I have posted today on arXiv a paper entitled Intertwining and commutation relations for birth-death processes, joint work with Aldéric Joulin.
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.
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
\[ P_t^x (y) = \mathbb{P}_x (X_t =y) = (\lambda_x t)\mathbf{1}_{y=x+1} +(\nu_x t)\mathbf{1}_{y=x-1} +(1- (\lambda_x + \nu_x) t)\mathbf{1}_{y=x} + t\varepsilon(t). \]
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
\[ \sum_{x=1}^\infty \frac{\lambda_0 \lambda_1 \cdots \lambda_{x-1}}{\nu_1 \nu_2 \cdots \nu_x} <\infty \quad\text{and}\quad \sum_{x=1}^\infty \left(\frac{1}{\lambda_x}+\frac{\nu_x}{\lambda_x\lambda_{x-1}} +\cdots+\frac{\nu_x\cdots\nu_1}{\lambda_x\cdots\lambda_1\lambda_0}\right) = \infty. \]
The unique stationary distribution \( {\mu} \) of the process is reversible and is given by
\[ \mu (x) = \mu (0) \prod_{y=1}^x \frac{\lambda_{y-1}}{\nu_y} ,\ x\in\mathbb{N} \quad \text{with} \quad \mu (0) := \left(1+\sum_{x=1}^\infty \frac{\lambda_0\lambda_1\cdots\lambda_{x-1}}{\nu_1\nu_2\cdots\nu_x}\right)^{-1} . \ \ \ \ \ (1) \]
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
\[ P_t f (x) = \mathbb{E}_x [f(X_t)] = \sum_{y=0}^\infty f(y) P_t^x (y). \]
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
\[ \mathcal{L} f(x) = \lambda_x \, \left( f(x+1) -f(x)\right) + \nu_x \, \left( f(x-1) -f(x)\right) = \lambda_x \, \partial f (x) + \nu_x \, \partial^* f(x), \]
where \( {\partial } \) and \( {\partial^*} \) are respectively the forward and backward discrete gradients on \( {\mathbb{N}} \):
\[ \partial f(x) := f(x+1)-f(x) \quad \text{and} \quad \partial^* f(x) := f(x-1)-f(x) . \]
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
\[ \mathcal{L} (X_t |X_0 = x) = \mathcal{B}_{x, e^{-\nu t}} \ast \mu_{\rho (1-e^{-\nu t})}. \ \ \ \ \ (2) \]
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
\[ \partial \mathcal{L} = \mathcal{L}^{V} \partial \ \ \ \ \ (3) \]
where \( {\mathcal{L}^{V}=\mathcal{L}-V} \) is the discrete Schrödinger operator with potential \( {V} \) given by
- \( {V(x) := \nu} \) in the case of the \( {M/M/\infty} \) queue
- \( {V(x) := \nu \mathbf{1}_{\{0\}}(x)} \) for the \( {M/M/1} \) queue.
