# Self-Duality of Markov Processes and Intertwining Functions

@article{Franceschini2018SelfDualityOM, title={Self-Duality of Markov Processes and Intertwining Functions}, author={Chiara Franceschini and Cristian Giardin{\`a} and Wolter G. M. Groenevelt}, journal={Mathematical Physics, Analysis and Geometry}, year={2018}, volume={21}, pages={1-21} }

We present a theorem which elucidates the connection between self-duality of Markov processes and representation theory of Lie algebras. In particular, we identify sufficient conditions such that the intertwining function between two representations of a certain Lie algebra is the self-duality function of a (Markov) operator. In concrete terms, the two representations are associated to two operators in interwining relation. The self-dual operator, which arise from an appropriate symmetric…

## 11 Citations

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## References

SHOWING 1-10 OF 53 REFERENCES

Duality and Hidden Symmetries in Interacting Particle Systems

- Mathematics
- 2009

In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in…

Orthogonal Stochastic Duality Functions from Lie Algebra Representations

- MathematicsJournal of statistical physics
- 2019

Stochastic duality functions for specific Markov processes are obtained using representation theory of Lie algebras using representations of the Heisenberg algebra andsu(1,1) for orthogonal (self-)duality functions in terms of hypergeometric functions for Specific interacting particle processes and interacting diffusion processes.

Stochastic Duality and Orthogonal Polynomials

- MathematicsSpringer Proceedings in Mathematics & Statistics
- 2019

For a series of Markov processes we prove stochastic duality relations with duality functions given by orthogonal polynomials. This means that expectations with respect to the original process (which…

Duality functions and stationary product measures

- Mathematics
- 2017

We investigate a general relation between stationary product measures and factorized (self-)duality functions. This yields a constructive approach to find (self-)duality functions from the stationary…

Duality for Stochastic Models of Transport

- Mathematics
- 2013

We study three classes of continuous time Markov processes (inclusion process, exclusion process, independent walkers) and a family of interacting diffusions (Brownian energy process). For each model…

Duality relations for asymmetric exclusion processes

- Mathematics
- 1997

We derive duality relations for a class ofUq[SU(2)]-symmetric stochastic processes, including among others the asymmetric exclusion process in one dimension. Like the known duality relations for…

Stochastic Higher Spin Vertex Models on the Line

- Mathematics
- 2015

We introduce a four-parameter family of interacting particle systems on the line, which can be diagonalized explicitly via a complete set of Bethe ansatz eigenfunctions, and which enjoy certain…

From duality to determinants for q-TASEP and ASEP

- Mathematics
- 2014

We prove duality relations for two interacting particle systems: the $q$-deformed totally asymmetric simple exclusion process ($q$-TASEP) and the asymmetric simple exclusion process (ASEP).…