Behavioural pseudometrics for continuous-time diffusions (2312.16729v2)
Abstract: Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in a discrete-time setting where the notion of a step is fundamental. In our setting we are considering "flow"-processes emphasizing that they evolve in continuous time. In such continuous-time settings, the concepts are not straightforward adaptations of their discrete-time analogues and we restrict our study to diffusions that do not lose mass over time and with additional regularity constraints. In previous work we proposed different definitions of behavioural equivalences for continuous-time stochastic processes where the evolution is a flow through time. That work only addressed equivalences. In this work, we aim at quantifying how differently processes behave. We present two pseudometrics for diffusion-like processes. These pseudometrics are fixpoints of two different functionals on the space of 1-bounded pseudometrics on the state space. We also characterize these pseudometrics in terms of real-valued modal logics; this is a quantitative analogue of the notion of logical characterization of bisimulation. These real-valued modal logics indicate that the two pseudometrics are different and thus yield different notions of behavioural equivalence.
- R. B. Ash. Real Analysis and Probability. Academic Press, 1972.
- Principles of Model Checking. MIT Press, 2008.
- P. Billingsley. Convergence of Probability Measures. Wiley Interscience, 2nd edition, 1999.
- Bisimulation for labelled Markov processes. In Proceedings of the Twelfth IEEE Symposium On Logic In Computer Science, Warsaw, Poland., 1997.
- Adam Bobrowski. Functional analysis for probability and stochastic processes: an introduction. Cambridge University Press, 2005.
- Bisimulation for feller-dynkin processes. Electronic Notes in Theoretical Computer Science, 347:45 – 63, 2019. Proceedings of the Thirty-Fifth Conference on the Mathematical Foundations of Programming Semantics.
- Towards a classification of behavioural equivalences in continuous-time Markov processes. Electronic Notes in Theoretical Computer Science, 2020. Proceedings of the Thirty-Sixth Conference on the Mathematical Foundations of Programming Semantics.
- Behavioural equivalences for continuous-time markov processes. Mathematical Structures in Computer Science, page 1–37, 2023.
- A logical characterization of bisimulation for labelled Markov processes. In proceedings of the 13th IEEE Symposium On Logic In Computer Science, Indianapolis, pages 478–489. IEEE Press, June 1998.
- Bisimulation for labeled Markov processes. Information and Computation, 179(2):163–193, Dec 2002.
- Metrics for labeled Markov systems. In Proceedings of CONCUR99, number 1664 in Lecture Notes in Computer Science. Springer-Verlag, 1999.
- A metric for labelled Markov processes. Theoretical Computer Science, 318(3):323–354, June 2004.
- A. Einstein. The theory of the brownian movement. Ann. der Physik, 17:549, 1905.
- Approximate reasoning for real-time probabilistic processes. Logical Methods in Computer Science, 2(1):paper 4, 2006.
- Approximate reasoning for real-time probabilistic processes. In The Quantitative Evaluation of Systems, First International Conference QEST04, pages 304–313. IEEE Press, 2004.
- Brownian motion and stochastic calculus, volume 113. Springer Science and Business Media, 2012.
- Alexander S. Kechris. Classical Descriptive set Theory, volume 156 of Graduate Texts in Mathematics. Springer-Verlag, 1995.
- K. G. Larsen and A. Skou. Bisimulation through probablistic testing. Information and Computation, 94:1–28, 1991.
- R. Milner. A Calculus for Communicating Systems, volume 92 of Lecture Notes in Computer Science. Springer-Verlag, 1980.
- D. Park. Concurrency and automata on infinite sequences. In Proceedings of the 5th GI Conference on Theoretical Computer Science, number 104 in Lecture Notes In Computer Science, pages 167–183. Springer-Verlag, 1981.
- Diffusions, Markov processes and martingales: Volume 1. Foundations. Cambridge university press, 2nd edition, 2000.
- HL Royden and PM Fitzpatrick. Real analysis 4th edition. Printice-Hall Inc, Boston, 2010.
- Davide Sangiorgi. On the origins of bisimulation and coinduction. ACM Transactions on Programming Languages and Systems (TOPLAS), 31(4):15, 2009.
- Franck van Breugel and James Worrell. A behavioural pseudometric for probabilistic transition systems. Theoretical Computer Science, 331(1):115 – 142, 2005.
- Cédric Villani. Optimal transport: old and new. Springer-Verlag, 2008.
- W. Whitt. An Introduction to Stochastic-Process Limits and their Applications to Queues. Springer Series in Operations Research. Springer-Verlag, 2002.