{"id":989,"date":"2010-11-10T16:16:59","date_gmt":"2010-11-10T14:16:59","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=989"},"modified":"2019-11-10T19:29:33","modified_gmt":"2019-11-10T18:29:33","slug":"intertwining-and-commutation-relations-for-birth-death-processes","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2010\/11\/10\/intertwining-and-commutation-relations-for-birth-death-processes\/","title":{"rendered":"Intertwining and commutation relations for birth-death processes"},"content":{"rendered":"

I have posted today on arXiv<\/a> a paper entitled Intertwining and commutation relations for birth-death processes<\/strong>, joint work with Ald\u00e9ric Joulin<\/a>.<\/p>\n

Given a birth-death process on \\( {\\mathbb{N}} \\) with semigroup \\( {(P_t)_{t\\geq 0}} \\) 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\\geq 0}} \\) is the Feynman-Kac semigroup with potential \\( {V_u} \\) of another birth-death process. We provide applications when \\( {V_u} \\) is positive and uniformly bounded from below, including Lipschitz contraction and Wasserstein curvature, various functional inequalities, and stochastic orderings. The proofs are remarkably simple and rely on interpolation, commutation, and convexity.<\/p>\n

Let us give the main ingredient. We consider a birth-death process \\( {(X_t)_{t\\geq 0}} \\) on the state space \\( {\\mathbb{N} := \\{ 0,1,2, \\ldots \\}} \\), i.e. a Markov process with transition probabilities given by<\/p>\n

\\[ P_t^x (y) = \\mathbb{P}_x (X_t =y) = (\\lambda_x t)\\mathbf{1}_{y=x+1} +(\\nu_x t)\\mathbf{1}_{y=x-1} +(1- (\\lambda_x + \\nu_x) t)\\mathbf{1}_{y=x} + t\\varepsilon(t). \\]<\/p>\n

The transition rates \\( {\\lambda} \\) and \\( {\\nu} \\) are respectively called the birth and death rates of the process \\( {(X_t)_{t\\geq 0}} \\). We assume that the process is irreducible, positive recurrent, and non-explosive. This holds when the rates satisfy to \\( {\\lambda>0} \\) on \\( {\\mathbb{N}} \\) and \\( {\\nu>0} \\) on \\( {\\mathbb{N}^*} \\) and \\( {\\nu_0 = 0} \\) and<\/p>\n

\\[ \\sum_{x=1}^\\infty \\frac{\\lambda_0 \\lambda_1 \\cdots \\lambda_{x-1}}{\\nu_1 \\nu_2 \\cdots \\nu_x} <\\infty \\quad\\text{and}\\quad \\sum_{x=1}^\\infty \\left(\\frac{1}{\\lambda_x}+\\frac{\\nu_x}{\\lambda_x\\lambda_{x-1}} +\\cdots+\\frac{\\nu_x\\cdots\\nu_1}{\\lambda_x\\cdots\\lambda_1\\lambda_0}\\right) = \\infty. \\]<\/p>\n

The unique stationary distribution \\( {\\mu} \\) of the process is reversible and is given by <\/a><\/p>\n

\\[ \\mu (x) = \\mu (0) \\prod_{y=1}^x \\frac{\\lambda_{y-1}}{\\nu_y} ,\\ x\\in\\mathbb{N} \\quad \\text{with} \\quad \\mu (0) := \\left(1+\\sum_{x=1}^\\infty \\frac{\\lambda_0\\lambda_1\\cdots\\lambda_{x-1}}{\\nu_1\\nu_2\\cdots\\nu_x}\\right)^{-1} . \\ \\ \\ \\ \\ (1) \\]<\/p>\n

Let us denote by \\( {\\mathcal{F}} \\) (respectively \\( {\\mathcal{F}_{\\!\\!+}} \\)) the space of real-valued (respectively positive) functions \\( {f} \\) on \\( {\\mathbb{N}} \\), and let \\( {b\\mathcal{F}} \\) be the subspace of bounded functions. The associated semigroup \\( {(P_t )_{t\\geq 0}} \\) is defined for any function \\( {f\\in b\\mathcal{F} \\cup \\mathcal{F}_+} \\) and \\( {x\\in\\mathbb{N}} \\) as<\/p>\n

