Characterizing Brownian Motion Through Integration

by Parts

Elton P. Hsu

Northwestern University, USA

Beijing Normal University August 13, 2004 Characterizing the Gaussian measure The basis of Stein’s method is Stein’s lemma: the equation

𝔼f′(X) = 𝔼Xf(X)

(xf(x) and f′(x) uniformly bounded) characterizes the standard normal distribution. Let μ be the standard Gaussian measure

μ(dx) = \frac{1}{√2π} e^{−x^{2}/2}.

Let

D = \frac{d}{dx}, D^{∗} = −\frac{d}{dx} + x.

Then D^{∗} is the adjoint of D with respect to μ:

⟨Dg, f⟩_{μ} = ⟨g, D^{∗}f⟩_{μ}.

We have 𝔼f′(X) = 𝔼Xf(X) ⟺ 𝔼D^{∗}f(X) = 0.

A proof by differentiation

D = d/dx generates the the translation group: T_{t} : x ↦ x + t.

Let μ^{X} be the law of X (standard Gaussian). Then

\frac{μ^{X+t} (dx)}{μ^{X}(dx)} = e^{tx−t^{2}/2}.

We have

𝔼f(X + t) = ∫_{ℝ} f(x)μ^{X+t} (dx)
= ∫_{ℝ} f(x)e^{tx−t^{2}/2}μ^{X}(dx)
= 𝔼 e^{tX−t^{2}/2}f(X).

Differentiating with respect to t and letting t = 0 we obtain

𝔼f′(X) = 𝔼Xf(X).

Adjoint operator

For any derivation operator D, the adjoint has the form D^{∗} = −D + D^{∗}1. If the law μ of a random variable X has a smooth density p_{X}, then

D^{∗}1(x) = −\frac{d}{dx} ln p_{X}(x).

𝔼f′(X) = 𝔼 Xf(X) can be written as

𝔼Df(X) = 𝔼D^{∗}1(X)f(X) = 𝔼 Xf(X).

It follows that

D^{∗}1(x) = x = −\frac{d}{dx} ln p_{X}(x),

which implies that

p_{X}(x) = \frac{1}{√2π} e^{−x^{2}/2}.

and X has the standard Gaussian distribution.

Stein’s lemma can be proved without assuming the existence of a density function, using Stein’s equation.

Stein’s equation

Φ(x) = \frac{1}{√2π} ∫_{−∞}^{x} e^{−t^{2}/2}dt = ℙ \{Z ≤ x\} .

[Z is standard Gaussian.] Stein’s equation

f′(x) − xf(x) = I_{(−∞,z]}(x) − Φ(z)

can be solved explicitly:

\frac{e^{−x^{2}/2}}{√2π}f(x) = \begin{cases}Φ(x)(1 − Φ(z)), & x ≤ z; \\ Φ(z)(1 − Φ(x)), & x ≥ z. \end{cases}

Both xf(x) and f′(x) are uniformly bounded. Using this f in 𝔼f′(X) = 𝔼 Xf(X) we have immediately

ℙ {X ≤ z} = Φ(z).

General Stein’s equation has the form

f′(x) − xf(x) = h(x) − 𝔼h(Z),

where Z is a standard Gaussian random variable. Special case: h(x) = I_{(−∞,z]}(x).

Infinite dimensional analog

Brownian motion B = {B_{s}, 0 ≤ s ≤ 1} is an infinite dimensional analog of a Gaussian random variable. What is the corresponding Stein’s characterization for Brownian motion B?

If we replace X by a Brownian motion B = {B_{s}, 0 ≤ s ≤ 1} , then the analog of D^{∗}1 is

D_{h}^{∗}1 = ∫_{0}^{1} ḣ_{s} dB_{s}, stochastic integral .

Here h takes from the infinite-dimensional space

ℋ = {h ∈ C[0, 1] : |h|_{ℋ} < ∞} ,

(the Cameron-Martin space). The Cameron-Martin norm is

|h|_{ℋ}^{2} = ∫_{0}^{1} |ḣ_{s}|^{2}ds.

