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
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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 $t$.References
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- M. Schilder, Some asymptotic formulas for Wiener integrals, Trans. Amer. Math. Soc. 125 (1966), 63–85. MR 201892, DOI 10.1090/S0002-9947-1966-0201892-6
- Martin Pincus, Gaussian processes and Hammerstein integral equations, Trans. Amer. Math. Soc. 134 (1968), 193–214. MR 231439, DOI 10.1090/S0002-9947-1968-0231439-1
- V. V. Baklan, Variational differential equations and Markov processes in Hilbert space, Dokl. Akad. Nauk SSSR 159 (1964), 707–710 (Russian). MR 0171316
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