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 N with semigroup (Pt)t≥0 and a discrete gradient ∂u depending on a positive weight u, we establish intertwining relations of the form ∂uPt=Qt∂u, where (Qt)t≥0 is the Feynman-Kac semigroup with potential Vu of another birth-death process. We provide applications when Vu 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 (Xt)t≥0 on the state space N:={0,1,2,…}, i.e. a Markov process with transition probabilities given by
Pxt(y)=Px(Xt=y)=(λxt)1y=x+1+(νxt)1y=x−1+(1−(λx+νx)t)1y=x+tε(t).
The transition rates λ and ν are respectively called the birth and death rates of the process (Xt)t≥0. We assume that the process is irreducible, positive recurrent, and non-explosive. This holds when the rates satisfy to λ>0 on N and ν>0 on N∗ and ν0=0 and
∞∑x=1λ0λ1⋯λx−1ν1ν2⋯νx<∞and∞∑x=1(1λx+νxλxλx−1+⋯+νx⋯ν1λx⋯λ1λ0)=∞.
The unique stationary distribution μ of the process is reversible and is given by
μ(x)=μ(0)x∏y=1λy−1νy, x∈Nwithμ(0):=(1+∞∑x=1λ0λ1⋯λx−1ν1ν2⋯νx)−1. (1)
Let us denote by F (respectively F+) the space of real-valued (respectively positive) functions f on N, and let bF be the subspace of bounded functions. The associated semigroup (Pt)t≥0 is defined for any function f∈bF∪F+ and x∈N as
Ptf(x)=Ex[f(Xt)]=∞∑y=0f(y)Pxt(y).
This family of operators is positivity preserving and contractive on Lp(μ), p∈[1,∞]. Moreover, the semigroup is also symmetric in L2(μ) since λxμ(x)=ν1+xμ(1+x) for any x∈N (detailed balance equation). The generator L of the process is given for any f∈F and x∈N by
Lf(x)=λx(f(x+1)−f(x))+νx(f(x−1)−f(x))=λx∂f(x)+νx∂∗f(x),
where ∂ and ∂∗ are respectively the forward and backward discrete gradients on N:
∂f(x):=f(x+1)−f(x)and∂∗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/∞ queues. The M/M/∞ queue has rates λx=λ and νx=νx for positive constants λ and ν. It is positive recurrent and its stationary distribution is the Poisson measure μρ with mean ρ=λ/μ. If Bx,p stands for the binomial distribution of size x∈N and parameter p∈[0,1], the M/M/∞ process satisfies for every x∈N and t≥0 to the Mehler type formula
L(Xt|X0=x)=Bx,e−νt∗μρ(1−e−νt). (2)
The M/M/1 queueing process has rates λx=λ and νx=ν1N∖{0} where 0<λ<ν are constants. It is a positive recurrent random walk on N reflected at 0. Its stationary distribution μ is the geometric measure with parameter ρ:=λ/ν given by μ(x)=(1−ρ)ρx for all x∈N. A remarkable common property shared by the M/M/1 and M/M/∞ processes is the intertwining relation
∂L=LV∂ (3)
where LV=L−V is the discrete Schrödinger operator with potential V given by
- V(x):=ν in the case of the M/M/∞ queue
- V(x):=ν1{0}(x) for the M/M/1 queue.
The operator LV is the generator of a Feynman-Kac semigroup (PVt)t≥0 given by
PVtf(x)=Ex[f(Xt)exp(−∫t0V(Xs)ds)].
The intertwining relation (3) is the infinitesimal version at time t=0 of the semigroup intertwining
∂Ptf(x)=PVt∂f(x)=Ex[∂f(Xt)exp(−∫t0V(Xs)ds)]. (4)
Conversely, one may deduce (4) from (3) by using a semigroup interpolation. Namely, if we consider
s∈[0,t]↦J(s):=PVs∂Pt−sf
with V as above, then (4) rewrites as J(0)=J(t) and (4) follows from (3) since
J′(s)=PVs(LV∂Pt−sf−∂LPt−sf)=0.
