stochastical 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: $$\int \phi \circ f^n_{\omega} \cdot \psi dm-\int \phi d \mu_{\sigma^n \omega} \int \psi dm \precsim C_{\epsilon, \phi, \psi}(\omega) \cdot \frac{1}{n^{\alpha}}.$$

I found two ways to define stochastically stable:

1. $$C_{\epsilon, \phi, \psi}(\omega)$$ 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)

So if we assume we have SRB $$\mu_f$$, stationary probability $$\mu_{\epsilon}$$, quasi-invariant probability $$\mu_{\omega}$$, and decay of correlation ( not loss of memory since it may depend on $$\omega$$ ), definition 1 and 2 are equivalent? or they are different terminologies? We all know the second definition is the official one. But why Baladi uses the same terminology? Thanks!

• These are certainly not equivalent. I believe that rotations by a random perturbation of an irrational number satisfy 2) but not 1). That is: $f(x)=x+\alpha$. – Anthony Quas Dec 6 '18 at 17:46
• I don’t think you get any decay of correlation: imagine $\phi$ and $\psi$ are supported on small intervals. – Anthony Quas Dec 7 '18 at 3:36