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. For example, $|b(x)| \le C/|x|^\sigma$ for some $\sigma > 0$ as $|x| \to \infty$. $w^1_t$ and $w^2_t$ are independent standard one-dimensional Brownian motion. We can solve $x_t$ by $$x_t = x_0 + w^1_t + \int_0^tb(y_0+w^2_s)ds.$$ Let $p(t, \mathbf{x}, \mathbf{y})$ denote the transition density of this process, that is, the probability density of this process starting at the point $\mathbf{x}$ at time $0$ and reaching the point $\mathbf{y}$ at time $t$.

**Question:** Can we get the decay estimate of the transition density $p(t, \mathbf{x}, \mathbf{y})$? For example, can we get Gaussian bound for the density similar to the situation when $b$ is identically zero? How does the decay rate change if we consider the drift function $b(\cdot)$ with different decay rates?

1more comment