I'm experimenting with this type of stochastic process and I'm wondering if there is a specific name for it. So far I've described it as a discrete-time continuous-state Markov process but curious to see if there's a special name. I have random variables $X_t$ where $t \in \mathbb{N}$ on the state-space $\mathbb{R}$, given by:
$X_{t+1}= \begin{cases} 0, & \text{with prob.}\ \alpha \\ X_t +\mu, & \text{with prob.}\ 1-\alpha \end{cases}$
where $\mu \in \mathbb{R}$. Thanks in advance!