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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Parabolic Itô equations


Author: Robert Marcus
Journal: Trans. Amer. Math. Soc. 198 (1974), 177-190
MSC: Primary 60H15
MathSciNet review: 0346909
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Abstract | References | Similar Articles | Additional Information

Abstract: A parabolic Itô equation is an equation of the form

$\displaystyle (\partial u/\partial t)(t,\omega ) = Lu(t,\omega ) + f(u(t,\omega )) + \alpha (t,\omega ),\quad u(0) = {u_0},{u_0},u \in H.$

$ H$ is a Hilbert space with scalar product $ u \cdot \upsilon $ and norm $ \vert \cdot \vert$. $ L$ is a linear time-independent negative-definite operator from $ H$ to $ H$. $ f$ is a Lipschitz continuous operator from $ H$ to $ H$. $ \alpha (t,\omega )$ is a white noise process in $ H$.

Under suitable technical conditions the following results are obtained:

I. A unique nonanticipating solution of (1) exists with $ {\sup _t}E\{ \vert u{\vert^2}\} < \infty $.

II. $ u(t,\omega ) = R(t,\omega ) + V(t,\omega )$ where $ R(t,\omega )$ is a stationary process and

$\displaystyle \mathop {\lim }\limits_{t \to \infty } E\{ \vert V(t,\omega ){\vert^2}\} = 0.$

III. If $ L$ is selfadjoint and $ f$ is the gradient of a smooth functional then an explicit expression is found for the stationary density of $ R(t,\omega )$ on $ H$.

IV. For the equation $ (\partial u/\partial t)(t,\omega ) = Lu(t,\omega ) + f(u(t,\omega )) + \varepsilon \alpha (t,\omega )$ an asymptotic expansion in $ \varepsilon $ is proven which holds uniformly in $ t$.


References [Enhancements On Off] (What's this?)

  • [1] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, N.J., 1964. MR 31 #6062. MR 0181836 (31:6062)
  • [2] M. Schilder, Some asymptotic formulas for Wiener integrals, Trans. Amer. Math. Soc. 125 (1966), 63-85. MR 34 #1770. MR 0201892 (34:1770)
  • [3] M. Pincus, Gaussian processes and Hammerstein integral equations, Trans. Amer. Math. Soc. 134 (1968), 193-214. MR 37 #6994. MR 0231439 (37:6994)
  • [4] V. V. Baklan, Variational differential equations and Markov processes in Hilbert space, Dokl. Akad. Nauk SSSR 159 (1964), 707-710 = Soviet Math. Dokl. 5 (1964), 1553-1556. MR 30 #1547. MR 0171316 (30:1547)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0346909-8
PII: S 0002-9947(1974)0346909-8
Keywords: Parabolic equations, Itô equations
Article copyright: © Copyright 1974 American Mathematical Society



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