TITLE: A singular stochastic differential equation
QUESTION [7 upvotes]: We consider the following SDE:
$$dX_t = 1(X_t = 0) \, dt + 1(X_t >0) \, dB_t, \quad X_0= x > 0,$$
where $(B_t, \, t \ge 0)$ is linear Brownian motion.
Let $\tau: = \inf\{t >0: X_t = 0\}$ be the first time at which $X$ hits $0$.
It is obvious that $(X_t, \, 0 \le t \le \tau)$ is Brownian motion up to the first hitting to $0$.
Question: Can we say something for $X$ after $\tau$? Is it well-defined?
It is clear that $X$ is not reflected Brownian motion but is also supported on $[0,\infty)$. It seems that there is accumulation of zeros after $\tau$ which would lead to local times.

REPLY [8 votes]: This is sticky reflecting Brownian motion, see for example this relatively recent paper. You can alternatively construct it by taking a reflected Brownian motion and then "stretching out" the local time accumulated at the origin, turning it into real time. Different constants in front of the $dt$ term yield different stretching factors.