\\[ P_t f (x) = \\mathbb{E}_x [f(X_t)] = \\sum_{y=0}^\\infty f(y) P_t^x (y). \\]<\/p>\n

This family of operators is positivity preserving and contractive on \\( {L^p (\\mu)} \\), \\( {p\\in [1,\\infty]} \\). Moreover, the semigroup is also symmetric in \\( {L^2(\\mu)} \\) since \\( {\\lambda_x\\mu(x) = \\nu_{1+x}\\mu(1+x)} \\) for any \\( {x\\in\\mathbb{N}} \\) (detailed balance equation). The generator \\( {\\mathcal{L}} \\) of the process is given for any \\( {f\\in \\mathcal{F}} \\) and \\( {x\\in\\mathbb{N}} \\) by<\/p>\n

\\[ \\mathcal{L} f(x) = \\lambda_x \\, \\left( f(x+1) -f(x)\\right) + \\nu_x \\, \\left( f(x-1) -f(x)\\right) = \\lambda_x \\, \\partial f (x) + \\nu_x \\, \\partial^* f(x), \\]<\/p>\n

where \\( {\\partial } \\) and \\( {\\partial^*} \\) are respectively the forward and backward discrete gradients on \\( {\\mathbb{N}} \\):<\/p>\n

\\[ \\partial f(x) := f(x+1)-f(x) \\quad \\text{and} \\quad \\partial^* f(x) := f(x-1)-f(x) . \\]<\/p>\n

Our approach is inspired from the remarkable properties of two special birth-death processes: the \\( {M\/M\/1} \\) and the \\( {M\/M\/\\infty} \\) queues. The \\( {M\/M\/\\infty} \\) queue has rates \\( {\\lambda_x=\\lambda} \\) and \\( {\\nu_x=\\nu x} \\) for positive constants \\( {\\lambda} \\) and \\( {\\nu} \\). It is positive recurrent and its stationary distribution is the Poisson measure \\( {\\mu_\\rho} \\) with mean \\( {\\rho=\\lambda\/\\mu} \\). If \\( {\\mathcal{B}_{x,p}} \\) stands for the binomial distribution of size \\( {x\\in\\mathbb{N}} \\) and parameter \\( {p \\in [0,1]} \\), the \\( {M\/M\/\\infty} \\) process satisfies for every \\( {x\\in\\mathbb{N}} \\) and \\( {t\\geq0} \\) to the Mehler type formula <\/a><\/p>\n

\\[ \\mathcal{L} (X_t |X_0 = x) = \\mathcal{B}_{x, e^{-\\nu t}} \\ast \\mu_{\\rho (1-e^{-\\nu t})}. \\ \\ \\ \\ \\ (2) \\]<\/p>\n

The \\( {M\/M\/1} \\) queueing process has rates \\( {\\lambda_x=\\lambda} \\) and \\( {\\nu_x=\\nu \\mathbf{1}_{\\mathbb{N}\\setminus\\{0\\}}} \\) where \\( {0<\\lambda<\\nu} \\) are constants. It is a positive recurrent random walk on \\( {\\mathbb{N}} \\) reflected at \\( {0} \\). Its stationary distribution \\( {\\mu} \\) is the geometric measure with parameter \\( {\\rho := \\lambda \/\\nu} \\) given by \\( {\\mu (x) = (1-\\rho)\\rho^x} \\) for all \\( {x\\in \\mathbb{N}} \\). A remarkable common property shared by the \\( {M\/M\/1} \\) and \\( {M\/M\/\\infty} \\) processes is the intertwining relation <\/a><\/p>\n

\\[ \\partial \\mathcal{L} = \\mathcal{L}^{V} \\partial \\ \\ \\ \\ \\ (3) \\]<\/p>\n

where \\( {\\mathcal{L}^{V}=\\mathcal{L}-V} \\) is the discrete Schr\u00f6dinger operator with potential \\( {V} \\) given by<\/p>\n