The condition 𝔼f′(X) = 𝔼 Xf(X) becomes

𝔼D_{h}F (B) = 𝔼 [F (X) ∫_{0}^{1} ḣ_{s} dB_{s}] , ∀h ∈ ℋ .

We will explain why this is the case.

A straightforward extension

An obvious extension is to consider the product Gaussian measure on ℝ^{ℤ_{+}} . We can find the analogue of 𝔼f′(X) = 𝔼 Xf(X) by simple calculations.

We consider the set 𝒞 of cylinder functions

F (x) = f(x^{0}, x^{1}, ⋯, x^{n}).

Here x = (x_{0}, x_{1}, x_{2}, ⋯). Consider the gradient

DF (x) = (f_{x_{0}}(x), f_{x_{1}}(x), ⋯).

More conveniently, consider the set of directional derivatives:

D_{l}F (x) = ⟨DF (x), l⟩ = ∞ ; ∑ ; i=0 l_{i}f_{x_{i}}(x).

Each l = (l_{1}, l_{2}, ⋯) is a direction of differentiation. We have

𝔼D_{l}F (X) = 𝔼⟨X, l⟩F (X).

Characterization The equations

𝔼D_{l}F (X) = 𝔼⟨X, l⟩F (X)

for F ∈ 𝒞 , l ∈ l^{2}(ℤ_{+} ) characterize the product Gaussian measure μ on ℝ^{ℤ_{+}} uniquely. This is easy to see. Let

Y_{l} = ⟨X, l⟩ = l_{0}X_{0} + ⋯ + l_{n}X_{n}

and F (x) = f(l_{0}x_{0} + l_{1}x_{1} + ⋯ + l_{n}x_{n}) = f(⟨x, l⟩). Then we have

|l|_{2}^{2} 𝔼f′(Y_{l}) = 𝔼Y_{l}f(Y_{l}).

Hence by the 1D result, Y_{l} ∼ N(0, |l|_{2}^{2}). Easy exercise: If Y_{l} = ⟨X, l⟩ has the law N(0, |l|_{2}^{2}) for all l ∈ l^{2}(ℤ_{+} ), then X = (X_{1}, X_{2}, …) is i.i.d. with N(0, 1).

Isometry by Fourier Analysis Consider the path space P_{o}(ℝ) = C([0, 1], ℝ). (Fourier analysis) Φ : P_{o}(ℝ) → ℝ^{ℤ_{+}} ,

Φ : path B ↦ {X_{n}} , X_{n} = ∫_{0}^{1} e_{n}dB_{t}. Here e_{0} = 1 and

e_{n} = √2 cos πnt, n = 1, 2, …. {e_{n}} is an orthonormal basis for L^{2}[0, 1]. Then

BM(ℝ) B ⟺ i.i.d {X_{n}} with N(0, 1). Inverse Φ^{−1} : ℝ^{ℤ_{+}} → P_{o}(ℝ^{1}),

B = h_{0}X_{0} + ∞ ; ∑ ; n=1 h_{n}X_{n}.

h_{0}(s) = t, h_{n}(s) = ∫_{0}^{s} e_{n}(u) du.

We have an isometry

Φ : (P_{o}(ℝ), ν) → (ℝ^{ℤ_{+}} , μ).

Here ν = μ ∘ Φ^{−1}.

Isometry by Fourier Analysis (continued)

Recall the equation on ℝ^{ℤ_{+}} :

𝔼D_{l}F (X) = 𝔼⟨X, l⟩F (X).

Rewrite this equation on the path space P_{o}(ℝ) using the isometry Φ : P_{o}(ℝ) → ℝ^{ℤ_{+}} , we rewrite the above relation in the path space P_{o}(ℝ). Namely, let

B = Φ^{−1}X and G = F ∘ Φ.

Then

𝔼D_{h}G(B) = 𝔼D̂_{h}^{∗}1(B) G(B),
D_{h} = Φ_{∗}^{−1}D_{l}, D_{h}^{∗}1(B) = ⟨X, l⟩.

