bibkey: chafaijoulin2013intertwining authors: Djalil Chafaï and Aldéric Joulin year: 2013 title: Intertwining and commutation relations for birth–death processes doi: 10.3150/12-BEJ433 url: https://arxiv.org/abs/1011.2331v4 claim: Positive weighted-gradient intertwining supplies monotonicity and spectral-gap estimates for birth–death processes. An actual reflected prime-two edge prevents radial monotonicity of the original even theta semigroup, so this scalar positive radial derivative interface cannot be transported directly. strata_touched: [] license: bibliographic-reference-only triage: anchor

Positive derivative transport and the reflected theta prime edge

Reuse the published theorem with its actual hypotheses

The journal reference is Bernoulli 19(5A) (2013), 1855–1879. The inspected arXiv electronic reprint has SHA-256 38a233ff5a9c4aa95876288b268c8f3919671a5373c14c7b3612c24d195cc990. Only bibliographic metadata and the model application are retained; no PDF, implementation or source text is copied into the project. The reprint explicitly has different pagination from the journal article; all theorem page locators below refer to that inspected reprint.

Section2, Theorem2.1, reprint p.4, concerns an irreducible, nonexplosive birth–death process on $\mathbb{N}$. For a positive weight $u$, its discrete derivative is ${\partial }_{u}f(x)=(f(x+1)-f(x))/u_x$. The modified process has birth rate ${\lambda }_x^u=(u_{x+1}/u_x){\lambda }_{x+1}$ and death rate ${\nu }_x^u=(u_{x-1}/u_x){\nu }_x$, with the usual zero boundary death rate. The potential is $V_u(x)={\nu }_{x+1}-{\nu }_x^u+{\lambda }_x-{\lambda }_x^u$. Under the section’s transition-rate and potential integrability assumptions, a lower bound on $V_u$ and bounded ${\partial }_{u}f$, it gives

$${\partial }_u{P}_{t}f=Q_t{\partial }_{u}f,Q_{t}g(x)={\mathbb{E}}_x\left[g(X_{u,t})\exp \left(-{\int }_0^t{V}_u(X_{u,s})ds\right)\right].\text{\hspace{1em}(R1)}$$

The Feynman–Kac weight is positive even if the lower-bounded potential has some negative values. Remark2.4, reprint p.6, obtains propagation of monotonicity from this identity. Corollary3.3, reprint p.11, gives a Poincare lower bound from a positive weighted Wasserstein curvature, using the same birth–death framework and its preceding assumptions. Those source results are reused, not reproved here. The original theta process has continuous states and nonlocal Gamma and prime-power jumps; it is not assigned the birth–death hypotheses.

The relevant proposed interface is weaker than total positivity: does the original even semigroup preserve functions nondecreasing in $r=|x|$? The existing ordered determinant test only excludes all-times order-two positivity. Its failure alone does not answer this weaker question. The test below uses a different actual edge and two ordered averaging probes.

Keep the original form, domain and all prime powers

Use the minimal mixed theta form, with $\rho (x)=2\Phi (x)\cosh (x/2)$, $d\nu =\rho dx$, $w_n=\Lambda (n)/\sqrt{n}$, and the symmetric conductance $J=J_{\Gamma }+J_p$ defined there. Let $A\ge 0$ be its energy operator on ${\mathcal{F}}_{\min ,\mathrm{even}}$ and $P_t=e^{-tA}$. Write ${\psi }_{\Gamma }(t)=e^{-t/2}/(1-e^{-2t})$ to distinguish the Gamma kernel from the golden conjugate coordinate. The real polarized form is

$$D(f,g)=\frac{1}{2}\iint (f(y)-f(x))(g(y)-g(x))J(dx,dy).$$

Set

$$r_0=\frac{1}{8},z_0=\log 2-r_0,0<\delta <\frac{1}{32}.$$

Choose a smooth nondecreasing function $H:\mathbb{R}\to [0,1]$, zero on $(-\infty ,-1/2]$ and one on $[1/2,\infty )$, and put

$$h_{\delta }(x)=H((|x|-z_0)/\delta ).$$

It is smooth and even, is nondecreasing in $|x|$, and equals one outside a compact set. Since $1\in {\mathcal{F}}_{\min }$ and $1-h_{\delta }\in C_c^{\infty }$, it belongs to the actual minimal even form domain. No increasing nonzero compactly supported input is requested.

Choose nonnegative even compact smooth $u_{L,\delta },u_{R,\delta }$, normalized by $\int u_{L,\delta }d\nu =\int u_{R,\delta }d\nu =1$. Their positive radial supports lie in the intervals of radius $\delta /4$ centered at $r_0-\delta$ and $r_0+\delta$, respectively. These intervals are strictly ordered, and $h_{\delta }=0$ on both probes. Write $v_{\delta }=u_{L,\delta }-u_{R,\delta }$; then $⟨v_{\delta },h_{\delta }{⟩}_{\nu }=0$.

