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| {{distinguish|Ornstein–Uhlenbeck process}}
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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]].
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| ==Introduction: the finite-dimensional picture==
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| ===The Laplacian===
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| Consider the [[gradient]] operator ∇ acting on scalar functions ''f'' : '''R'''<sup>''n''</sup> → '''R'''; the gradient of a scalar function is a [[vector field]] ''v'' = ∇''f'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>. The [[divergence]] operator div, acting on vector fields to produce scalar fields, is the [[adjoint operator]] to ∇. The Laplace operator Δ is then the [[function composition|composition]] of the divergence and gradient operators:
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| :<math>\Delta = \mathrm{div} \circ \nabla</math>,
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| acting on scalar functions to produce scalar functions. Note that ''A'' = −Δ is a positive operator, whereas Δ is a [[dissipative operator]].
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| Using [[spectral theory]], one can define a [[square root]] (1 − Δ)<sup>1/2</sup> for the operator (1 − Δ). This square root satisfies the following relation involving the [[Sobolev space|Sobolev ''H''<sup>1</sup>-norm]] and [[Lp space|''L''<sup>2</sup>-norm]] for suitable scalar functions ''f'':
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| :<math>\big\| f \big\|_{H^{1}}^{2} = \big\| (1 - \Delta)^{1/2} f \big\|_{L^{2}}^{2}.</math>
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| ===The Ornstein–Uhlenbeck operator===
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| Often, when working on '''R'''<sup>''n''</sup>, 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 space|separable]] [[Banach space]] ''E'', what does make sense is a notion of [[Gaussian measure]]; in particular, the [[abstract Wiener space]] construction makes sense.
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| To get some intuition about what can be expected in the infinite-dimensional setting, consider standard Gaussian measure ''γ''<sup>''n''</sup> on '''R'''<sup>''n''</sup>: for Borel subsets ''A'' of '''R'''<sup>''n''</sup>,
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| :<math>\gamma^{n} (A) := \int_{A} (2 \pi)^{-n/2} \exp ( - | x |^{2} / 2) \, \mathrm{d} x.</math>
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| This makes ('''R'''<sup>''n''</sup>, ''B''('''R'''<sup>''n''</sup>), ''γ''<sup>''n''</sup>) into a [[probability space]]; '''E''' will denote [[expected value|expectation]] with respect to ''γ''<sup>''n''</sup>.
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| The '''gradient operator''' ∇ acts on a (differentiable) function ''φ'' : '''R'''<sup>''n''</sup> → '''R''' to give a [[vector field]] ∇''φ'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>.
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| The '''divergence operator''' ''δ'' (to be more precise, ''δ''<sub>''n''</sup>, since it depends on the dimension) is now defined to be the [[adjoint operator|adjoint]] of ∇ in the [[Hilbert space]] sense, in the Hilbert space ''L''<sup>2</sup>('''R'''<sup>''n''</sup>, ''B''('''R'''<sup>''n''</sup>), ''γ''<sup>''n''</sup>; '''R'''). In other words, ''δ'' acts on a vector field ''v'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> to give a scalar function ''δv'' : '''R'''<sup>''n''</sup> → '''R''', and satisfies the formula
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| :<math>\mathbb{E} \big[ \nabla f \cdot v \big] = \mathbb{E} \big[ f \delta v \big].</math>
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| 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''<sup>''i''</sup>, ''i'' = 1, ..., ''n'', as follows:
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| :<math>\delta v (x) = \sum_{i = 1}^{n} \left( x_{i} v^{i} (x) - \frac{\partial v^{i}}{\partial x_{i}} (x) \right).</math>
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| 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.
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| The (finite-dimensional) '''Ornstein–Uhlenbeck operator''' ''L'' (or, to be more precise, ''L''<sub>''m''</sub>) is defined by
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| :<math>L := - \delta \circ \nabla,</math>
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| with the useful formula that for any ''f'' and ''g'' smooth enough for all the terms to make sense,
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| :<math>\delta ( f \nabla g) = - \nabla f \cdot \nabla g - f L g.</math>
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| The Ornstein–Uhlenbeck operator ''L'' is related to the usual Laplacian Δ by
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| :<math>L f (x) = \Delta f (x) - x \cdot \nabla f (x).</math>
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| ==The Ornstein–Uhlenbeck operator for a separable Banach space==
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| 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''<sup>2</sup>(''E'', ''γ''; '''R''') into ''L''<sup>2</sup>(''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''<sup>2</sup>(''E'', ''γ''; '''R'''), but is a [[dense set|dense]] [[linear subspace]], the Watanabe-Sobolev space, often denoted by <math>\mathbb{D}^{1,2}</math> (once differentiable in the sense of Malliavin, with derivative in ''L''<sup>2</sup>).
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| 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
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| :<math>\mathbb{E} \big[ \langle \mathrm{D} F, v \rangle_{H} \big] = \mathbb{E} \big[ F \delta v \big]</math>
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| for all ''F'' in <math>\mathbb{D}^{1,2}</math> and ''v'' in the domain of ''δ''.
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| Then the '''Ornstein–Uhlenbeck operator''' for ''E'' is the operator ''L'' defined by
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| :<math>L := - \delta \circ \mathrm{D}.</math>
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| ==References==
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| * {{cite book
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| | last = Ocone
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| | first = Daniel L.
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| | chapter = A guide to the stochastic calculus of variations
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| | title = Stochastic analysis and related topics (Silivri, 1986)
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| | series = Lecture Notes in Math. 1316
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| | pages = 1–79
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| | publisher = Springer
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| | location = Berlin
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| | year = 1988
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| }} {{MathSciNet|id=953793}}
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| * {{cite web
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| | last = Sanz-Solé
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| | first = Marta
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| | title = Applications of Malliavin Calculus to Stochastic Partial Differential Equations (Lectures given at Imperial College London, 7–11 July 2008)
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| | year = 2008
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| | url = http://www.ma.ic.ac.uk/~dcrisan/lecturenotes-london.pdf
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| | accessdate = 2008-07-09
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| }}
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| {{DEFAULTSORT:Ornstein-Uhlenbeck operator}}
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| [[Category:Operator theory]]
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| [[Category:Stochastic calculus]]
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Thus, it is imperative that you must Hire Word - Press Developers who have the expertise and proficiency in delivering theme integration and customization services. If a newbie missed a certain part of the video then they could always rewind. If you have any thoughts pertaining to wherever and how to use wordpress backup plugin, you can contact us at our web site. This plugin is a must have for anyone who is serious about using Word - Press. You can up your site's rank with the search engines by simply taking a bit of time with your site. For a Wordpress website, you don't need a powerful web hosting account to host your site.
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Internet is not only the source for information, it is also one of the source for passive income. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Just download it from the website and start using the same. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. The 2010 voting took place from July 7 through August 31, 2010.