## Bernoulli

• Bernoulli
• Volume 19, Number 5A (2013), 1855-1879.

### Intertwining and commutation relations for birth–death processes

#### Abstract

Given a birth–death process on $\mathbb{N}$ with semigroup $(P_{t})_{t\geq0}$ and a discrete gradient ${\partial}_{u}$ depending on a positive weight $u$, we establish intertwining relations of the form ${\partial}_{u}P_{t}=Q_{t}\,{\partial}_{u}$, where $(Q_{t})_{t\geq0}$ is the Feynman–Kac semigroup with potential $V_{u}$ of another birth–death process. We provide applications when $V_{u}$ is nonnegative and uniformly bounded from below, including Lipschitz contraction and Wasserstein curvature, various functional inequalities, and stochastic orderings. Our analysis is naturally connected to the previous works of Caputo–Dai Pra–Posta and of Chen on birth–death processes. The proofs are remarkably simple and rely on interpolation, commutation, and convexity.

#### Article information

Source
Bernoulli, Volume 19, Number 5A (2013), 1855-1879.

Dates
First available in Project Euclid: 5 November 2013

https://projecteuclid.org/euclid.bj/1383661206

Digital Object Identifier
doi:10.3150/12-BEJ433

Mathematical Reviews number (MathSciNet)
MR3129037

Zentralblatt MATH identifier
1286.60084

#### Citation

Chafaï, Djalil; Joulin, Aldéric. Intertwining and commutation relations for birth–death processes. Bernoulli 19 (2013), no. 5A, 1855--1879. doi:10.3150/12-BEJ433. https://projecteuclid.org/euclid.bj/1383661206

#### References

• [1] Ané, C. (2001). Clark–Ocone formulas and Poincaré inequalities on the discrete cube. Ann. Inst. Henri Poincaré Probab. Stat. 37 101–137.
• [2] Bakry, D. (1997). On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. In New Trends in Stochastic Analysis (Charingworth, 1994) 43–75. River Edge, NJ: World Sci. Publ.
• [3] Bakry, D. and Émery, M. (1985). Diffusions hypercontractives. In Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Math. 1123 177–206. Berlin: Springer.
• [4] Barbour, A.D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. New York: The Clarendon Press/Oxford Univ. Press.
• [5] Barbour, A.D. and Xia, A. (2006). On Stein’s factors for Poisson approximation in Wasserstein distance. Bernoulli 12 943–954.
• [6] Bobkov, S.G. and Ledoux, M. (1998). On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 347–365.
• [7] Bobkov, S.G. and Tetali, P. (2006). Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19 289–336.
• [8] Brègman, L.M. (1967). The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics 7 200–217.
• [9] Brown, T.C. and Xia, A. (2001). Stein’s method and birth–death processes. Ann. Probab. 29 1373–1403.
• [10] Caputo, P., Dai Pra, P. and Posta, G. (2009). Convex entropy decay via the Bochner–Bakry–Emery approach. Ann. Inst. Henri Poincaré Probab. Stat. 45 734–753.
• [11] Chafaï, D. (2004). Entropies, convexity, and functional inequalities: On $\Phi$-entropies and $\Phi$-Sobolev inequalities. J. Math. Kyoto Univ. 44 325–363.
• [12] Chafaï, D. (2006). Binomial-Poisson entropic inequalities and the $M/M/\infty$ queue. ESAIM Probab. Stat. 10 317–339 (electronic).
• [13] Chen, M. (1996). Estimation of spectral gap for Markov chains. Acta Math. Sinica (N.S.) 12 337–360.
• [14] Chen, M.F. (2004). From Markov Chains to Non-equilibrium Particle Systems, 2nd ed. River Edge, NJ: World Scientific.
• [15] Chen, M.F. (2010). Speed of stability for birth–death processes. Front. Math. China 5 379–515.
• [16] Chen, M.F. and Wang, F.Y. (1997). Estimation of spectral gap for elliptic operators. Trans. Amer. Math. Soc. 349 1239–1267.
• [17] Gao, F., Guillin, A. and Wu, L. (2010). Bernstein type’s concentration inequalities for symmetric Markov processes. Preprint.
• [18] Guillin, A., Léonard, C., Wu, L. and Yao, N. (2009). Transportation-information inequalities for Markov processes. Probab. Theory Related Fields 144 669–695.
• [19] Helffer, B. (2002). Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics. Series in Partial Differential Equations and Applications 1. River Edge, NJ: World Scientific.
• [20] Joulin, A. (2009). A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature. Bernoulli 15 532–549.
• [21] Joulin, A. and Privault, N. (2004). Functional inequalities for discrete gradients and application to the geometric distribution. ESAIM Probab. Stat. 8 87–101.
• [22] Ledoux, M. (2000). The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6) 9 305–366.
• [23] Liu, W. and Ma, Y. (2009). Spectral gap and convex concentration inequalities for birth–death processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 58–69.
• [24] Ma, Y., Wang, R. and Wu, L. (2011). Transportation-information inequalities for continuum Gibbs measures. Electron. Commun. Probab. 16 600–613.
• [25] Malrieu, F. and Talay, D. (2006). Concentration inequalities for Euler schemes. In Monte Carlo and Quasi-Monte Carlo Methods 2004 355–371. Berlin: Springer.
• [26] Robert, P. (2003). Stochastic Networks and Queues, french ed. Applications of Mathematics (New York) 52. Berlin: Springer.
• [27] Schuhmacher, D. (2009). Stein’s method and Poisson process approximation for a class of Wasserstein metrics. Bernoulli 15 550–568.
• [28] Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Chichester: Wiley.
• [29] van Doorn, E.A. (2003). On associated polynomials and decay rates for birth–death processes. J. Math. Anal. Appl. 278 500–511.
• [30] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Berlin: Springer.
• [31] Wu, L. (2006). Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition. Ann. Probab. 34 1960–1989.