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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.

Introduction: the finite-dimensional picture

The Laplacian

Consider the gradient operator ∇ acting on scalar functions f : Rⁿ → R; the gradient of a scalar function is a vector field v = ∇f : Rⁿ → Rⁿ. 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:

$\mathrm{\Delta }=\mathrm{d}\mathrm{i}\mathrm{v}\circ \mathrm{\nabla }$,

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)^1/2 for the operator (1 − Δ). This square root satisfies the following relation involving the Sobolev H¹-norm and L²-norm for suitable scalar functions f:

$\Vert f{\Vert }_{H^1}^2=\Vert (1-\mathrm{\Delta }{)}^{1/2}f{\Vert }_{L^2}^2.$

The Ornstein–Uhlenbeck operator

Often, when working on Rⁿ, 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 γⁿ on Rⁿ: for Borel subsets A of Rⁿ,

${\gamma }^n(A):=\underset{A}{\int }(2\pi {)}^{-n/2}\exp (-|x{|}^2/2)\mathrm{d}x.$

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

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

The divergence operator δ (to be more precise, δ_{n, since it depends on the dimension) is now defined to be the adjoint of ∇ in the Hilbert space sense, in the Hilbert space L²(Rⁿ, B(Rⁿ), γⁿ; R). In other words, δ acts on a vector field v : Rⁿ → Rⁿ to give a scalar function δv : Rⁿ → R, and satisfies the formula}

$\mathbb{𝔼}[\mathrm{\nabla }f\cdot v]=\mathbb{𝔼}\left[f\delta v\right].$

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 vⁱ, i = 1, ..., n, as follows:

$\delta v(x)=\sum _{i=1}^n(x_i{v}^i(x)-\frac{\partial v^i}{\partial x_i}(x)).$

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, L_m) is defined by

$L:=-\delta \circ \mathrm{\nabla },$

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

$\delta (f\mathrm{\nabla }g)=-\mathrm{\nabla }f\cdot \mathrm{\nabla }g-fLg.$

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

$Lf(x)=\mathrm{\Delta }f(x)-x\cdot \mathrm{\nabla }f(x).$

The Ornstein–Uhlenbeck operator for a separable Banach space

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 L²(E, γ; R) into L²(E, γ; H) – in some sense, it measures “how random” a function on E is. The domain of D is not the whole of L²(E, γ; R), but is a dense linear subspace, the Watanabe-Sobolev space, often denoted by ${\mathbb{𝔻}}^{1,2}$ (once differentiable in the sense of Malliavin, with derivative in L²).

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

$\mathbb{𝔼}[⟨\mathrm{D}F,v{⟩}_H]=\mathbb{𝔼}\left[F\delta v\right]$

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

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

$L:=-\delta \circ \mathrm{D}.$

References

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