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!