Intertwining and duality for consistent Markov processes

Simone Floreani; Sabine Jansen; Frank Redig; Stefan Wagner

doi:10.1214/24-EJP1124

2024 Intertwining and duality for consistent Markov processes

Simone Floreani, Sabine Jansen, Frank Redig, Stefan Wagner

Author Affiliations +

Electron. J. Probab. 29: 1-34 (2024). DOI: 10.1214/24-EJP1124

Abstract

In this paper we derive intertwining relations for a broad class of conservative particle systems both in discrete and continuous setting. Using the language of point process theory, we are able to derive a new framework in which duality and intertwining can be formulated for particle systems evolving in general spaces. These new intertwining relations are formulated with respect to factorial and orthogonal polynomials.

Our novel approach unites all the previously found self-dualities in the context of discrete consistent particle systems and provides new duality results for several interacting systems in the continuum, such as interacting Brownian motions. We also introduce a process that we call generalized inclusion process, consisting of interacting random walks in the continuum, for which our method applies and yields generalized Meixner polynomials as orthogonal self-intertwiners.

References

1.

L. Accardi and A. Boukas, Quantum probability, renormalization and infinite-dimensional*-Lie algebras, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 5 (2009), Paper 056. MR2506156L. Accardi and A. Boukas, Quantum probability, renormalization and infinite-dimensional*-Lie algebras, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 5 (2009), Paper 056. MR2506156

2.

L. Accardi, U. Franz, and M. Skeide, Renormalized Squares of White Noise and Other Non-Gaussian Noises as Lévy Processes on Real Lie Algebras, Communications in Mathematical Physics 228 (2002), no. 1, 123–150. MR1911251L. Accardi, U. Franz, and M. Skeide, Renormalized Squares of White Noise and Other Non-Gaussian Noises as Lévy Processes on Real Lie Algebras, Communications in Mathematical Physics 228 (2002), no. 1, 123–150. MR1911251

3.

W.A. Al-Salam and T.S. Chihara, Convolutions of orthonormal polynomials, SIAM Journal on Mathematical Analysis 7 (1976), no. 1, 16–28. MR0399537W.A. Al-Salam and T.S. Chihara, Convolutions of orthonormal polynomials, SIAM Journal on Mathematical Analysis 7 (1976), no. 1, 16–28. MR0399537

4.

M. Ayala, G. Carinci, and F. Redig, Quantitative Boltzmann–Gibbs principles via orthogonal polynomial duality, Journal of Statistical Physics 171 (2018), no. 6, 980–999. MR3805584M. Ayala, G. Carinci, and F. Redig, Quantitative Boltzmann–Gibbs principles via orthogonal polynomial duality, Journal of Statistical Physics 171 (2018), no. 6, 980–999. MR3805584

5.

M. Ayala, G. Carinci, and F. Redig, Higher order fluctuation fields and orthogonal duality polynomials, Electronic Journal of Probability 26 (2021), 1–35. MR4235478M. Ayala, G. Carinci, and F. Redig, Higher order fluctuation fields and orthogonal duality polynomials, Electronic Journal of Probability 26 (2021), 1–35. MR4235478

6.

Yu.M. Berezansky, Infinite-dimensional non-gaussian analysis and generalized shift operators, Functional Analysis and Its Applications 30 (1996), no. 4, 61–65. MR1444463Yu.M. Berezansky, Infinite-dimensional non-gaussian analysis and generalized shift operators, Functional Analysis and Its Applications 30 (1996), no. 4, 61–65. MR1444463

7.

Yu.M. Berezansky, Pascal measure on generalized functions and the corresponding generalized Meixner polynomials, Methods of Functional Analysis and Topology 8 (2002), no. 1, 1–13. MR1903124Yu.M. Berezansky, Pascal measure on generalized functions and the corresponding generalized Meixner polynomials, Methods of Functional Analysis and Topology 8 (2002), no. 1, 1–13. MR1903124

8.

R.M. Blumenthal and R.K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR0264757R.M. Blumenthal and R.K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR0264757

9.

D. Brockington and J. Warren, The Bethe Ansatz for Sticky Brownian Motions, Stochastic Processes and their Applications 162 (2023), 1–48. MR4584437D. Brockington and J. Warren, The Bethe Ansatz for Sticky Brownian Motions, Stochastic Processes and their Applications 162 (2023), 1–48. MR4584437

10.

