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

Recently I asked the following question about MO Attractors in Random Dynamics.

To let $ Delta $ be the interval $ (-1.1) $Then we can look at the probability space $ ( Delta, mathcal {B} ( delta), nu) $, from where $ mathcal {B} ( Delta) $ is the Borel $ sigma $algebra and $ nu $ is equal to half the Lebesgue measure.

Then we can equip the room $ 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 measurement $ nu ^ { mathbb {N}} $ in measurable space$ ( Delta ^ { mathbb {N}}, mathcal {B} ( Delta ^ { mathbb {N}})) $, so 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 it go $ sigma> 2 / (3 sqrt {3}) $ be a real number and define

$$ x _- ^ * ( sigma) = text {The unique real root of the polynomial} x ^ 3+ sigma = x, $$

$$ x _ + ^ * ( sigma) = text {The unique real root of the polynomial} x ^ 3- sigma = x, $$

that's easy to see $ 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}. $

That's how we are for everyone $ x in mathbb {R} $ and $ ( omega_n) _n in Delta ^ mathbb {N} $Define the following order

$$ left {x, sqrt (3) {x + sigma omega_1}, sqrt (3) { sqrt (3) {x + sigma omega_1} + sigma w_2}, sqrt ( 3) { sqrt (3) { sqrt (3) {x + sigma omega_1} + sigma w_2} + sigma w_3}, ldots right }. $$

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 the $ ((x _- ^ * ( sigma), x _ + ^ * ( sigma)), mathcal {B} ((x _- ^ * ( sigma), x _ + ^ * ( sigma) )) $ is a called stationary measure, if

$$ mu (A) = int _ {(x _- ^ * ( sigma), x _ + ^ * ( sigma))} P_1 (x, A) text {d} mu (x) ; forall A in mathcal {B} ((x _- ^ * ( sigma), x _ + ^ * ( sigma))), $$

from where $ mathcal {B} ((x _- ^ * ( sigma), x _ + ^ * ( sigma))) $ is the Borel $ sigma $-Algebra. Besides, once $ (x _- ^ * ( sigma), x _ + ^ * ( sigma)) $ it is easy to prove that there is at least one stakionäre measure.

The answer I received to MO suggests that there is only one stationary measure.

**Does anyone know if that's true?** An indication of such a result is sufficient for my purposes.