Ornstein–Uhlenbeck Processes
Ornstein–Uhlenbeck may refer to: * Ornstein–Uhlenbeck operator * Ornstein–Uhlenbeck process {{disambiguation ...
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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. Introduction: the finite-dimensional picture The Laplacian 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 \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 −& ...
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