×

zbMATH — the first resource for mathematics

Stabilization of partial differential equations by noise. (English) Zbl 0962.60052
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.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnold, L., 1974. Stochastic Differential Equations: Theory and Applications, A Wiley - Interscience Publication, Wiley, New York. · Zbl 0278.60039
[2] Arnold, L., 1979. A new example of an unstable system being stabilized by random parameter noise. Inform. Comm. Math. Chem. 133-140. · Zbl 0424.93059
[3] Arnold, L., Crauel, H., Eckmann, J.-P. (Eds.), 1991. Lyapunov exponents, Lecture Notes in Mathematics 1486. Springer, Berlin.
[4] Arnold, L.; Crauel, H.; Wihstutz, V., Stabilization of linear systems by noise, SIAM J. control optim., 21, 451-461, (1983) · Zbl 0514.93069
[5] Arnold, L.; Kloeden, P., Lyapunov exponents and rotation number of two-dimensional systems with telegraphic noise, SIAM J. appl. math., 49, 1242-1274, (1989) · Zbl 0684.60046
[6] Arnold, L., Wihstutz, V. (Eds.), 1986. Lyapunov exponents, Lecture Notes in Mathematics 1186. Springer, Berlin. · Zbl 0599.60061
[7] Brzeźniak, Z., Flandoli, F., 1992. Regularity of solutions and random evolution operator for stochastic parabolic equations. In: Da Prato, G., Tubaro, L. (Eds.), Stochastic Partial Differential Equations and Applications. Pitman Research Notes in Mathematics Series 268. Longman Scientific & Technical, Harlow, pp. 54-71.
[8] Carmona, R.A., Molchanov, S.A., 1994. Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 (518). · Zbl 0925.35074
[9] Da Prato, G.; Iannelli, M.; Tubaro, L., Some results on linear stochastic differential equations in Hilbert spaces, Stochastics, 6, 105-116, (1982) · Zbl 0475.60041
[10] Flandoli, F., 1991. Stochastic flow and Lyapunov exponents for abstract stochastic PDE’s of parabolic type. In: Lecture Notes in Mathematics 1486 (see Arnold et al., 1991), pp. 196-205. · Zbl 0746.60066
[11] Flandoli, F.; Schaumlöffel, K.-U., Stochastic parabolic equations in bounded domains: random evolution operator and Lyapunov exponents, Stochastics stochastic rep., 29, 461-485, (1990) · Zbl 0704.60060
[12] Lindemann, I., Stability of hyperbolic partial differential equations with random loads, SIAM J. appl. math., 52, 347-367, (1992) · Zbl 0751.60055
[13] Pardoux, E.; Wihstutz, V., Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion, SIAM J. appl. math., 48, 442-457, (1988) · Zbl 0641.60065
[14] Pardoux, E.; Wihstutz, V., Lyapunov exponents of linear stochastic systems with large diffusion term, Stochastic process. appl., 40, 289-308, (1992) · Zbl 0749.60061
[15] Schaumlöffel, K.-U.; Flandoli, F., A multiplicative ergodic theorem with applications to a first order stochastic hyperbolic equation in a bounded domain, Stochastics stochastic rep., 34, 241-255, (1991) · Zbl 0724.60072
[16] Taylor, M.E., 1996. Partial Differential Equations I. Basic Theory. Applied Mathematical Sciences 115, Springer, New York. · Zbl 0869.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.