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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)
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References:
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