White noise

Quantum mechanics

Engineering

Mathematical physics

Stochastic partial differential equation

Statistics

Stable process

Differential equation

Brownian motion

Stochastic process

Stochastic differential equation

Brownian noise

Computer science

Statistical physics

Mechanical engineering

Wiener process

Physics

Mathematics

Optics

Image (mathematics)

Invariant (physics)

Econometrics

Mathematical analysis

Artificial intelligence

Ordinary differential equation

Invariant measure

Limiting

Partial differential equation

Intensity (physics)

Noise (video)

Ergodic theory

Continuous-time stochastic process

Science China Mathematics

The limiting behavior of stochastic evolution processes with small noise intensity ϵ is investigated in distribution-based approaches. Let μϵ be a stationary measure for stochastic process Xϵ with small ϵ and X0 be a semiflow on a Polish space. Assume that {μϵ: 0 < ϵ ⩽ ϵ0} is tight. Then all their limits in the weak sense are X0-invariant and their supports are contained in the Birkhoff center of X0. Applications are made to various stochastic evolution systems, including stochastic ordinary differential equations, stochastic partial differential equations, and stochastic functional differential equations driven by Brownian motion or Lévy processes.

On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity