ALMOST SURE AND MOMENT STABILITY OF STOCHASTIC PARTIAL
DIFFERENTIAL EQUATIONS
Abstract: We study the almost sure and moment stability of a class of stochastic partial
differential equations and we present an infinite-dimensional version of a theorem proved for
stochastic ordinary differential equations by Arnold, Oeljeklaus and Pardoux. We also
investigate how adding a term with white noise influences the stability of a deterministic
system. The outcome is quite surprising. It turns out that regardless whether the deterministic
system was stable or unstable, after adding a term with sufficiently large noise, it
becomes pathwise exponentially stable and unstable in the -th mean for
1991 AMS Mathematics Subject Classification: Primary: 60H15, 35K10; Secondary:
65N25.
Key words and phrases: Stochastic partial differential equation, almost sure stability,
moment stability, deterministic partial differential equation, stabilization by noise,
destabilization by noise.