We need to calculate D_{h} and D_{h}^{∗}1 on the path space P_{o}(ℝ) (in terms of h and B).

Relation with stochastic integrals Simple calculations show that:

(1) D_{h} is the directional derivative:

D_{h}G(B) = lim_{t→0} \frac{G(B + th) − G(B)}{t} ,

where

h(t) = ∞ ; ∑ ; n=0 l_{n}h_{n}(t) = ∞ ; ∑ ; n=0 l_{n} ∫_{0}^{t} e_{n}(s) ds.

(2) D_{h}^{∗}1 is a stochastic integral, D_{h}^{∗}1 = ∫_{0}^{1} ḣ(s) dB_{s} = ⟨h, B⟩_{ℋ} ,

Summarize: in the path space P_{o}(ℝ), 𝔼D_{h}G(B) = 𝔼⟨h, B⟩_{ℋ} G(B).

Relation with stochastic integrals (continued)

How does the stochastic integral appear?

⟨X, l⟩ = ∞ ; ∑ ; n=0 l_{n}X_{n}

= ∞ ; ∑ ; n=0 l_{n} ∫_{0}^{1} e_{n}(s)dB_{s}

= ∫_{0}^{1} ḣ(s) dB_{s},

where

h(t) = ∞ ; ∑ ; n=0 l_{n} ∫_{0}^{t} e_{n}(s) ds.

Recall that the norm in the Cameron-Martin space ℋ is

⟨h_{1}, h_{2}⟩_{ℋ} = ∫_{0}^{1} ḣ_{1}(s)ḣ_{2}(s) ds.

We can write symbolically

D_{h}^{∗}1(B) = ⟨h, B⟩_{ℋ} = ∫_{0}^{1} ḣ_{s} dB_{s}.

It is clear from here that the natural domain for h is the Cameron-Martin space ℋ .

Another point of view

Cameron-Martin-Maruyama-Girsanov theorem:

\frac{dμ^{B+th}}{dμ^{B}} = exp [t⟨h, B⟩_{ℋ} − \frac{t^{2}}{2}|h|_{ℋ}^{2} ] .

Hence

𝔼 G(B + th) = 𝔼 G(B) exp [t⟨h, B⟩_{ℋ} − \frac{t^{2}}{2}|h|_{ℋ}^{2} ] .

Differentiating with respect to t and then letting t = 0 we have

𝔼D_{h}G(B) = 𝔼⟨h, B⟩_{ℋ} G(B).

This is the integration by parts formula in the path space P_{o}(ℝ). It is valid for all h ∈ ℋ and nice functions G (e.g., cylinder functions).

Note that h does not have to be deterministic. It can be taken to be ℬ_{∗}-adapted, where ℬ_{∗} = {ℬ_{s}, 0 ≤ s ≤ 1} is the standard Borel filtration on the path space P_{o}(ℝ).

Characterization of Euclidean BM

Theorem 1 Let B be a real-valued ℱ_{∗}- semimartingale. It is a Brownian motion if and only if the integration by parts formula

𝔼D_{h}G(B) = 𝔼⟨h, B⟩_{ℋ} G(B)

holds for all ℱ_{∗}-adapted h ∈ ℋ and all cylinder functions G.

First proof. Go back to the product space ℝ^{ℤ_{+}} via the map Φ (Fourier analysis). This proof will not extend to non-flat spaces (manifolds).

This proof uses the product structure and is not good for nonflat path space, i.e., Brownian motion on a Riemannian manifolds.

Second proof We use Lévy’s criterion: B is a Brownian motion if and only if

(1) B is a (local) martingale;

(2) ⟨B⟩_{t} = t. We have 𝔼D_{h}G(B) = 𝔼⟨h, B⟩_{ℋ} G(B). Take G = 1, we have D_{h}G = 0, hence

𝔼⟨h, B⟩_{ℋ} = 𝔼 ∫_{0}^{1} ḣ_{s}dB_{s} = 0

for all ℱ_{∗}-adapted ḣ. This implies that B is a (local) martingale. Take G(B) = B, then

D_{h}G(B) = lim_{t→0} \frac{G(B + th) − G(B)}{t} = h.

