Ornstein–Uhlenbeck operator

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:

${\displaystyle \Delta =\mathrm {div} \circ \nabla }$,

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)^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:

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

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,

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

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

${\displaystyle \mathbb {E} {\big [}\nabla f\cdot v{\big ]}=\mathbb {E} {\big [}f\delta v{\big ]}.}$

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 vⁱ, i = 1, ..., n, as follows:

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

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, L_m) is defined by

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

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

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

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

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