A particle continuously moves in $\mathbb{R}^+ = \{x \in \mathbb{R}, x\ge 0\}$. It starts from $0$ at time $0$, and it has a restart probability defined by a pdf on $\mathbb{R}^+$. If it is at $0$ at time $t$, it samples a restart delay $\delta t$ according to this pdf; it is again at $0$ at time $t+\delta t$ (restart), and it is at $x$ at time $t+x$ for all $x$ in $[0,\delta t)$.
I am looking for the master equation of the position of this particle over time, without any assumption on the restart pdf.
I am also interested in a discrete-time, discrete-space random process that converges to the same distribution when the discrete steps go to $0$.
References and suggestions are welcome too!