Second proof (continued) The integration by parts formula

𝔼D_{h}G(B) = 𝔼⟨h, B⟩_{ℋ} G(B)

becomes

𝔼h_{1} = 𝔼⟨h, B⟩_{ℋ} B_{1}.

The right side is

𝔼 [B_{1} ∫_{0}^{1} ḣ_{s}dB_{s}] = 𝔼 ∫_{0}^{1} ḣ_{s}d⟨B⟩_{s}. Hence, equating the expressions, we have

𝔼 ∫_{0}^{1} ḣ_{s}ds = 𝔼 ∫_{0}^{1} ḣ_{s}d⟨B⟩_{s}.

This holds for all ℱ_{∗}-adapted h, and we therefore must have ⟨B⟩_{t} = t. B is a Brownian motion. We next consider Brownian motion on a Riemannian manifold.

Brownian motion on Riemannian manifold M B – Brownian motion on ℝ^{n}. J : P_{o}(ℝ^{n}) → P_{o}(M) = rolling without slipping . Differential geometry:

J(straight lines ) = geodesics

Stochastic differential geometry:

J(ℝ^{n} − BM ) = M − BM

J (Itô map) is invertible and is defined by a stochastic differential equation! The Wiener measure on P_{o}(M) is defined by

ν = the law of Brownian motion on  M
= μ ∘ J^{−1}.

X is Brownian motion on M if and only if the development B = J^{−1}X is a Brownian motion on ℝ^{n}.

Gradient operator D_{h} in the path space Tangent space at X ∈ P_{o}(M) is

T_{X}P_{o}(M) = ℋ .

Each h ∈ ℋ determines a vector field D_{h} on P_{o}(M) by

D_{h}(X)_{s} = U(X)_{s}h_{s},

where U(X) is the parallel transport along X[0, 1]. [picture on D_{h}] Note that U = U(X) is the solution of an SDE on the frame bundle 𝒪(M) driven by X.

Integration by parts in the path space P_{o}(M)

An explicit integration by parts formula for D_{h} is equivalent to an explicit expression for the adjoint D_{h}^{∗}. From

D_{h}^{∗}G = −D_{h}G + (D_{h}^{∗}1)G,

it is enough to specify D_{h}^{∗}1.

Theorem 2 (Bismut (1984), Driver (1992))

D_{h}^{∗}1(X) = ∫_{0}^{1}⟨ḣ_{s} − \frac{1}{2}Ric_{U(X)_{s}} h_{s}, dB_{s}⟩.

Here B = J^{−1}X is the anti-development of X and Ric_{u} : ℝ^{n} → ℝ^{n} is the Ricci transform.

If the Ricci curvature vanishes, the above formula reduces the usual integration by parts formula we derived before

D_{h}^{∗}1 = ∫_{0}^{1}⟨ḣ_{s}, dB_{s}⟩.

One-dimension vs. nonflat infnite dimensions

Suppose that X is a standard Gaussian random variable. Then

𝔼f′(X) = 𝔼 Xf(X).

Suppose that X is a Brownian motion on a compact Riemannian manifold M. Then

𝔼D_{h}G(X) = 𝔼G(X) [∫_{0}^{1}⟨ḣ_{s} − \frac{1}{2}Ric_{U(X)_{s}} h_{s}, dB_{s}⟩]

for all h ∈ ℋ .

Questions: Does this relation characterize Brownian motion uniquely among M-valued semimartingales?

Characterization of Brownian motion on M

Theorem 3 Let X be an M-valued semimartingale. Then it is a Brownian motion on M if and only if

𝔼D_{h}G(X) = 𝔼G(X) [∫_{0}^{1}⟨ḣ_{s} − \frac{1}{2}Ric_{U(X)_{s}} h_{s}, dB_{s}⟩]

for all cylinder functions G ∈ 𝒞 and all adapted h ∈ ℋ . Here B and U are the anti-development of X in ℝ^{n} and the horizontal lift of X in 𝒪(M), respectively.