The operator \( {\mathcal{L} ^{V}} \) is the generator of a Feynman-Kac semigroup \( {(P_t^{V})_{t\geq 0}} \) given by
\[ P_t^{V} f(x) = \mathbb{E}_x \left[ f(X_t) \exp \left(-\int_0^t V(X_s) ds \right) \right]. \]
The intertwining relation (3) is the infinitesimal version at time \( {t=0} \) of the semigroup intertwining
\[ \partial P_t f (x) = P_t^{V} \partial f (x) = \mathbb{E}_x \left[ \partial f(X_t) \, \exp \left( – \int_0^t V(X_s) \, ds \right)\right] . \ \ \ \ \ (4) \]
Conversely, one may deduce (4) from (3) by using a semigroup interpolation. Namely, if we consider
\[ s\in[0,t]\mapsto J(s) := P_s^{V} \partial P_{t-s} f \]
with \( {V} \) as above, then (4) rewrites as \( {J(0) = J(t)} \) and (4) follows from (3) since
\[ J'(s) = P_s^{V} \left( \mathcal{L}^{V} \partial P_{t-s} f – \partial \mathcal{L} P_{t-s} f \right) =0. \]
Let us fix some \( {u \in \mathcal{F}_{\!\!+}} \). The \( {u} \)-modification of the original process \( {(X_t)_{t\geq 0}} \) is a birth-death process \( {(X_{u, t})_{t\geq 0}} \) with semigroup \( {(P_{u,t})_{t\geq 0}} \) and generator \( {\mathcal{L}_u} \) given by
\[ \mathcal{L}_u f(x) = \lambda^u_x \, \partial f (x) + \nu^u_x \, \partial^* f(x), \]
where the birth and death rates are respectively given by
\[ \lambda^u_x := \frac{u_{x+1}}{u_x} \, \lambda_{x+1} \quad\text{and}\quad \nu^u_x := \frac{u_{x-1}}{u_x} \, \nu_x . \]
One can check that the measure \( {\lambda u^2\mu} \) is reversible for \( {(X_{u,t})_{t\geq0}} \). As consequence, the process \( {(X_{u,t})_{t\geq0}} \) is positive recurrent if and only if \( {\lambda u^2} \) is \( {\mu} \)-integrable. We define the discrete gradient \( {\partial_u} \) and the potential \( {V_u} \) by
\[ \partial_u := (1/u)\partial \quad\text{and}\quad V_u (x) := \nu_{x+1} – \nu^u_x +\lambda_x – \lambda^u_x. \]
Let \( {\varphi : \mathbb{R}\rightarrow\mathbb{R}_+} \) be a smooth convex function such that for some constant \( {c>0} \), and for all \( {r\in\mathbb{R}} \),
\[ \varphi ‘(r)r \geq c\varphi (r). \ \ \ \ \ (5) \]
In particular, \( {\varphi} \) vanishes at \( {0} \), is non-increasing on \( {(-\infty , 0)} \) and non-decreasing on \( {(0,\infty)} \). Moreover, the behavior at infinity is at least polynomial of degree \( {c} \). Note that one can easily find a sequence of such functions converging pointwise to the absolute value \( {\left|\cdot\right|} \).
Theorem 1 (Intertwining and sub-commutation) Assume that for every \( {x\in\mathbb{N}} \) and \( {t\geq0} \), we have
\[ \mathbb{E} _x \left[ \exp \left( – \int_0^t V_u (X_{u,s}) \, ds \right)\right] <\infty . \]
Then for every \( {f\in b\mathcal{F}} \), \( {x\in\mathbb{N}} \), \( {t\geq0} \),
\[ \partial_u P_t f (x) \, = \, P_{u,t}^{V_u} \partial_u f (x) \, = \, \mathbb{E} _x \left[ \partial_u f(X_{u,t}) \, \exp \left( – \int_0^t V_u (X_{u,s}) \, ds \right)\right]. \ \ \ \ \ (6) \]
Moreover, if \( {V_u\geq0} \) then for every \( {f\in b\mathcal{F}} \), \( {x\in\mathbb{N}} \), \( {t\geq0} \),
\[ \varphi \left( \partial_u P_t f \right)(x) \leq \mathbb{E}_x \left[ \varphi( \partial_u f) (X_{u,t}) \, \exp \left( – \int_0^t c V_u ( X_{u,s}) \, ds \right)\right] . \ \ \ \ \ (7) \]
Proof: Let us prove (7). If we define
