# stochastically stable

Given dynamic $$f: S^1 \to S^1$$ with Lebegue measure $$dm$$ on $$S^1$$. Assume it has unique SRB probability measure $$\frac{d\mu_f}{dm} dm$$. Given left shift space $$([-\epsilon, \epsilon]^{\otimes \mathbb{N}}, \theta_{\epsilon}^{\otimes \mathbb{N}}, \sigma )$$, where $$\theta_{\epsilon}$$ is probability on $$[-\epsilon,\epsilon ]$$.

For any $$\omega \in [-\epsilon, \epsilon]^{\otimes \mathbb{N}}$$, define $$f^n_{\omega}:=f_{\omega_{n-1} } \circ f_{\omega_{n-2} } \circ…\circ f_{\omega_{0} }$$ where $$f^1_{\omega}=f_{w_0}:=f+\omega_0$$. Assume for each $$\epsilon$$, we have stationary probability $$\mu_{\epsilon}$$ on $$S^1$$, which means $$\mu_{\epsilon}E=\int 1_{E} \circ f^1_{\omega}(x) d\mu_{\epsilon}(x) d\theta_{\epsilon}(\omega).$$

Assume we have quasi-invariant probability $$\mu_{\omega}:= h_{\omega}dm$$ s.t. $$(f^1_{\omega})_{*} \mu_{\omega}=\mu_{\sigma(\omega)}$$. Assume we have decay of correlation: $$\int \phi \circ f^n_{\omega} \cdot \psi dm-\int \phi d \mu_{\sigma^n \omega} \int \psi dm \precsim C_{\epsilon, \phi, \psi} \cdot \frac{1}{n^{\alpha}}.$$

With the setting above, I found two ways to define stochastically stable:

1. $$C_{\epsilon, \phi, \psi}$$ does not blow up when $$\epsilon \to 0$$. See Page 5 second paragraph of Almost Sure Rates of Mixing for I.i.d. Unimodal Maps (1999) Baladi (http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.8601)

2. $$\mu_{\epsilon} \to \mu_f$$ in weak * topology as $$\epsilon \to 0$$. See page 2 of On stochastic stability of expanding circle maps with neutral fixed points, Sebastian van Strien (https://arxiv.org/abs/1212.5671)

are they equivalent? or different definition? Many Thanks!