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 ||x||2 Take f (x) = e 2 in the change of variables formula and write φ(x) = x + K(x). Then we obtain Z Z − 1 ||y||2
− 1 |x|2 So, denoting by γ the Gaussian measure dγ = e 2 dx, we have Z
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) ∈ 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 < ∞.
Equip H with the inner product
Z T 0 0 < h, k >≡ ht kt dt, h, k ∈ H. 0
Then P(w(t1),..., w(tn) ∈ B) = Z 1 − 1 ||h(x)||2 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
{σ : [0, T ] 7→ R, σ0 = 0}.
We define the Wiener measure γ on C0 to be the law of Brownian motion w. i.e. γ(S) = P(w ∈ 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 ∈ L(H, Rn) be surjective. Then P induces an inner n n product < ., . >P on R : for x, y ∈ R , < x, y >P ≡< x˜, y˜ >H , where P(˜x) = x, P(˜y) = y, x˜, y˜ ∈ (ker P)⊥.
For any Borel set B ⊂ Rn we define
Z ||x||2 −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 P∈ L(H, Rn), B ⊂ Rn, Q ∈ L(H, Rm), C ⊂ Rm, then µ˜(P−1(B)) =µ ˜(Q−1(C)).
Thusµ ˜ defines a ”measure” on the ring of cylinder sets {P−1(B)} in H.
µ˜ is finitely but not countably additive (in the case dim H = ∞).
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 ∈ 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 ||K(x)||2 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 ∈ 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 < ∞ . 0 Z T 0 Lh = ht dw, h ∈ 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 = {K K : E 7→ E ∗ is C 1}
(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 ∈ 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 ∈ 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 Itˆointegral
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.
Denis Bell Change of variables in infinite-dimensional space