How is called a stationnary stochastic process consisting in Poissonian jumps between Gaussian random variables?

Question

I have a (physical) signal $x(t)$ that is well described by a random process consisting in constant values $x_i$ during random times $\tau_i$. The time intervals $\tau_i$ between jumps are exponentially distributed (so the jumps constitute a Poisson process) and at every jump $i$, a new value $x_i$ is picked from a Gaussian distribution (so without correlation to any previous value). The whole process is therefore stationnary, and has exponentially decreasing autocorrelation - but this is not a Gauss-Markov (or Ornstein–Uhlenbeck) process. I did not find studies of such process, and would not even know how to name it. Is it a kind of continuous-time Markov chain? (But it has continuous values.) Maybe some sort of renewal process. I would be grateful if someone knows the name of such random signal, and possibly some documentation about it.