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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 \(x\in M\).

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