Stabilization of partial differential equations by noise
type: Item
instance of: scholarly article
title: Stabilization of partial differential equations by noise
zbMATH Open document ID: 0962.60052
DOI: 10.1016/S0304-4149(98)00080-5
author: Anna A. Kwiecińska
published in: Stochastic Processes and their Applications
publication date: 2001-01-17T00:00:00Z
review text: Let \(\mathcal O\subset \mathbb R^{d}\), \(d\leq 3\), be a bounded domain with a \(C^\infty \)-boundary. The heat equation \[ \partial u/\partial t = \Delta u + \alpha u\tag{1} \] with a Dirichlet boundary condition \(u = 0\) on \(\partial \mathcal O\) and an initial condition \(u(0) = f \in H^{1}_{0}(\mathcal O)\cap H^{2}(\mathcal O)\) and its stochastic counterpart \[ dv = (\Delta v + \beta v) dt + \gamma v dW\tag{2} \] with the same boundary and initial data are considered, \(\alpha \), \(\beta \), \(\gamma \) being arbitrary constants and \(W\) denoting the standard real Wiener process. The Lyapunov exponent of (1) is defined by \(\lambda ^{u}(f) = \limsup _{t\to \infty} t^{-1}\log \|u(t)\|_{L^2}\); analogously, the Lyapunov exponent of (2) is defined pathwise by \(\lambda ^{v}(f,\omega) = \limsup _{t\to \infty} t^{-1}\log \|v(t,\omega)\|_{L^2}\). Let \(\{\lambda _{i}\}_{i\geq 0}\) be the eigenvalues of the Laplacian \(\Delta \) with the Dirichlet boundary conditions on \(\mathcal O\), let \(\{e_{i}\}_{i\geq 0}\) be the corresponding eigenvectors and denote by \(j_0\) the least integer \(j\geq 0\) such that \(\langle f,e_{j}\rangle \neq 0\). It is proved by a direct computation that \(\lambda ^{u}(f) = \lambda _{j_0} + \alpha \), while \(\lambda ^{v}(f) = \lambda ^{u}(f) + (\beta -\alpha) - \frac 12\gamma ^{2}\) almost surely. It has been known for a long time that adding a noise to an ordinary differential equation may change the stability properties dramatically; this example shows that the same phenomenon may occur for partial differential equations as well.
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 1554280
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: stabilization by noise
zbMATH Keywords: heat equation
MaRDI profile type: Publication
cites work: Q4404205
cites work: Q3859690
cites work: Lyapunov exponents. Proceedings of a conference, held in Oberwolfach, Germany, May 28-June 2, 1990
cites work: Stabilization of Linear Systems by Noise
cites work: Lyapunov Exponents and Rotation number of Two-Ddimensional Systems with Telegraphic Noise
cites work: Q3734804
cites work: Q4279405
cites work: Parabolic Anderson problem and intermittency
cites work: On the path regularity of a stochastic process in a hilbert space, defined by the ito integral
cites work: Q3979036
cites work: Stochastic parabolic equations in bounded domains: random evolution operator and lyapunov exponents
cites work: Stability of Hyperbolic Partial Differential Equations with Random Loads
cites work: Lyapunov Exponent and Rotation Number of Two-Dimensional Linear Stochastic Systems with Small Diffusion
cites work: Lyapounov exponent of linear stochastic systems with large diffusion term
cites work: A multiplicative ergodic theorem with applications to a first order stochastic hyperbolic equation in a bounded domain
cites work: Q4889891
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description: scientific article; zbMATH DE number 1554280