Where the probes are supported, symmetry of $J$ and $h_{\delta }=0$ give

$$-D(u,h_{\delta })=\int u(x)b_{\delta }(x)d\nu (x),$$

with the following even row function:

$$\begin{array}{rl}{b}_{\delta }(r)=\frac{1}{2\cosh (r/2)}[& {\int }_{\mathbb{R}}\Phi (y){\psi }_{\Gamma }(|r-y|)h_{\delta }(y)dy\\ & +\sum _{n\ge 2}{w}_n(\Phi (r+\log n)h_{\delta }(r+\log n)+\Phi (r-\log n)h_{\delta }(r-\log n))].\end{array}\text{\hspace{1em}(R2)}$$

This is the row normalization $J/\nu$ for the Markov generator $-A$. It is twice the FOT kernel normalization $J/(2\nu )$; the one-half in $D$ is removed by the two symmetric cross terms. Formula(R2) is only used away from the diagonal where the input vanishes, and does not assert operator-domain membership for $h_{\delta }$.

The reflected prime-two atom switches while the background agrees

For every $r$ in either probe interval, $h_{\delta }(r+\log n)=1$ for all $n\ge 2$ and $h_{\delta }(r-\log n)=1$ for all $n\ge 3$. The sole switched destination is $r-\log 2$:

$$\{\begin{array}{ll}{h}_{\delta }(r-\log 2)=1,& r\in {\mathrm{supp}}_{\mathrm{rad}}{u}_{L,\delta },\\ h_{\delta }(r-\log 2)=0,& r\in {\mathrm{supp}}_{\mathrm{rad}}{u}_{R,\delta }.\end{array}\text{\hspace{1em}(R3)}$$

Indeed, its radius $\log 2-r$ is at least $z_0+3\delta /4$ on the left and at most $z_0-3\delta /4$ on the right. The source intervals are below the transition; the plus destinations and the reflected $n\ge 3$ destinations are above it, with a fixed positive margin. The coefficient of the switched atom is

$$a_2(r)=\frac{\log 2}{2\sqrt{2}\cosh (r/2)}\Phi (\log 2-r)>0.\text{\hspace{1em}(R4)}$$

All other prime-power row terms form the same continuous background on both probes. The original local tail bound $\Phi (r\pm \log n)\le C_K{n}^{-2}$ on a fixed compact $K$ bounds their series by $C_K{\sum }_{n\ge 2}(\log n)n^{-5/2}<\infty$. Uniform convergence therefore proves continuity, retaining every prime power and requiring neither a cutoff nor PNT.

For the Gamma background, restrict the sources to $|r-r_0|<1/16$. The support where $h_{\delta }$ is nonzero is separated from these sources by a fixed positive distance. Thus ${\psi }_{\Gamma }(|r-y|)$ is uniformly bounded there, and $C\Phi (y)$ is an integrable majorant. As $\delta \downarrow 0$ and $r\to r_0$, $h_{\delta }(y)\to 1_{\{|y|>z_0\}}$ except at two Lebesgue-null endpoints. Dominated convergence gives the same Gamma row limit for both concentrating probes. It also gives uniform convergence on their shrinking supports; otherwise a sequence of points in those supports would contradict the same joint limit. The Gamma diagonal singularity never enters this computation.

Subtracting the two normalized probe averages in(R2) now yields

$$I_{\delta }:=-D(v_{\delta },h_{\delta })⟶a_2(r_0)>0(\delta \downarrow 0).\text{\hspace{1em}(R5)}$$

The coefficient uses both reflected source halves through normalization against the even probability measure; no additional factor two is added.

Short time reverses the ordered averages

Reuse the form-semigroup derivative (O2). For a fixed sufficiently small $\delta$ with $I_{\delta }>0$, it gives

$$⟨v_{\delta },P_t{h}_{\delta }{⟩}_{\nu }=t{I}_{\delta }+o(t)>0\text{for all sufficiently small }t>0.\text{\hspace{1em}(R6)}$$

The order of limits matters: first choose $\delta$, then let $t$ tend to zero. There is no uniform short-time remainder claim as the probes concentrate. All inputs are in the form domain, as required by(O2).

If $P_t{h}_{\delta }$ had a representative nondecreasing in $|x|$, its average against the normalized left probe could not exceed its average against the normalized right probe. Equation(R6) reverses that inequality. Under the inherited original-form and semigroup premises, the argument therefore excludes all-times radial monotonicity of the original even semigroup.

In particular, a scalar positive weighted radial first-derivative intertwiner, valid on these plateau tests and implying preservation of this monotonicity cone, cannot realize(R1) for this model. That implication is essential to the exclusion: no claim is made about arbitrary positive intertwiners, other partial orders, signed or vector transports, or fixed-time comparisons. Ordinary Markov positivity is unaffected.

The remaining all-input estimate is unchanged

The application isolates a specific failure of the published method’s proposed radial transport, with all original long edges present. It does not produce a subthreshold eigenvector or decide the sign of the critical remainder. The still-needed estimate is $D(r)\ge \Vert r{\Vert }_{\nu }^2/2$ on that remainder, or actual cofinal low-center signs sufficient for the same original half-bound. A five-mode FIB tiling may label the sources and destinations, but those labels do not remove the reflected map $r\mapsto |\log 2-r|$.

The source mapping and equations(R2)–(R6) are conditional paper mathematics, not Lean certification, a new general method or a priority claim. No fixed-target action, moment, Gram or numerical producer is replayed. Cofinal positivity, the half-bound, full Robin and RH remain unresolved.