G. Carinci, C. Franceschini, C. Giardinà, W. Groenevelt, and F. Redig, Orthogonal dualities of Markov processes and unitary symmetries, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 15 (2019), Paper 053. MR3980548G. Carinci, C. Franceschini, C. Giardinà, W. Groenevelt, and F. Redig, Orthogonal dualities of Markov processes and unitary symmetries, SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 15 (2019), Paper 053. MR3980548

11.

G. Carinci, C. Giardinà, C. Giberti, and F. Redig, Dualities in population genetics: A fresh look with new dualities, Stochastic Processes and their Applications 125 (2015), no. 3, 941–969. MR3303963G. Carinci, C. Giardinà, C. Giberti, and F. Redig, Dualities in population genetics: A fresh look with new dualities, Stochastic Processes and their Applications 125 (2015), no. 3, 941–969. MR3303963

12.

G. Carinci, C. Giardinà, and F. Redig, Consistent particle systems and duality, Electronic Journal of Probability 26 (2021), 1–31. MR4320949G. Carinci, C. Giardinà, and F. Redig, Consistent particle systems and duality, Electronic Journal of Probability 26 (2021), 1–31. MR4320949

13.

D.J. Daley and D. Vere-Jones, An introduction to the theory of point processes. Vol. I: Elementary theory and methods, second ed., Probability and its Applications, Springer, New York, 2003. MR1950431D.J. Daley and D. Vere-Jones, An introduction to the theory of point processes. Vol. I: Elementary theory and methods, second ed., Probability and its Applications, Springer, New York, 2003. MR1950431

14.

D.A. Dawson and A. Greven, Duality for spatially interacting Fleming-Viot processes with mutation and selection, Preprint. arXiv: 1104.1099, 2011. MR3155790D.A. Dawson and A. Greven, Duality for spatially interacting Fleming-Viot processes with mutation and selection, Preprint. arXiv: 1104.1099, 2011. MR3155790

15.

A. De Masi and E. Presutti, Mathematical methods for hydrodynamic limits, Lecture Notes in Mathematics, Springer, Berlin, 1991. MR1175626A. De Masi and E. Presutti, Mathematical methods for hydrodynamic limits, Lecture Notes in Mathematics, Springer, Berlin, 1991. MR1175626

16.

G. Di Nunno, B. Øksendal, and F. Proske, White noise analysis for Lévy processes, Journal of Functional Analysis 206 (2004), no. 1, 109–148. MR2024348G. Di Nunno, B. Øksendal, and F. Proske, White noise analysis for Lévy processes, Journal of Functional Analysis 206 (2004), no. 1, 109–148. MR2024348

17.

A. Etheridge, An Introduction to Superprocesses, AMS, University Lecture Series, 2000. MR1779100A. Etheridge, An Introduction to Superprocesses, AMS, University Lecture Series, 2000. MR1779100

18.

S.N. Ethier and T.G. Kurtz, Markov Processes, Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics, 1986. MR0838085S.N. Ethier and T.G. Kurtz, Markov Processes, Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics, 1986. MR0838085

19.

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 2 ed., vol. Volume 2, Wiley, 1971. MR0270403W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 2 ed., vol. Volume 2, Wiley, 1971. MR0270403

20.

D. Finkelshtein, Y. Kondratiev, E. Lytvynov, and M.J. Oliveira, Stirling operators in spatial combinatorics, Journal of Functional Analysis (2021), Paper 109285, 45. MR4334681D. Finkelshtein, Y. Kondratiev, E. Lytvynov, and M.J. Oliveira, Stirling operators in spatial combinatorics, Journal of Functional Analysis (2021), Paper 109285, 45. MR4334681

21.

S. Floreani, C. Giardinà, F. den Hollander, S. Nandan, and F. Redig, Switching interacting particle systems: scaling limits, uphill diffusion and boundary layer, Journal of Statistical Physics 186 (2022), Paper 33. MR4372572S. Floreani, C. Giardinà, F. den Hollander, S. Nandan, and F. Redig, Switching interacting particle systems: scaling limits, uphill diffusion and boundary layer, Journal of Statistical Physics 186 (2022), Paper 33. MR4372572

22.

S. Floreani, F. Redig, and F. Sau, Hydrodynamics for the partial exclusion process in random environment, Stochastic Processes and their Applications 142 (2021), 124–158. MR4314096S. Floreani, F. Redig, and F. Sau, Hydrodynamics for the partial exclusion process in random environment, Stochastic Processes and their Applications 142 (2021), 124–158. MR4314096

23.

