Existence and uniqueness of a stationary measure

Question

This same question was also posted on MSE https://math.stackexchange.com/questions/3327007/existence-and-uniqueness-of-a-stationary-measure.

Recently I have posted the following question on MO Attractors in random dynamics.

Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal half of the Lebesgue measure.

Then we can endow the space $\Delta^{\mathbb{N}}:= \{ (\omega_n)_{n\in \mathbb{N}};\ \omega_n \in \Delta, \ \forall \ n\in \mathbb{N}\}$ with the $\sigma$-algebra $\mathcal{B}(\Delta^{\mathbb{N}})$ (Borel $\sigma$-algebra of $\Delta^{\mathbb{N}}$ induced by the product topology) and the probability measuare $\nu^{\mathbb{N}}$ in the measurable space$(\Delta^{\mathbb{N}},\mathcal{B}(\Delta^{\mathbb{N}}))$, such that $$\nu^{\mathbb{N}} \left(A_1\times A_2\times \ldots \times A_n \times \prod_{i=n+1}^{\infty} \Delta\right)=\nu(A_1) \cdot \ldots\cdot \nu(A_n). $$

Now, let $\sigma>2/(3\sqrt{3})$ be a real number, and define $$x_-^*(\sigma) = \text{The unique real root of the polynomial }x^ 3+ \sigma = x, $$ $$x_+^*(\sigma) = \text{The unique real root of the polynomial }x^ 3- \sigma = x, $$ it is easy to see that $x_+^*(\sigma) = -x_-^*(\sigma)$.

We can then define the function $$h:\mathbb{N}\times \Delta^ \mathbb{N}\times [x_-^*(\sigma),x_+^*(\sigma)]\to [x_-^*(\sigma),x_+^*(\sigma)], $$ in the following recursive way,

• $h(0,(\omega_n)_{n},x) = x$, $\forall\ (\omega_n)_n\in \mathbb{N}$ and $\forall\ x\in \mathbb{R}$;
• $h(i+1,(\omega_n)_{n},x) = \sqrt[3]{h(i,(\omega_n)_{n},x) + \sigma \omega_i}.$

This way we are for, every $x \in \mathbb{R}$ and $(\omega_n)_n\in\Delta^\mathbb {N}$, defining the following sequence $$\left\{x, \sqrt[3]{x + \sigma \omega_1},\sqrt[3]{\sqrt[3]{x + \sigma \omega_1}+\sigma w_2},\sqrt[3]{\sqrt[3]{\sqrt[3]{x + \sigma \omega_1}+\sigma w_2} + \sigma w_3}, \ldots \right\}.$$

Now, define the following family of Markov kernels $$P_n(x,A) = \nu^{\mathbb{N}}\left(\left\{(\omega)_{n\in \mathbb{N}} \in \Delta^{\mathbb{N}};\ h(n,(\omega_n)_{n\in\mathbb{N}} ,x)\in A \right\}\right). $$

A probability measure $\mu$ in $([x_-^*(\sigma),x_+^*(\sigma)], \mathcal{B}([x_-^*(\sigma),x_+^*(\sigma)])$ is a called stationary measure if

$$\mu(A) = \int_{[x_-^*(\sigma),x_+^*(\sigma)]} P_1(x,A)\text{d}\mu(x);\ \forall \ A\in \mathcal{B}([x_-^*(\sigma),x_+^*(\sigma)]),$$ where $\mathcal{B}([x_-^*(\sigma),x_+^*(\sigma)])$ is the Borel $\sigma$-algebra. Moreover, once $[x_-^*(\sigma),x_+^*(\sigma)]$ it is easy to prove that there exists at least one stacionary measure.

The answer that I received on MO suggests that there exists only one stationary measure.

Does anyone know if this is true? A reference to such a result is enough for mine purposes.

Answer 1

This conclusion is almost certainly true, and the argument is morally true, but I don’t understand the precise argument given. In particular, what variables are the supremum and infimum being taken over? Also the argument given fails if all of the maps are the identity map (so the argument is not correct in the generality claimed).

What you’re looking at is an example of an iterated function system: you have $x_{n+1}$ is $h_{\omega_n}(x_n)$. I haven’t looked in detail, but I assume that there is an interval $J$ such that $h_\omega(x)\in J$ for all $\omega \in[-1,1]$? Assuming this, what remains is to show (ideally) that each $h_\omega$ is contracting on all of $J$. This would allow one to complete the argument given. Failing this, it would suffice to show that there exists a sequence $\delta_n\to 0$ such that $h_{\omega_1}\circ\ldots h_{\omega_n}(J)$ is of length at most $\delta_n$ for each choice of $\omega_1,\ldots\omega_n$. Failing this, I would try for an argument based on negativity of the Lyapunov Exponent.

Assuming that one of these conditions holds, you can find the unique invariant measure by looking at the distribution of $\lim_{n\to\infty}h_{\omega_{-1}}\circ\ldots\circ h_{\omega_{-n}}(0)$. The conditions above are to ensure that the limit exists.