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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: 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$.

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Keywords: Parabolic equations, Itô equations
Article copyright: © Copyright 1974 American Mathematical Society