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Ornstein–Uhlenbeck operator

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Not to be confused with Ornstein–Uhlenbeck process.

In mathematics, the Ornstein–Uhlenbeck operator is a generalization of the Laplace operator to an infinite-dimensional setting. The Ornstein–Uhlenbeck operator plays a significant role in the Malliavin calculus. The operator is named after Leonard Ornstein and George Eugene Uhlenbeck.

Introduction: the finite-dimensional picture

Laplacian properties

Consider the gradient operator ∇ acting on scalar functions f : R → R; the gradient of a scalar function is a vector field v = ∇f : R → R. The divergence operator div, acting on vector fields to produce scalar fields, is the adjoint operator to ∇. The Laplace operator Δ is then the composition of the divergence and gradient operators:

Δ = d i v ∘ ∇ {\displaystyle \Delta =\mathrm {div} \circ \nabla } ,

acting on scalar functions to produce scalar functions. Note that A = −Δ is a positive operator, whereas Δ is a dissipative operator.

Using spectral theory, one can define a square root (1 − Δ) for the operator (1 − Δ). This square root satisfies the following relation involving the Sobolev H-norm and L-norm for suitable scalar functions f:

‖ f ‖ H 1 2 = ‖ ( 1 − Δ ) 1 / 2 f ‖ L 2 2 . {\displaystyle {\big \|}f{\big \|}_{H^{1}}^{2}={\big \|}(1-\Delta )^{1/2}f{\big \|}_{L^{2}}^{2}.}

Definition of the operator

Often, when working on R, one works with respect to Lebesgue measure, which has many nice properties. However, remember that the aim is to work in infinite-dimensional spaces, and it is a fact that there is no infinite-dimensional Lebesgue measure. Instead, if one is studying some separable Banach space E, what does make sense is a notion of Gaussian measure; in particular, the abstract Wiener space construction makes sense.

To get some intuition about what can be expected in the infinite-dimensional setting, consider standard Gaussian measure γ on R: for Borel subsets A of R,

γ n ( A ) := ∫ A ( 2 π ) − n / 2 exp ⁡ ( − | x | 2 / 2 ) d x . {\displaystyle \gamma ^{n}(A):=\int _{A}(2\pi )^{-n/2}\exp(-|x|^{2}/2)\,\mathrm {d} x.}

This makes (R, B(R), γ) into a probability space; E will denote expectation with respect to γ.

The gradient operator ∇ acts on a (differentiable) function φ : R → R to give a vector field ∇φ : R → R.

The divergence operator δ (to be more precise, δn, since it depends on the dimension) is now defined to be the adjoint of ∇ in the Hilbert space sense, in the Hilbert space L(R, B(R), γ; R). In other words, δ acts on a vector field v : R → R to give a scalar function δv : R → R, and satisfies the formula

E [ ∇ f ⋅ v ] = E [ f δ v ] . {\displaystyle \mathbb {E} {\big }=\mathbb {E} {\big }.}

On the left, the product is the pointwise Euclidean dot product of two vector fields; on the right, it is just the pointwise multiplication of two functions. Using integration by parts, one can check that δ acts on a vector field v with components v, i = 1, ..., n, as follows:

δ v ( x ) = ∑ i = 1 n ( x i v i ( x ) − ∂ v i ∂ x i ( x ) ) . {\displaystyle \delta v(x)=\sum _{i=1}^{n}\left(x_{i}v^{i}(x)-{\frac {\partial v^{i}}{\partial x_{i}}}(x)\right).}

The change of notation from "div" to "δ" is for two reasons: first, δ is the notation used in infinite dimensions (the Malliavin calculus); secondly, δ is really the negative of the usual divergence.

The (finite-dimensional) Ornstein–Uhlenbeck operator L (or, to be more precise, Lm) is defined by

L := − δ ∘ ∇ , {\displaystyle L:=-\delta \circ \nabla ,}

with the useful formula that for any f and g smooth enough for all the terms to make sense,

δ ( f ∇ g ) = − ∇ f ⋅ ∇ g − f L g . {\displaystyle \delta (f\nabla g)=-\nabla f\cdot \nabla g-fLg.}

The Ornstein–Uhlenbeck operator L is related to the usual Laplacian Δ by

L f ( x ) = Δ f ( x ) − x ⋅ ∇ f ( x ) . {\displaystyle Lf(x)=\Delta f(x)-x\cdot \nabla f(x).}

For separable Banach spaces

Consider now an abstract Wiener space E with Cameron-Martin Hilbert space H and Wiener measure γ. Let D denote the Malliavin derivative. The Malliavin derivative D is an unbounded operator from L(E, γ; R) into L(E, γ; H) – in some sense, it measures "how random" a function on E is. The domain of D is not the whole of L(E, γ; R), but is a dense linear subspace, the Watanabe-Sobolev space, often denoted by D 1 , 2 {\displaystyle \mathbb {D} ^{1,2}} (once differentiable in the sense of Malliavin, with derivative in L).

Again, δ is defined to be the adjoint of the gradient operator (in this case, the Malliavin derivative is playing the role of the gradient operator). The operator δ is also known the Skorokhod integral, which is an anticipating stochastic integral; it is this set-up that gives rise to the slogan "stochastic integrals are divergences". δ satisfies the identity

E [ ⟨ D F , v ⟩ H ] = E [ F δ v ] {\displaystyle \mathbb {E} {\big }=\mathbb {E} {\big }}

for all F in D 1 , 2 {\displaystyle \mathbb {D} ^{1,2}} and v in the domain of δ.

Then the Ornstein–Uhlenbeck operator for E is the operator L defined by

L := − δ ∘ D . {\displaystyle L:=-\delta \circ \mathrm {D} .}

References

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