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Regularity property of Donsker's delta function

Regularity property of Donsker's delta function LetL be the space of rapidly decreasing smooth functions on ℝ andL * its dual space. Let (L 2)+ and (L 2)− be the spaces of test Brownian functionals and generalized Brownian functionals, respectively, on the white noise spaceL * with standard Gaussian measure. The Donsker delta functionδ(B(t)−x) is in (L 2)− and admits the series representation $$\delta (B(t) - x) = (2\pi t)^{ - 1/2} \exp ( - x^2 /2t)\sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} H_n (x/\sqrt {2t} )} \times H_n (B(t)/\sqrt {2t} )$$ , whereH n is the Hermite polynomial of degreen. It is shown that forφ in (L 2)+,g t,φ(x)≡⟨δ(B(t)−x), φ⟩ is inL and the linear map takingφ intog t,φ is continuous from (L 2)+ intoL. This implies that forf inL * is a generalized Brownian functional and admits the series representation $$f(B(t)) = (2\pi t)^{ - 1/2} \sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} \langle f,\xi _{n, t} \rangle } H_n (B(t)/\sqrt {2t} )$$ , whereξ n,t is the Hermite function of degreen with parametert. This series representation is used to prove the Ito lemma forf inL *, $$f(B(t)) = f(B(u)) + \int_u^t {\partial _s^ * } f'(B(s)) ds + (1/2)\int_u^t {f''} (B(s)) ds$$ , where∂ s * is the adjoint of $$\dot B(s)$$ -differentiation operator∂ s . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Regularity property of Donsker's delta function

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References (9)

Publisher
Springer Journals
Copyright
Copyright © 1984 by Springer-Verlag New York Inc.
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/BF01449036
Publisher site
See Article on Publisher Site

Abstract

LetL be the space of rapidly decreasing smooth functions on ℝ andL * its dual space. Let (L 2)+ and (L 2)− be the spaces of test Brownian functionals and generalized Brownian functionals, respectively, on the white noise spaceL * with standard Gaussian measure. The Donsker delta functionδ(B(t)−x) is in (L 2)− and admits the series representation $$\delta (B(t) - x) = (2\pi t)^{ - 1/2} \exp ( - x^2 /2t)\sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} H_n (x/\sqrt {2t} )} \times H_n (B(t)/\sqrt {2t} )$$ , whereH n is the Hermite polynomial of degreen. It is shown that forφ in (L 2)+,g t,φ(x)≡⟨δ(B(t)−x), φ⟩ is inL and the linear map takingφ intog t,φ is continuous from (L 2)+ intoL. This implies that forf inL * is a generalized Brownian functional and admits the series representation $$f(B(t)) = (2\pi t)^{ - 1/2} \sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} \langle f,\xi _{n, t} \rangle } H_n (B(t)/\sqrt {2t} )$$ , whereξ n,t is the Hermite function of degreen with parametert. This series representation is used to prove the Ito lemma forf inL *, $$f(B(t)) = f(B(u)) + \int_u^t {\partial _s^ * } f'(B(s)) ds + (1/2)\int_u^t {f''} (B(s)) ds$$ , where∂ s * is the adjoint of $$\dot B(s)$$ -differentiation operator∂ s .

Journal

Applied Mathematics and Optimization – Springer Journals

Published: Mar 27, 2005

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