In stochastic processes, like homogeneous Markov processes, Poisson processes, Queueing systems etc., the functions that represent (transition) probabilities are continuous over time. This is also very convenient mathematically, since it allows further useful analysis to be done, such as transition rates in Markov processes.

I recently read a paper ( https://arxiv.org/abs/1811.07401 ) where a stochastic process is described, with probabilities being discontinuous at an infinite (!!) number of points. According to the paper, it shows the problematic nature of a certain class of algorithms with which the process is associated.

Are there any continuous-time stochastic processes with discontinuous transition probabilities or is it fundamentally incompatible with mathematical/physical reality?? I would like an answer for stochastic processes describing real phenomena/systems, not for theoretically/artificially constructed processes.


Certainly there are. In the abstract, what defines a Markov process is that the transition probabilities (or rates, in continuous time) do not depend on the history; this is usually abbreviated by stating that the probabilities only depend on the current state, but they can also depend on time.

As a very concrete example of a system with this property, you can take the Markov chain that is used to describe the interaction of a atomic wave function with the external electromagnetic field, as described by Jean Dalibard, Yvan Castin, and Klaus Mølmer, "Wave-function approach to dissipative processes in quantum optics," Phys. Rev. Lett. 68, 580 (1992). The complicated density matrix for a multi-level atom, evolving in time according to the optical Bloch equations, is replaced with a wave function that evolves according to a Markov process. Under the stochastic evolution, spontaneous decay appears (just as it seems to appear with real atoms) as a random process. This formalism can easily accommodate external forcing by one or more lasers, which may be turned on or off at any given time. And obviously, switching off the external pump lasers will change the transition rates between different state in a discontinuous fashion.

  • $\begingroup$ Thanks for your answer, Buzz! It's always about quantum mechanics, isn't it? $\endgroup$ – Robert_Lewis Dec 19 '18 at 0:24
  • $\begingroup$ Let me just ask a couple of questions: doesn't turning the lasers on or off bring a deterministic element to the process, thus affecting the randomness? and when you say "will change the transition rates between different state in a discontinuous fashion", what about transition probabilities? $\endgroup$ – Robert_Lewis Dec 19 '18 at 0:35
  • $\begingroup$ @Robert_Lewis Yes, turning on a laser changes both the deterministic and probabilistic parts of the time evolution. $\endgroup$ – Buzz Dec 19 '18 at 3:43

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