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Stream: event: MIT Categories Seminar

Topic: September 3: Prakash Panangaden's talk

Paolo Perrone (Sep 01 2020 at 18:43):

Hi all! This is the thread of discussion for Prakash' talk, "Projected limits of Markov processes".
When: Thursday September 3, 12 noon EDT (Boston time).
(Note that the time of the talk is back to usual.)

Paolo Perrone (Sep 01 2020 at 18:43):

Zoom meeting:
https://mit.zoom.us/j/280120646
Meeting ID: 280 120 646

Youtube live stream:
https://youtu.be/editZPH4lZk

Paolo Perrone (Sep 01 2020 at 18:46):

Note also that this is the third talk in a series, albeit it should be self-contained.

• The first talk has been given in the ACT@UCR seminar (thread here, video here);
  
• The second talk has been given at the Categorical Probability and Statistics workshop (thread here, video here).
  

Paolo Perrone (Sep 03 2020 at 15:40):

Hello! 20 minutes to start!

Paolo Perrone (Sep 03 2020 at 22:56):

Here's the video!
https://youtu.be/ApDkCro7-v8

Tobias Fritz (Sep 04 2020 at 11:20):

Hi @Prakash Panangaden! Unfortunately I've had to leave early yesterday, so let me ask my question here.

I wonder to what extent your result can be generalized to arbitrary diagram shapes, such as the case of continuous time. In other words, since a Markov process is a functor from $\mathbf{B}\mathbb{N}$, the category with a single object and endomorphism monoid $\mathbb{N}$, to Stoch, the Kleisli category of the Giry monad. A labelled Markov process is likewise a functor $\mathbf{B}List(L) \to \mathtt{Stoch}$, where $List(L)$ is the free monoid generated by the set of labels $L$. So does your theory of bisimulation and approximation generalize to considering categories of functors $D \to \mathtt{Stoch}$, where $D$ is an arbitrary small category, and one can define bisimulation and prove approximations for these in the same way?

I'm asking this because it could potentially be of interest in the theory of discrete groups, where approximation properties of a measure-theoretical flavour (such as amenability) are of interest, and perhaps there's some relation to your approximation results?

Prakash Panangaden (Sep 04 2020 at 11:26):

Hi Tobias,
I hope this reply reaches you. I am very interested in the continuous-time case but there are all kinds of new phenomena in that case and I am not sure if a generic argument would capture all that. I can’t really answer your question without thinking hard about it but I would think that bisimulation should generalize to the case you want.
Perhaps we can discuss this over the coming months.
Best wishes,
Prakash

Tobias Fritz (Sep 05 2020 at 06:05):

Thanks, @Prakash Panangaden! That makes sense to me. I understand that the case of continuous time comes with many other subtleties that a generic treatment would not see, so I had mentioned it only to give an obvious example. I think that the case of discrete group actions could be more interesting.