Inhomogeneous Burke Property

• The inhomogeneous Burke property is a refined invariance principle that extends Burke’s theorem to systems with time or space-varying parameters, ensuring output distributions remain consistent.
• It underpins diverse areas such as time-inhomogeneous Markov chains, stochastic diffusions, Diophantine approximations, and random graph models, demonstrating widespread applications.
• Techniques like ergodicity, representation theorems, coupling methods, and algebraic symmetries are used to guarantee stable, invariant outputs despite dynamic changes.

The inhomogeneous Burke property is a rigorous invariance phenomenon originally motivated by queueing theory but now recognized as a powerful structural principle across disparate areas of probability, stochastic processes, statistical physics, and dynamical systems. While Burke’s classical theorem asserts that in a stationary $M/M/1$ queue, the departure process retains the same (Poisson) distribution as the arrival process, inhomogeneous Burke properties articulate related invariance or “memory-loss” results in settings where parameters or dynamics vary with time, location, or additional structure. The central unifying theme is that under suitable conditions—often loss of memory (ergodicity/merging), existence of stable measures, or algebraic/integrable symmetries—the output process exhibits distributional invariance reflective of the input statistics, even in the presence of inhomogeneity.

1. Merging and Stability in Time-Inhomogeneous Markov Chains

A foundational abstraction of the inhomogeneous Burke property is found in the merging and stability phenomena for time-inhomogeneous Markov chains (Saloff-Coste et al., 2010). Given a sequence of Markov kernels $\{K_i\}_{i\geq 1}$, the distribution after $n$ steps, $\mu_n = \mu_0 K_1K_2\cdots K_n$, exhibits merging (weak ergodicity) if for any two initial measures $\mu_0, \nu_0$,

$$\lim_{n\to\infty} \|\mu_0 K_1\cdots K_n - \nu_0 K_1\cdots K_n\|_{TV} = 0,$$

where $\|\cdot\|_{TV}$ is total variation distance. This formalizes memory-loss: the chain "forgets" initial conditions, analogous to Burke's classical output invariance.

Stability is the existence of a measure $T$ and constant $c\geq 1$ such that from sufficiently large $n$,

$$c^{-1} T(x) \leq (\mu_0 K_1\cdots K_n)(x) \leq c T(x),$$

for all states $x$. This guarantees the chain’s output distribution remains comparable to a “rough invariant shape,” mirroring the statistical invariance of queue departures in the Burke theorem.

Both properties together yield an inhomogeneous Burke property: despite time-dependent transitions, the Markov chain’s asymptotics are governed by an input-independent distribution $T$, with precise rates controlled by singular value decay,

$$\|\mu_n(x,\cdot)-T\|_{TV} \leq \frac{1}{2} \sqrt{\frac{T(x)}{\mu_0(x)}\prod_{i=1}^n \sigma_i},$$

where $\sigma_i$ is the second largest singular value of $K_i$.

2. Representation and Comparison in Time-Inhomogeneous Markov Processes

Comparison results for time-inhomogeneous Markov processes extend the Burke property to systems where local characteristics (drift, diffusion, jump measures) may vary (Rueschendorf et al., 2015). The infinitesimal generator $A_s$ incorporates time-dependent coefficients, and comparison is achieved by generator ordering: $A_s f \leq B_s f$ for a function class $\mathcal{F$. The associated representation theorem for solutions of inhomogeneous evolution problems,

$F_t(s) = T_{s,t}F_t(t) - \int_s^t T_{s,r}G(r) dr,$

integrates local orderings into a global stochastic order on expectations, $E[f(X_t)] \leq E[f(Y_t)]$.

This is an “infinitesimal” version of the Burke property: whenever the evolution operator preserves $\mathcal{F}$ and generator ordering holds, the output process’s distribution remains stochastically ordered relative to the input—even when dynamics are nonstationary. These results apply to diffusions, processes with independent increments, and Lévy driven diffusions, underpinning ordering and invariance properties in inhomogeneous stochastic systems.

3. Invariant Functionals for Inhomogeneous Diffusion

In time-inhomogeneous diffusion processes, invariance emerges via the Onsager-Machlup functional (Coulibaly-Pasquier, 2011). For generators of the form $L_t = \Delta_{g(t)} + Z(t, \cdot)$ (Laplace–Beltrami operator with time-dependent metric and drift), the Onsager–Machlup functional $H$ governs the asymptotic cost of small deviations: $$P_{x_0}\left[\sup_t d(t,X_t, p(t)) \leq \epsilon\right] \sim C \epsilon^\alpha \exp\left\{ -\int_0^T H(t,p(t),\dot{p}(t)) dt \right\},$$ with $H$ including contributions from drift, metric variation, and curvature (e.g., $$|Z(t,x) - v|_{g(t)}^2 + \operatorname{div}_{g(t)}Z(t,x) - R_{g(t)}(x) + \tfrac{1}{12}\operatorname{tr}_{g(t)}\dot{g}(t)$$). Under metric evolution (e.g., Ricci flow), this functional is analogous to the $\mathcal{L}_0$ distance of Lott, thus connecting probabilistic invariance phenomena to geometric flows. The extra metric variation term is the haLLMark of inhomogeneity, and the resulting action–minimization principle remains manifestly invariant under time-dependent geometry.