Let us fix some u∈F+. The u-modification of the original process (Xt)t≥0 is a birth-death process (Xu,t)t≥0 with semigroup (Pu,t)t≥0 and generator Lu given by
Luf(x)=λux∂f(x)+νux∂∗f(x),
where the birth and death rates are respectively given by
λux:=ux+1uxλx+1andνux:=ux−1uxνx.
One can check that the measure λu2μ is reversible for (Xu,t)t≥0. As consequence, the process (Xu,t)t≥0 is positive recurrent if and only if λu2 is μ-integrable. We define the discrete gradient ∂u and the potential Vu by
∂u:=(1/u)∂andVu(x):=νx+1−νux+λx−λux.
Let φ:R→R+ be a smooth convex function such that for some constant c>0, and for all r∈R,
φ′(r)r≥cφ(r). (5)
In particular, φ vanishes at 0, is non-increasing on (−∞,0) and non-decreasing on (0,∞). 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 |⋅|.
Theorem 1 (Intertwining and sub-commutation) Assume that for every x∈N and t≥0, we have
Ex[exp(−∫t0Vu(Xu,s)ds)]<∞.
Then for every f∈bF, x∈N, t≥0,
∂uPtf(x)=PVuu,t∂uf(x)=Ex[∂uf(Xu,t)exp(−∫t0Vu(Xu,s)ds)]. (6)
Moreover, if Vu≥0 then for every f∈bF, x∈N, t≥0,
φ(∂uPtf)(x)≤Ex[φ(∂uf)(Xu,t)exp(−∫t0cVu(Xu,s)ds)]. (7)
Proof: Let us prove (7). If we define
s∈[0,t]↦J(s):=PcVuu,sφ(∂uPt−sf)
then (7) rewrites as J(0)≤J(t). Hence it suffices to show that J is non-decreasing. We have the intertwining relation
∂uL=LVuu∂u, (8)
where Lu is the generator of the u-modification process (Xu,t)t≥0 and where
LVuu:=Lu−Vu.
Now
J′(s)=PcVuu,s(T)whereT=LcVuuφ(∂uPt−sf)−φ′(∂uPt−sf)∂uLPt−sf.
Letting gu=∂uPt−sf, we obtain, by using (8),
T=LcVuuφ(gu)−φ′(gu)LVuugu
and thus
T=λu(∂φ(gu)−φ′(gu)∂gu)+νu(∂∗φ(gu)−φ′(gu)∂∗gu)+Vu(φ′(gu)gu−cφ(gu)).
Now (5) and Vu≥0 give T≥0. Since the Feynman-Kac semigroup (PcVuu,t)t≥0 is positivity preserving, we get (7). The proof of (6) is similar but simpler (T is identically zero).
The identity (6) implies a propagation of monotonicity: if f is non-increasing then Ptf is also non-increasing.
Actually, the intertwining relations above have their counterpart in continuous state space. Let A be the generator of a one-dimensional real-valued diffusion (Xt)t≥0 of the type
Af=σ2f″+bf′,
where f and the two functions σ,b are sufficiently smooth. Given a smooth positive function a on R, the gradient of interest is ∇af=af′. Denote (Pt)t≥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:
∇aPtf(x)=Ex[∇af(Xa,t)exp(−∫t0Va(Xa,s)ds)].
Here (Xa,t)t≥0 is a new diffusion process with generator
Aaf=σ2f″+baf′
and drift ba and potential Va given by
ba:=2σσ′+b−2σ2a′aandVa:=σ2a″a−b′+a′aba.
In particular, if the weight a=σ, where σ is assumed to be positive, then the two processes above have the same distribution and by Jensen's inequality, we obtain
|∇σPtf(x)|≤Ex[|∇σf(Xt)|exp(−∫t0(σσ″−b′+bσ′σ)(Xs)ds)].
Hence under the assumption that there exists a constant ρ such that
infσσ″−b′+bσ′σ≥ρ,
then we get |∇σPtf|≤e−ρtPt|∇σf|. This type of sub-commutation relation is at the heart of the Bakry-Emery calculus for diffusions.
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