Recall: X is a Brownian motion on M if and only if the continuous semimartingale B = J^{−1}X is a Brownian motion on ℝ^{n}.

We need to verify that the semimartingale B satisfies Lévy’s criterion.

Sketch of proof (step 1)

(1) B is a (local) martingale. Note that

𝔼D_{h}G(X) = 𝔼G(X) [∫_{0}^{1}⟨ḣ_{s} − \frac{1}{2}Ric_{U(X)_{s}} h_{s}, dB_{s}⟩] .

Let g ∈ ℋ and consider the equation

ḣ_{s} − \frac{1}{2}Ric_{U(X)_{s}} h_{s} = ġ_{s}, 0 ≤ s ≤ 1.

This equation is solvable for g ∈ ℋ . Let G(X) = 1, we have

𝔼 ∫_{0}^{1}⟨ġ_{s}, dB_{s}⟩ = 0

for all uniformly bounded, adapted ġ_{s}. This shows that B is a (local) martingale.

Sketch of proof (step 2)

(2) ⟨B, B⟩_{t} = identity matrix · t.

We let G : P_{o}(M) → ℝ^{n} be the function

G(X) = (J^{−1}X)_{1} = B_{1}.

The key step is to compute the derivative D_{h}B_{1}. We have

D_{h}B_{1} = D_{h}(J^{−1}X)_{1} = (J_{∗}^{−1}D_{h})_{1}.

Note that B is obtained from X by solving an SDE, therefore we can compute the pullback (J_{∗}^{−1}D_{h} on P_{o}(ℝ^{n}) of the vector field D_{h} on P_{o}(M). The answer is

(J_{∗}^{−1}D_{h})_{1} = ∫_{0}^{1} [ḣ_{s} − \frac{1}{2}Ric_{U(X)_{s}} h_{s}] ds

+ martingale _{1}.

Again let

ġ_{s} = ḣ_{s} − \frac{1}{2}Ric_{U(X)_{s}} h_{s}.

Consider the equation

𝔼D_{h}G(X) = 𝔼G(X) [∫_{0}^{1}⟨ḣ_{s} − \frac{1}{2}Ric_{U(X)_{s}} h_{s}, dB_{s}⟩] .

On the left side we have

𝔼(J_{∗}^{−1}D_{h})_{1} = 𝔼 ∫_{0}^{1} ġ_{s}ds.

On the right side we have

𝔼 [B_{1} ∫_{0}^{1}⟨ġ_{s}, dB_{s}⟩] = 𝔼 ∫_{0}^{1} ġ_{s} d⟨B^{∗}, B⟩_{s}.

It follows that

𝔼 ∫_{0}^{1} ġ_{s} ds = 𝔼 ∫_{0}^{1} ġ_{s} d⟨B^{∗}, B⟩_{s}.

Since {ġ_{s}, 0 ≤ s ≤ 1} can be an arbitrary adapted process, we must have

⟨B^{∗}, B⟩_{t} = identity matrix · t.

Therefore B is a Brownian motion on ℝ^{n} by Lévy’s criterion.

Conclusion: X is a Brownian motion on M.

Main point of the talk

Inspired by Stein’s observation that

𝔼f′(X) = 𝔼Xf(X)

characterizes the standard Gaussian law uniquely, we can show that the well known integration by parts formula in the path space characterizes uniquely Brownian motion (Wiener measure) on a Riemannian manifold.

Remark Stein’s method can be used to estimate the distance between a random variable and the standard Gaussian random variable (central limit theorem).

Can we develop a theory to describe the distance between a manifold-valued martingale and a Brownian motion on a Riemannian manifold?

Is there a class of Stein’s equations on path spaces?

Credits Stein’s method

– Stein, L. Chen. Integration by parts for path groups – Airault-Malliavin. Brownian motion on manifolds

– Elworthy, Malliavin. Gradient operator on path spaces

– Malliavin, Gross. Integration by parts for Riemannian BM – Bismut, Driver.

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