Stabilization of partial differential equations by noise
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.
stochastic partial differential equations
scientific article; zbMATH DE number 1554280
scientific article