4. Invariant and Winning Sets in Inhomogeneous Diophantine Approximation

Inhomogeneous Burke properties also govern robustness of approximation properties. The sets of weighted inhomogeneous badly approximable vectors,

$$\mathbf{Bad}_\theta(\mathbf{w}) = \left\{ \mathbf{x} \in \mathbb{R}^d : \liminf_{\substack{q\in\mathbb{Z}\setminus\{0\} \ |q|\to\infty}} \max_{1\le i\le d}\left| q x_i - \theta_i \right|^{1/w_i} |q| > 0 \right\},$$

are shown to be hyperplane absolute winning (Datta et al., 9 Apr 2025) and absolute winning on nondegenerate analytic curves (Datta et al., 2023). This property is robust under countable intersections and smooth transformations, thus representing a statistical invariance across a wide class of inhomogeneous settings.

For analytic manifolds, almost every point is not weighted inhomogeneous badly approximable, i.e., the set is null for Lebesgue measure. The proofs employ duality and quantitative nondivergence estimates from homogeneous dynamics, establishing transference between Diophantine inequalities and the behavior of associated lattice flows—a conceptual parallel to stationarity and invariance seen in queueing models.

5. Invariance in Inhomogeneous Polymer and Percolation Models

Recent advances highlight algebraic and combinatorial invariance in random polymer and percolation settings. In last-passage percolation (LPP) with inhomogeneous exponential weights,

$$\omega_{(i,j)} \sim \mathrm{Exp}(a_i + b_j), \quad a_i + b_j > 0,$$

the joint distribution of Busemann increments is invariant under finite permutations of the $a_i$ or $b_j$ (Bates et al., 14 Jun 2025). The invariance is proven via explicit coupling and application of the Burke property: transformations that swap rates in the underlying queueing system do not affect passage times, and the Busemann process admits an exact characterization via a finite-dimensional model.

Further, the discrete periodic Pitman transform acts on sequences of weights in polymer models; under the log–inverse–gamma, geometric, or exponential product measures,

$$(X_1, X_2) \overset{d}{=} (T(X_1, X_2), D(X_1, X_2)),$$

where $T$ and $D$ are Pitman transforms involving cyclic sums (in positive– and zero–temperature limits), and this invariance extends to multi-path partition functions under arbitrary permutations of column parameters (Engel et al., 7 Aug 2025). The transforms satisfy braid relations and are involutions, supporting the invariance and the explicit combinatorial description of invariant measures for the associated Markov chain.

6. Local Versus Global Control in Inhomogeneous Random Graphs

In inhomogeneous random graph models generated by a graphon $W:\Omega^2 \to [0,1]$ (Hladký et al., 2023), global connectivity is governed by two critical properties:

• The graphon $W$ is connected: no measurable set $X \subset \Omega$ disconnects the space.
• The measure of low-degree vertices decays fast: $$\lim_{\alpha\searrow 0} \frac{g_W(\alpha)}{\alpha} = 0$$ where $g_W(\alpha)$ denotes the measure of vertices of $W$-degree $<\alpha$.

Under these, the random graph $G(n,W)$ is robustly connected with high probability, and the only obstructions to connectivity are local (isolated vertices, small components). The "inhomogeneous Burke property" in this context posits that the global property (connectivity) is controlled by vanishing local defects—once local degrees are sufficient, global invariance emerges, exactly paralleling the disappearance of isolated vertices at the connectivity threshold in homogeneous models.

7. Operator-Level Generalizations and Fractional Evolution

In the analysis of nonlinear stochastic PDEs, such as generalized inhomogeneous Burgers equations (Maltese, 7 Nov 2024), invariance emerges through operator-level generalization. Classical derivatives are replaced by linear operators $O_t$, $N_x$, $M_x$, and the multidimensional evolution: $$O_t u - \sum_{d=1}^m A_d(t) M_d(u L_d(u)) = \sum_{d=1}^m N_d M_d u + \left(\sum_{d=1}^m A_d(t)\right) u$$ admits exact solutions via transformations (e.g., generalized Cole–Hopf, Hermite polynomials) and the method of invariant spaces. Even when the time evolution is governed by Caputo-type fractional derivatives, memory effects modify but do not disrupt invariance in the structure of the solution, a dynamical analogy to Burke properties in probabilistic settings.

Conclusion

The inhomogeneous Burke property encapsulates a fundamental invariance principle: under suitable ergodicity, stable shape, and/or algebraic symmetries, remarkably robust output distributions persist in complex stochastic, algebraic, and geometric settings—notwithstanding significant temporal, spatial, or parametric inhomogeneity. Its formal manifestations range from loss of memory and stable profiles in Markov chains, invariant partition functions in stochastic polymers, winning sets in Diophantine approximation, to global connectivity in random graphs. Methods used to establish this property include singular value decay, representation theorems for evolution systems in Banach spaces, coupling and queueing arguments, algebraic braid relations, and transference in dynamics. These results underpin a wide array of theoretical and applied models, providing guarantees of predictable long-term behavior and explicit sampling procedures even under highly non-stationary conditions.