A Wasserstein rate dictionary for local-update generators

Definitions and mathematical statement

The source asks for Wasserstein convergence rates for local jump generators \[ (LA)(x)=\sum_i\int\omega_i(x,{\,\mathrm{d}} y)[A(y)-A(x)], \] whose dependence and updates use finitely many coordinates near site $i$. It does not specify a state space, metric, rates or irreducibility. The aid adopts a finite-state single-site interpretation, which is a special case rather than every generator in that display.

Let $E=\prod_{i=1}^nE_i$, with each $E_i$ a finite metric space of metric $d_i$. For fixed rates $r_i>0$ and probability kernels $K_i(x,{\,\mathrm{d}} z)$ on $E_i$, set \[ (LA)(x)=\sum_i r_i\int[A(x^{i\leftarrow z})-A(x)]K_i(x,{\,\mathrm{d}} z). \] Suppose a nonnegative influence matrix $C$ satisfies \[ W_{d_i}(K_i(x,\cdot),K_i(y,\cdot))\leq\sum_jC_{ij}d_j(x_j,y_j) \quad\text{for all }x,y,i. \] Here $W_d$ is the minimum expected metric distance over couplings. A common-clock optimal update coupling gives, for $u_i(t)={\mathbb{E}} d_i(X_i(t),Y_i(t))$ and $R=\operatorname{diag}(r_i)$, the reported coordinatewise bound \[ u(t)\leq e^{tR(C-I)}u(0). \] If $w_i>0$ and $w^{\mathsf T}R(C-I)\leq-\lambda w^{\mathsf T}$ entrywise for $\lambda>0$, then the weighted metric $D_w(x,y)=\sum_iw_id_i(x_i,y_i)$ satisfies \[ W_{D_w}(\mu P_t,\eta P_t)\leq e^{-\lambda t}W_{D_w}(\mu,\eta). \] Thus there is a unique stationary law and exponential convergence in this metric. For common rate $r$ and spectral radius $\rho(C)<1$, positive left subeigenvectors give every rate $r(1-q)$ with $\rho(C)<q<1$, approaching $r(1-\rho(C))$. Attainment by a strictly positive Perron vector requires an appropriate extra condition, such as irreducibility of $C$.

For a second collection of update kernels $\widetilde K_i$ with the same site rates, define \[ \varepsilon_i=\sup_xW_{d_i}(K_i(x,\cdot),\widetilde K_i(x,\cdot)). \] When $\rho(C)<1$, the aid reports a stationary coupling, for any stationary laws $\pi,\widetilde\pi$ of the two systems, with coordinate discrepancies bounded by \[ u\leq(I-C)^{-1}\varepsilon, \qquad W_{D_w}(\pi,\widetilde\pi)\leq w^{\mathsf T}(I-C)^{-1}\varepsilon. \] This is a local-kernel perturbation bound; the nonnegative resolvent is $\sum_{k\geq0}C^k$.

For explicit spatial consequences, assume common rate $r$, row sums at most $0<q<1$, and finite interaction range. Let $B$ be the set supporting initial discrepancies or kernel perturbations, and let $h_i$ be the minimum number of directed influence steps from $B$ to $i$, with edge $j\to i$ when $C_{ij}>0$. For initial coordinate discrepancies at most $D_*$, zero outside $B$, a precise Poisson-tail version is \[ u_i(t)\leq D_*e^{-rt}\sum_{k\geq h_i}\frac{(rqt)^k}{k!} =D_*e^{-r(1-q)t}\Pr\{\operatorname{Poisson}(rqt)\geq h_i\}. \] For perturbations $\varepsilon_i\leq\varepsilon_*$ supported on $B$, the corresponding stationary bound is \[ u_i\leq\frac{\varepsilon_*q^{h_i}}{1-q}. \] Unreachable sites have zero bounds. The case $q=0$ is the immediate no-propagation limit. Finite interaction range translates influence-step distance into spatial separation; the Poisson bound is a small-tail estimate, not a strict deterministic light cone. The summary does not specify all constants or distance conventions, so these formulas state an explicit version of its reported consequences.

Finally, on a cyclic set of sites with states $\{0,1\}$, take $K_i(x,\cdot)=\delta_{x_{i-1}}$, indices modulo $n\geq2$. Its influence matrix is the cyclic shift, with $\rho(C)=1$, and both constant configurations are absorbing. This reported nearest-neighbor copying obstruction shows locality alone cannot guarantee a common limiting law or strict Wasserstein mixing.

Short English statement

For finite-state single-site updates, the aid turns a directed influence matrix into Wasserstein contraction, stationary sensitivity and spatial decay bounds. A local copying cycle with spectral radius one shows why locality needs an additional contraction hypothesis.

Sources

• Local registry: permanent UnsolvedMath ID $20001414$, exact public code catalogue entry; aim-dynamical-systems-notes.json, notes[71].
• AIM workshop "Stochastic methods for non-equilibrium dynamical systems", Problem 29: https://aimath.org/pastworkshops/noneqdynsysproblems.pdf. The canonical local-jump generator and its finite-neighborhood qualification are preserved above. No proposer, metric specification or authoritative reference is supplied; the literature field is empty.
• Research attempt 1: partial_result, status at run partially_solved, rigor label proved, novelty confidence low, classification PARTIAL-PROGRESS. It reports common-clock coupling, the influence propagator, Perron-weighted rates, stationary resolvent sensitivity, Poisson/geometric spatial bounds and the copying-cycle obstruction. The current registry status is partially_solved. No separate full bibliography is supplied.