S. Floreani, F. Redig, and F. Sau, Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations, Annales de l’Institut Henri Poincaré – Probabilités et Statistiques 58 (2022), no. 1, 220–247. MR4374677S. Floreani, F. Redig, and F. Sau, Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations, Annales de l’Institut Henri Poincaré – Probabilités et Statistiques 58 (2022), no. 1, 220–247. MR4374677

24.

C. Franceschini and C. Giardinà, Stochastic duality and orthogonal polynomials, Sojourns in Probability Theory and Statistical Physics-III, Springer, 2019, pp. 187–214. MR4044393C. Franceschini and C. Giardinà, Stochastic duality and orthogonal polynomials, Sojourns in Probability Theory and Statistical Physics-III, Springer, 2019, pp. 187–214. MR4044393

25.

C. Giardinà, J. Kurchan, and F. Redig, Duality and exact correlations for a model of heat conduction, Journal of Mathematical Physics 48 (2007), no. 3, 033301. MR2314497C. Giardinà, J. Kurchan, and F. Redig, Duality and exact correlations for a model of heat conduction, Journal of Mathematical Physics 48 (2007), no. 3, 033301. MR2314497

26.

C. Giardinà, J. Kurchan, F. Redig, and K. Vafayi, Duality and Hidden Symmetries in Interacting Particle Systems, Journal of Statistical Physics 135 (2009), 25–55. MR2505724C. Giardinà, J. Kurchan, F. Redig, and K. Vafayi, Duality and Hidden Symmetries in Interacting Particle Systems, Journal of Statistical Physics 135 (2009), 25–55. MR2505724

27.

W. Groenevelt, Orthogonal Stochastic Duality Functions from Lie Algebra Representations, Journal of Statistical Physics 174 (2019), no. 1, 97–119. MR3904511W. Groenevelt, Orthogonal Stochastic Duality Functions from Lie Algebra Representations, Journal of Statistical Physics 174 (2019), no. 1, 97–119. MR3904511

28.

C. Howitt and J. Warren, Consistent families of Brownian motions and stochastic flows of kernels, The Annals of Probability 37 (2009), no. 4, 1237–1272. MR2546745C. Howitt and J. Warren, Consistent families of Brownian motions and stochastic flows of kernels, The Annals of Probability 37 (2009), no. 4, 1237–1272. MR2546745

29.

O. Kallenberg, Random Measures, Theory and Applications, Springer, 2017. MR3642325O. Kallenberg, Random Measures, Theory and Applications, Springer, 2017. MR3642325

30.

J.F.C. Kingman, Completely random measures, Pacific Journal of Mathematics 21 (1967), no. 1, 59 – 78. MR0210185J.F.C. Kingman, Completely random measures, Pacific Journal of Mathematics 21 (1967), no. 1, 59 – 78. MR0210185

31.

J.F.C. Kingman, Poisson processes, Oxford Studies in Probability, vol. 3, The Clarendon Press, Oxford University Press, New York, 1993. MR1207584J.F.C. Kingman, Poisson processes, Oxford Studies in Probability, vol. 3, The Clarendon Press, Oxford University Press, New York, 1993. MR1207584

32.

C. Kipnis, C. Marchioro, and E. Presutti, Heat flow in an exactly solvable model, Journal of Statistical Physics 27 (1982), no. 1, 65–74. MR0656869C. Kipnis, C. Marchioro, and E. Presutti, Heat flow in an exactly solvable model, Journal of Statistical Physics 27 (1982), no. 1, 65–74. MR0656869

33.

R. Koekoek, P.A. Lesky, and R.F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer, Heidelberg, 2010. MR2656096R. Koekoek, P.A. Lesky, and R.F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer, Heidelberg, 2010. MR2656096

34.

Y. Kondratiev, E. Lytvynov, and M. Röckner, Equilibrium Kawasaki Dynamics Of Continuous Particle Systems, Infinite Dimensional Analysis, Quantum Probability and Related Topics 10 (2007), no. 02, 185–209. MR2337519Y. Kondratiev, E. Lytvynov, and M. Röckner, Equilibrium Kawasaki Dynamics Of Continuous Particle Systems, Infinite Dimensional Analysis, Quantum Probability and Related Topics 10 (2007), no. 02, 185–209. MR2337519

35.

