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Almost sure exponential behaviour for a parabolic SPDE on a manifold. (English) Zbl 1058.60053
Let $$M$$ be a compact complete $$d$$-dimensional Riemannian manifold. Denote by $$\Delta$$ the Laplace-Beltrami operator on $$M$$, by $$\{e_ {n}\}_ {n\geq 1}$$ the orthonormal basis of $$L^ 2(M)$$ consisting of the eigenvectors of $$-\Delta$$ and by $$\{\lambda _ {n}\}_ {n\geq 1}$$ the associated eigenvalues. Suppose that there exists a constant $$c>0$$ such that $$\| \nabla e_ {i}\| _ \infty \leq c\lambda _ {i} ^ {1/2}\| e_ {i}\| _ \infty$$ for all $$i\geq 1$$. Let $$W$$ be a cylindrical Wiener process on $$L^2(M)$$ defined formally by $$W(t) = \sum _ {i\geq 1} q_ {i}^ {1/2}\beta _ {i}(t)e_ {i}$$, where $$\{\beta _ {i}\}_ {i\geq 1}$$ is a family of independent standard real Wiener processes and $$\{q_ {i}\}_ {i\geq 1}$$ are positive numbers satisfying $$\sum _ {i\geq 1} q_ {i}\| e_ {i}\| ^ 2_ \infty (1+\lambda _ {i}) ^ \alpha <\infty$$ for some $$\alpha >0$$. Let $$\kappa >0$$. The long-time behaviour of mild solutions to a linear Stratonovich stochastic parabolic equation with a multiplicative noise, $du = \kappa \Delta u\,dt + u\circ dW, \quad u(0,\cdot ) = 1,$ is studied. The main result states that there exists a constant $$C$$ such that, for small $$\kappa$$, $\limsup _ {t\to \infty } \frac 1{t}\log u(t,x) \leq {C\over \log (1/\kappa)}$ holds almost surely for every $$x\in M$$.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
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