# Ornstein–Uhlenbeck operator

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$^n$ → R; the gradient of a scalar function is a vector field v = ∇f : R$^n$ → R$^n$. 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:
    $\Delta =\mathrm {div} \circ \nabla$,
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)$^{1/2}$ for the operator (1 − Δ). This square root satisfies the following relation involving the Sobolev H$^1$-norm and L$^2$-norm for suitable scalar functions f:
    ${\big \|}f{\big \|}_{H^{1}}^{2}={\big \|}(1-\Delta )^{1/2}f{\big \|}_{L^{2}}^{2}.$

### Definition of the operator

Often, when working on R$^n$, 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$^n$ on R$^n$: for Borel subsets A of R$^n$,
    $\gamma ^{n}(A):=\int _{A}(2\pi )^{-n/2}\exp(-|x|^{2}/2)\,\mathrm {d} x.$
This makes (R$^n$, B(R$^n$), γ$^n$) into a probability space; E will denote expectation with respect to γ$^n$.
The gradient operator ∇ acts on a (differentiable) function φ : R$^n$ → R to give a vector field ∇φ : R$^n$ → R$^n$.
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 L$^2$(R$^n$, B(R$^n$), γ$^n$; R). In other words, δ acts on a vector field v : R$^n$ → R$^n$ to give a scalar function δv : R$^n$ → R, and satisfies the formula
    $\mathbb {E} {\big\nabla f\cdot v{\big ]}=\mathbb {E} {\bigf\delta v{\big ]}.$
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$^i$, i = 1, ..., n, as follows:
    $\delta v(x)=\sum _{i=1}^{n}\left(x_{i}v^{i}(x)-{\frac {\partial v^{i}}{\partial x_{i}}}(x)\right).$
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, L$_m$) is defined by
    $L:=-\delta \circ \nabla ,$
with the useful formula that for any f and g smooth enough for all the terms to make sense,
    $\delta (f\nabla g)=-\nabla f\cdot \nabla g-fLg.$
The Ornstein–Uhlenbeck operator L is related to the usual Laplacian Δ by
    $Lf(x)=\Delta f(x)-x\cdot \nabla f(x).$

## 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$^2$(E, γ; R) into L$^2$(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$^2$(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 L$^2$).
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
    $\mathbb {E} {\big\langle \mathrm {D} F,v\rangle _{H}{\big ]}=\mathbb {E} {\bigF\delta v{\big ]}$
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
    $L:=-\delta \circ \mathrm {D} .$