\[ s\in[0,t]\mapsto J(s) := P_{u,s}^{cV_u} \varphi (\partial_u P_{t-s} f) \]
then (7) rewrites as \( {J(0) \leq J(t)} \). Hence it suffices to show that \( {J} \) is non-decreasing. We have the intertwining relation
\[ \partial_u \mathcal{L} = \mathcal{L}_u^{V_u} \partial_u, \ \ \ \ \ (8) \]
where \( {\mathcal{L}_u} \) is the generator of the \( {u} \)-modification process \( {(X_{u,t})_{t\geq 0}} \) and where
\[ \mathcal{L}_u^{V_u}:=\mathcal{L}_u-V_u. \]
Now
\[ J'(s) = P_{u,s} ^{cV_u} (T) \quad\text{where}\quad T = \mathcal{L}_u^{cV_u} \varphi (\partial_u P_{t-s} f) – \varphi ‘(\partial_u P_{t-s} f)\, \partial_u \mathcal{L} P_{t-s}f . \]
Letting \( {g_u = \partial_u P_{t-s} f} \), we obtain, by using (8),
\[ T = \mathcal{L}_u^{cV_u} \varphi (g_u) – \varphi ‘(g_u) \mathcal{L}_u^{V_u} g_u \]
and thus
\[ T = \lambda^u \left( \partial \varphi (g_u) – \varphi ‘(g_u)\partial g_u \right) + \nu^u \left( \partial^* \varphi (g_u) – \varphi ‘(g_u)\partial^* g_u \right) + V_u \left( \varphi ‘(g_u) g_u – c\varphi (g_u)\right). \]
Now (5) and \( {V_u\geq0} \) give \( {T\geq0} \). Since the Feynman-Kac semigroup \( {(P_{u,t}^{cV_u})_{t\geq 0}} \) is positivity preserving, we get (7). The proof of (6) is similar but simpler (\( {T} \) is identically zero). $\Box$
The identity (6) implies a propagation of monotonicity: if \( {f} \) is non-increasing then \( {P_tf} \) is also non-increasing.
Actually, the intertwining relations above have their counterpart in continuous state space. Let \( {\mathcal{A}} \) be the generator of a one-dimensional real-valued diffusion \( {(X_{t})_{t\geq 0}} \) of the type
\[ \mathcal{A} f = \sigma ^2 f”+ bf’, \]
where \( {f} \) and the two functions \( {\sigma,b} \) are sufficiently smooth. Given a smooth positive function \( {a} \) on \( {\mathbb{R}} \), the gradient of interest is \( {\nabla_a f = a\, f’} \). Denote \( {(P_t)_{t\geq 0}} \) the associated diffusion semigroup. Then it is not hard to adapt to the continuous case the argument of theorem~1 to show that the following intertwining relation holds:
\[ \nabla_a P_tf (x) = \mathbb{E}_x \left[ \nabla_a f(X_{a,t}) \, \exp \left( – \int_0^t V_a ( X_{a,s}) \, ds \right)\right] . \]
Here \( {(X_{a,t})_{t\geq 0}} \) is a new diffusion process with generator
\[ \mathcal{A} _a f = \sigma ^2 f” + b_a f’ \]
and drift \( {b_a} \) and potential \( {V_a} \) given by
\[ b_a := 2\sigma \sigma ‘ +b – 2\sigma ^2 \, \frac{a’}{a} \quad\text{and}\quad V_a := \sigma ^2 \, \frac{a”}{a} – b’ + \frac{a’}{a} \, b_a. \]
In particular, if the weight \( {a=\sigma} \), where \( {\sigma} \) is assumed to be positive, then the two processes above have the same distribution and by Jensen’s inequality, we obtain
\[ \vert \nabla_\sigma P_tf (x) \vert \leq \mathbb{E}_x \left[ \vert \nabla_\sigma f (X_{t}) \vert \, \exp \left( – \int_0^t \left( \sigma \sigma” -b’ + b\, \frac{\sigma ‘}{\sigma} \right) ( X_{s}) \, ds \right)\right] . \]
Hence under the assumption that there exists a constant \( {\rho} \) such that
\[ \inf \, \sigma \sigma” -b’ + b\, \frac{\sigma ‘}{\sigma} \geq \rho, \]
then we get \( {\vert \nabla_\sigma P_tf \vert \leq e^{-\rho t} \, P_t \vert \nabla_\sigma f \vert} \). This type of sub-commutation relation is at the heart of the Bakry-Emery calculus for diffusions.