Consider the following SDE system

$$dx_t = b(y_t)dt + dw^1_t, quad dy_t = dw^2_t.$$

Here the drift $b(cdot)$ is a smooth function that may decay slowly. $w^1_t$ and $w^2_t$ are independent standard one-dimensional Brownian motion. We can solve $x_t$ by

$$x_t = x_0 + w^1_t + int_0^tb(y_0+w^2_s)ds.$$

The invariant measure of the stochastic system exists and is absolutely continuous with respect to the Lebesgue measure of $mathbb{R}^2$.

**Question:** Can we get the decay estimate of the density of the invariant measure? How does the decay rate change if we consider the drift function $b(cdot)$ with different decay rates?