Characterization of stochastic processes which stabilize linear companion form systems

The class of stochastic processes is characterized which, as multiplicative noise with large intensity, stabilizes a linear system with companion form dxd-matrix. This includes the characterization of parametric noise which stabilizes the damped inverse pendulum. The proof yields also an expansion of the top Lyapunov exponent in terms of the noise intensity as well as a criterion for a stationary diffusion process permitting a stationary integral and it shows that stabilizing noise averages the Lyapunov spectrum.