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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Parabolic Itô equations
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by Robert Marcus
Trans. Amer. Math. Soc. 198 (1974), 177-190
DOI: https://doi.org/10.1090/S0002-9947-1974-0346909-8

Abstract:

A parabolic Itô equation is an equation of the form \[ (\partial u/\partial t)(t,\omega ) = Lu(t,\omega ) + f(u(t,\omega )) + \alpha (t,\omega ),\quad u(0) = {u_0},{u_0},u \in H.\] $H$ is a Hilbert space with scalar product $u \cdot \upsilon$ and norm $| \cdot |$. $L$ 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 ${\sup _t}E\{ |u{|^2}\} < \infty$. II. $u(t,\omega ) = R(t,\omega ) + V(t,\omega )$ where $R(t,\omega )$ is a stationary process and \[ \lim \limits _{t \to \infty } E\{ |V(t,\omega ){|^2}\} = 0.\] 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 $t$.
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Bibliographic Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 198 (1974), 177-190
  • MSC: Primary 60H15
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0346909-8
  • MathSciNet review: 0346909