Y.G. Kondratiev, T. Kuna, M.J. Oliveira, J.L. da Silva, and L. Streit, Hydrodynamic limits for the free Kawasaki dynamics of continuous particle systems, Preprint. arXiv: 0912.1312, 2009. MR2337519Y.G. Kondratiev, T. Kuna, M.J. Oliveira, J.L. da Silva, and L. Streit, Hydrodynamic limits for the free Kawasaki dynamics of continuous particle systems, Preprint. arXiv: 0912.1312, 2009. MR2337519

36.

T.J. Kozubowski and K. Podgórski, Distributional properties of the negative binomial Lévy process, Probability and Mathematical Statistics 29 (2009), no. 1, 43–71. MR2553000T.J. Kozubowski and K. Podgórski, Distributional properties of the negative binomial Lévy process, Probability and Mathematical Statistics 29 (2009), no. 1, 43–71. MR2553000

37.

G. Last, Stochastic analysis for Poisson processes, Stochastic analysis for Poisson point processes. Malliavin calculus, Wiener-Itô chaos expansions and stochastic geometry (Giovanni Peccati and Matthias Reitzner, eds.), Bocconi Springer Ser., vol. 7, Bocconi Univ. Press, 2016, pp. 1–36. MR3585396G. Last, Stochastic analysis for Poisson processes, Stochastic analysis for Poisson point processes. Malliavin calculus, Wiener-Itô chaos expansions and stochastic geometry (Giovanni Peccati and Matthias Reitzner, eds.), Bocconi Springer Ser., vol. 7, Bocconi Univ. Press, 2016, pp. 1–36. MR3585396

38.

G. Last and M. Penrose, Lectures on the Poisson Process, Institute of Mathematical Statistics Textbooks, Cambridge University Press, 2017. MR3791470G. Last and M. Penrose, Lectures on the Poisson Process, Institute of Mathematical Statistics Textbooks, Cambridge University Press, 2017. MR3791470

39.

G. Last and M.D. Penrose, Poisson process Fock space representation, chaos expansion and covariance inequalities, Probability Theory and Related Fields 150 (2011), no. 3, 663–690. MR2824870G. Last and M.D. Penrose, Poisson process Fock space representation, chaos expansion and covariance inequalities, Probability Theory and Related Fields 150 (2011), no. 3, 663–690. MR2824870

40.

Y. Le Jan and O. Raimond, Flows, coalescence and noise, The Annals of Probability 32 (2004), no. 2, 1247 – 1315. MR2060298Y. Le Jan and O. Raimond, Flows, coalescence and noise, The Annals of Probability 32 (2004), no. 2, 1247 – 1315. MR2060298

41.

A. Lenard, States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures, Archive for Rational Mechanics and Analysis 59 (1975), no. 3, 241–256. MR0391831A. Lenard, States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures, Archive for Rational Mechanics and Analysis 59 (1975), no. 3, 241–256. MR0391831

42.

T.M. Liggett, Interacting particle systems, 2 ed., Classics in mathematics, Springer, Berlin, 2005. MR2108619T.M. Liggett, Interacting particle systems, 2 ed., Classics in mathematics, Springer, Berlin, 2005. MR2108619

43.

A. Løkka and F.N. Proske, Infinite dimensional analysis of pure jump Lévy processes on the Poisson space, Mathematica Scandinavica 98 (2006), no. 2, 237–261. MR2243705A. Løkka and F.N. Proske, Infinite dimensional analysis of pure jump Lévy processes on the Poisson space, Mathematica Scandinavica 98 (2006), no. 2, 237–261. MR2243705

44.

E. Lytvynov, Orthogonal Decompositions For Lévy Processes With An Application To The Gamma, Pascal, And Meixner Processes, Infinite Dimensional Analysis, Quantum Probability and Related Topics (2003), 73–102. MR1976871E. Lytvynov, Orthogonal Decompositions For Lévy Processes With An Application To The Gamma, Pascal, And Meixner Processes, Infinite Dimensional Analysis, Quantum Probability and Related Topics (2003), 73–102. MR1976871

45.

E. Lytvynov, Polynomials of Meixner’s type in infinite dimensions—Jacobi fields and orthogonality measures, Journal of Functional Analysis 200 (2003), no. 1, 118–149. MR1974091E. Lytvynov, Polynomials of Meixner’s type in infinite dimensions—Jacobi fields and orthogonality measures, Journal of Functional Analysis 200 (2003), no. 1, 118–149. MR1974091

46.

P. Meyer, Quantum Probability for Probabilists, 2 ed., Lecture Notes in Mathematics, vol. 1538, Springer, Berlin, 1995. MR1222649P. Meyer, Quantum Probability for Probabilists, 2 ed., Lecture Notes in Mathematics, vol. 1538, Springer, Berlin, 1995. MR1222649

47.

