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.