell Change of variables in infinite-dimensional space The Gaussian cylinder set measure on a Hilbert space Let H be an (infinite-dimensional) Hilbert space. Suppose P 2 L(H; Rn) be surjective. Then P induces an inner n n product < :; : >P on R : for x; y 2 R ; < x; y >P ≡< x~; y~ >H , where P(~x) = x; P(~y) = y; x~; y~ 2 (ker P)?. For any Borel set B ⊂ Rn we define Z jjxjj2 −1 1 − P µ~(P (B)) ≡ e 2 dx n=2 (2π) B where x = (x1;::: xn) and dx is the Lebesgue measure corresponding to < :; : >P . Denis Bell Change of variables in infinite-dimensional space The following consistency condition can be shown: If P−1(B) = Q−1(C), where P2 L(H; Rn); B ⊂ Rn; Q 2 L(H; Rm); C ⊂ Rm, then µ~(P−1(B)) =µ ~(Q−1(C)). Thusµ ~ defines a "measure" on the ring of cylinder sets fP−1(B)g in H. µ~ is finitely but not countably additive (in the case dim H = 1). Denis Bell Change of variables in infinite-dimensional space Let i : H 7! E denote an embedding of H into a separable Banach space E. Then there is an induced cylinder-set measureγ ~ on E defined by γ~(P−1(B)) ≡ µ~((P ◦ i)−1(B)); P 2 L(E; Rn); B ⊂ Rn: Theorem If the norm on H is sufficiently stronger than the norm on E, then γ~ is countably additive on the cylinder sets of E and hence extends to a measure γ on E. Denis Bell Change of variables in infinite-dimensional space Abstract Wiener space The previous theorem is due to L. Gross (1967). It generalizes Wiener's original theorem. Gross called this construction an abstract Wiener space. The result is a measure on an arbitrary Banach space E whose finite-dimensional images are Gaussian measures. It is convenient to think of E ∗ as a subset of E via the maps i ∗ i E ∗ 7! H∗ =∼ H 7! E: Denis Bell Change of variables in infinite-dimensional space Change of variables formula in abstract Wiener space The following result is due to H-H.Kuo. Theorem Let B be an open subset of E and φ : B 7! φ(B) ⊂ E be a diffeomorphism of the form φ = I + K; where K is a C 1 map from B to E ∗. Then Z (K(x);x)− 1 jjK(x)jj2 γ(φ(B)) = e 2 H detH Dφ(x)dγ: B Denis Bell Change of variables in infinite-dimensional space Divergence operators Definition Let E be a normed vector space equipped with a measure µ.A linear operator L with domain D (a subset of the set of vector fields on E) is a divergence operator if L satisfies the integration-by-parts formula Z Z DZ fdµ = f · (LZ)dµ E E for a dense class of C 1 real-valued functions f on E, and Z 2 D. Denis Bell Change of variables in infinite-dimensional space Examples of divergence operators In euclidean space: Let E = Rn and µ the Lebesgue measure. Then L is the classical n n divergence operator: for Z = (Z1;:::; Zn): R 7! R , n X @Zi LZ = − @xi i=1 defined on C 1 vector fields. Denis Bell Change of variables in infinite-dimensional space Divergence Theorems Theorem There is a divergence operator L for the Wiener space, with domain D = Z T n 0 2 o h : [0; T ] 7! R adapted, absolutely continuous; hs ds < 1 : 0 Z T 0 Lh = ht dw; h 2 D: 0 (Gaveau &Trauber) Denis Bell Change of variables in infinite-dimensional space Theorem Let µ be a Gaussian measure on E arising from the abstract Wiener space i : H 7! E. There is the divergence operator (LK)(x) = (K(x); x) − traceH DK(x) with domain . D = fK K : E 7! E ∗ is C 1g (L. Gross) Denis Bell Change of variables in infinite-dimensional space Change of measure under the flow of a vector field Let K be a vector field on E. For x 2 E, consider the ODE dα = K(α(t)): dt α(0) = x: The map φt : x 7! α(t) is called the flow of K. Denis Bell Change of variables in infinite-dimensional space Theorem Suppose a measure µ on E admits a divergence operator (L; D). Let K 2 D and let φt denote the associated flow. Then (under appropriate conditions) there is the change of variables formula Z R t − (LK)(φs (x))ds µ(φt (B)) = e 0 dµ. B (B., Comptes Rendus, Paris, 2006) Denis Bell Change of variables in infinite-dimensional space An example In general, it is not possible to compute the flow φt explicitly, but there is a case when it is trivial, i.e. when K is a constant vector in E. Suppose this is the case. Then the differential equation dφ t = K(φ ); φ = x dt t 0 has the solution φt (x) = x + Kt: Denis Bell Change of variables in infinite-dimensional space Suppose we are in the setting of the classical Wiener space. Let K be a Cameron-Martin path. We compute the change of measure formula in this case. Recall that the divergence of K is given by the It^ointegral Z T 0 (LK)(w) = Kudw: 0 Denis Bell Change of variables in infinite-dimensional space Thus Z T 0 (LK)(φs (w)) = Kud(w + sK) 0 Z T Z T 0 0 2 = Kudw + s Ku du: 0 0 Hence we have Z − R t (LK)(φ (w))ds γ(B + tK) = e 0 s dγ B 2 Z − t R T K 0 dw+ t R T K 0 2du = e 0 u 2 0 u dγ: B This formula is known as the Cameron-Martin theorem.
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