T. Meyer-Brandis, Differential equations driven by Lévy white noise in spaces of Hilbert space-valued stochastic distributions, Stochastics. An International Journal of Probability and Stochastic Processes 80 (2008), no. 4, 371–396. MR2427538T. Meyer-Brandis, Differential equations driven by Lévy white noise in spaces of Hilbert space-valued stochastic distributions, Stochastics. An International Journal of Probability and Stochastic Processes 80 (2008), no. 4, 371–396. MR2427538

48.

D. Nualart and W. Schoutens, Chaotic and predictable representations for Lévy processes, Stochastic Processes and their Applications 90 (2000), no. 1, 109–122. MR1787127D. Nualart and W. Schoutens, Chaotic and predictable representations for Lévy processes, Stochastic Processes and their Applications 90 (2000), no. 1, 109–122. MR1787127

49.

J. Pitman, Combinatorial stochastic processes, Lecture Notes in Mathematics, vol. 1875, Springer, Berlin, 2006, Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard. MR2245368J. Pitman, Combinatorial stochastic processes, Lecture Notes in Mathematics, vol. 1875, Springer, Berlin, 2006, Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard. MR2245368

50.

F. Redig and F. Sau, Factorized Duality, Stationary Product Measures and Generating Functions, Journal of Statistical Physics 172 (2018), no. 4, 980–1008. MR3830295F. Redig and F. Sau, Factorized Duality, Stationary Product Measures and Generating Functions, Journal of Statistical Physics 172 (2018), no. 4, 980–1008. MR3830295

51.

E. Schertzer, R. Sun, and J.M. Swart, The Brownian web, the Brownian net, and their universality, Advances in disordered systems, random processes and some applications, Cambridge Univ. Press, Cambridge, 2017, pp. 270–368. MR3644280E. Schertzer, R. Sun, and J.M. Swart, The Brownian web, the Brownian net, and their universality, Advances in disordered systems, random processes and some applications, Cambridge Univ. Press, Cambridge, 2017, pp. 270–368. MR3644280

52.

W. Schoutens, Stochastic processes and orthogonal polynomials, Lecture Notes in Statistics, vol. 146, Springer, New York, 2000. MR1761401W. Schoutens, Stochastic processes and orthogonal polynomials, Lecture Notes in Statistics, vol. 146, Springer, New York, 2000. MR1761401

53.

R.F. Serfozo, Point processes, Stochastic models, Handbooks in operations research and management science, vol. 2, North-Holland, Amsterdam, 1990, pp. 1–93. MR1100748R.F. Serfozo, Point processes, Stochastic models, Handbooks in operations research and management science, vol. 2, North-Holland, Amsterdam, 1990, pp. 1–93. MR1100748

54.

D. Surgailis, On multiple Poisson stochastic integrals and associated Markov semigroups, Probability and Mathematical Statistics 3 (1984), no. 2, 217–239. MR0764148D. Surgailis, On multiple Poisson stochastic integrals and associated Markov semigroups, Probability and Mathematical Statistics 3 (1984), no. 2, 217–239. MR0764148

55.

A. Yablonski, The Calculus of Variations for Processes with Independent Increments, Rocky Mountain Journal of Mathematics 38 (2008), no. 2, 669 – 701. MR2401574A. Yablonski, The Calculus of Variations for Processes with Independent Increments, Rocky Mountain Journal of Mathematics 38 (2008), no. 2, 669 – 701. MR2401574

56.

S. Wagner, Orthogonal Intertwiners for Infinite Particle Systems In The Continuum, Stochastic Processes and their Applications 168 (2024), Paper 104269. MR4672704S. Wagner, Orthogonal Intertwiners for Infinite Particle Systems In The Continuum, Stochastic Processes and their Applications 168 (2024), Paper 104269. MR4672704

Creative Commons Attribution 4.0 International License.

Citation Download Citation

Simone Floreani, Sabine Jansen, Frank Redig, and Stefan Wagner "Intertwining and duality for consistent Markov processes," Electronic Journal of Probability 29(none), 1-34, (2024). https://doi.org/10.1214/24-EJP1124

Received: 17 November 2022; Accepted: 15 April 2024; Published: 2024

Access the abstract

JOURNAL ARTICLE
34 PAGES

DOWNLOAD PDF + SAVE TO MY LIBRARY