Existence of a fixed point for the stochastic Goodwin-Lotka-Volterra return map

Let $(x_t,y_t)$ be a solution of the stochastic Goodwin-Lotka-Volterra system with initial condition

$$(x_0,y_0)=(y/\tilde\theta,y),\qquad y\in(0,\tilde y),$$

and let $S$ be the finite stopping time defined by the cited theorem. Define the expected return map

$$\mathcal S:y\in(0,\tilde y)\longmapsto \mathbb E[y_S]\in(0,\tilde y).$$

Fixed-point conjecture. The map $\mathcal S$ has at least one fixed point in $(0,\tilde y)$.

The conjecture asserts the existence of a stochastic orbit whose expected crossing value after one loop equals its initial value. The surrounding discussion motivates this through numerical simulations suggesting convergence of the expected return values and periods, while a proof would require suitable finiteness and recurrence estimates for the stopping time.