A Continuous Semigroup Approach to the Distributional Stability of Nonlinear Models

We prove the existence of an invariant measure for the continuous semigroup associate with a nonlinear model under the compact set Lyapunov condition. Further,adding the ergodicity of the semigroup operator, we prove the asymptotic stability in distribution for the semigroup. We give a criteria of the asymptotic stability in distribution for the type of evolution equation having a linear generator. Our method is based on continuous semigroup and its generator.We illustrate the result by the Lorenz chaotic model and prove the existence of the natural invariant measure for Lorenz chaotic model.