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. 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. 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: 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: 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, 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, δn, 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