Lévy’s stochastic area formula and Brownian motion on compact Lie groups

Shinzo Watanabe
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Shinzo Watanabe: Kyoto University, Department of Mathematics, Graduate School of Science

A chapter in Itô’s Stochastic Calculus and Probability Theory, 1996, pp 401-411 from Springer

Abstract: Abstract Let (W, P) be the d-dimensional Wiener space: W be the space of continuous paths W = {w ∈ C([0, ∞) → R d )|w(0) = 0} and P be the standard d-dimensional Wiener measure on W. Then w = (w k(t)) k=1 d in W is a canonical realization of a d-dimensional Wiener process. Lévy’s stochatic area is defined on the two-dimensional Wiener space by Itô’s stochastic integral as follows: $$S\left( t.\omega \right)=\frac{1}{2}\int{_{0}^{t}}{{w}^{1}}\left( s \right)d{{w}^{2}}\left( s \right)-{{w}^{2}}\left( s \right)d{{w}^{1}}\left( s \right)$$ .

Keywords: Brownian Motion; Riemannian Manifold; Stochastic Differential Equation; Heat Kernel; Index Theorem (search for similar items in EconPapers)
Date: 1996
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DOI: 10.1007/978-4-431-68532-6_26

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