Papers

We study slow-fast stochastic systems in which a fast process, evolving within several ergodic components, drives the transition rates of a slow jump process switching between these components. Considering a differential equation whose vector field depends on both processes, we show that, under suitable ergodicity and regularity conditions, its solutions converge in law in $D([0,T],\mathbb R^d)$ to an averaged system depending only on the slow process. The averaged drift is obtained by averaging the original vector field over the invariant measure of the fast process within each component. Our results extend classical averaging theory to systems where the vector field is explicitly influenced by a fast random environment with multi-component dynamics.

Creator economy platforms face a strategic dilemma: allocating traffic to established stars for immediate ad revenue versus nurturing emerging creators to build a follower base for future monetization. We develop a continuous-time dynamic optimization model to characterize the optimal traffic allocation policy for a platform managing heterogeneous creators with dual revenue streams (advertising and direct follower contributions). We characterize the optimal policy analytically, revealing a “most-valuable-creator-first" rule driven by a forward-looking activation set. Under Bass diffusion dynamics, this policy exhibits a sophisticated “conditional reversal" strategy, where the platform temporarily prioritizes lagging creators to capitalize on word-of-mouth effects. Regarding the ecosystem structure, we find the optimal policy acts as a selective gatekeeper. Unlike myopic policies that lead to a harsh “winner-take-all" market, or naive fairness-driven heuristics that foster inefficient “indiscriminate growth," the optimal policy imposes a strict capability threshold that screens which creators receive platform traffic. Furthermore, it enforces disciplined growth for successful entrants, capping their follower bases at an optimal ceiling to prevent over-investment. Finally, we demonstrate that simple heuristics can lead to significant revenue losses (up to 25%) and propose a practical “follower-growth adjusted" heuristic that achieves near-optimal performance by leveraging observed growth momentum.

The narrow escape problem is a prototypical example for studying entropic metastability, motivated by the analysis of biological and chemical systems. The problem concerns the determination of the exit time and position of a Brownian particle trapped in a domain with a reflecting boundary pierced by narrow holes. Our goal is to investigate this problem in a general domain in any dimension (greater than or equal to two), using the quasi-stationary distribution approach to metastability. In particular, we derive first order asymptotic expansions with error bounds of the mean exit time and the law of the exit position in the limit where the hole sizes tend to zero. Our analytical predictions are illustrated by numerical simulations, using dedicated Monte Carlo techniques.

Integral operators on the real unit interval are constructed as transported from ones on the $p$-adic unit disc via the Monna map. This gives rise to strong Markov processes on $[0, 1]$, whose paths are right-continuous and have no discontinuities other than jumps. The spectra of the corresponding diffusion operators, whose kernel functions depend on a $p$-adic distance, are calculated. Further, the solution to the Cauchy problem for their heat equations are approximated via continuous- time random walks on finite sets coming from a hierarchical partition of the unit interval induced by the $p$-adic distance. The transport of $p$-adic diffusion to the real domain via the Monna map gives rise to a simple visualisation method. Illustrations of concrete examples are undertaken in the end.

Let $\mathcal H=(V,E)$ be a locally finite simple hypergraph, equip $V$ with the hyperpath metric, and consider the equal-edges random walk \cite{CoupetteEtAl2023}. For adjacent vertices $x$ and $y$, we prove that the idleness function $\alpha\mapsto\kappa_\alpha^{\mathcal H}(x,y)$ is piecewise affine and with no more than three affine pieces. A separate mass-balance argument gives linearity on $[1/2,1]$ for every locally finite simple hypergraph and, consequently, a limit-free expression for the Lin–Lu–Yau curvature. In the $r$-uniform linear case, the hypergraph walk agrees exactly with the simple random walk on its 2-section. This reduction transfers the sharp endpoint intervals of Bourne, Cushing, Liu, Münch, and Peyerimhoff \cite{BourneEtAl2018}.

We establish exponential mixing for the randomly forced and weakly damped KdV equation in $L^2(\mathbb{T})$. The noise is bounded, localized, and degenerate in high frequencies. Our proof relies on a general probabilistic framework in [11,33], nonlinear smoothing for KdV and its linearization via normal form transformation, and stabilization of the system by localized force. This paper continues a series of works connecting asymptotic compactness, control theory, and ergodicity and mixing for randomly forced dispersive PDEs.

We derive explicit characterizations of Gibbs-measures related to the Aviles-Giga functional. We build on the variational approach presented by Barashkov and Gubinelli (2020) in the context of the $\Phi^4_3$-model.

In this paper, we show existence and uniqueness of solutions to the viscous Burgers' equation in $\mathbb{R}^d$, when the initial condition $u_0$ is in Hermite-Sobolev space of index $p$, for suitable non-negative integers $p$. Our solutions are local in time. We also have a regularity result, viz. if $u_0$ belongs to Schwartz space, then so does the solution.

We study the partial-sum behavior of tempered linear processes \[ X_{N,n}=\sum_{j=1}^{\infty}e^{-\lambda_Nj}\frac{\ell(j)}{j}\varepsilon_{n-j}, \qquad \lambda_N\downarrow0, \] where $\ell$ is slowly varying and the innovations belong to the domain of attraction of an $\alpha$-stable law with $1<\alpha\leq2$. The filter $j^{-1}\ell(j)$ represents the logarithmic boundary between summable and power-law long-memory coefficients. Let \[ Q_N=\sum_{j=1}^{N}e^{-\lambda_Nj}\frac{\ell(j)}{j}, \qquad L(N)=\sum_{j=1}^{N}\frac{\ell(j)}{j}. \] We prove that the partial-sum process, normalized by $B_NQ_N$, converges to the stable Lévy motion associated with the innovations. Moreover, \[ Q_N\sim L(N)\quad\text{if }N\lambda_N=O(1), \qquad Q_N\sim L(1/\lambda_N)\quad\text{if }N\lambda_N\to\infty. \] Then weak, moderate, and strong tempering have the same first-order Lévy limit but different normalizations. In the weakly and moderately tempered regimes, the second-order remainder, normalized by $B_N\ell(N)$, converges in finite-dimensional distributions to a logarithmically tempered stable process; in the Gaussian finite-moment case the convergence is functional. These results extend the untempered logarithmic-boundary theorem and complement existing invariance principles for tempered linear processes.

Erdős Problem 725 asks for an asymptotic formula for the number $L_{k,n}$ of ordered, labelled $k\times n$ Latin rectangles. Godsil and McKay proved that $L_{k,n}\sim (n!)^k((n)_k/n^k)^n(1-k/n)^{-n/2}e^{-k/2}$ for $k=o(n^{6/7})$. We provide a partial solution to Erdős Problem 725 by proving this asymptotic for every $k=o(n)$. More precisely, set $\widetilde A_{k,n}=(n!)^k((n)_k/n^k)^n\exp\{[n(H_n-H_{n-k})-k]/2\}$. For every $K(n)=o(n)$, uniformly for $0\leq k\leq K(n)$, we prove $\log(L_{k,n}/\widetilde A_{k,n})=O(k^2/n^2)$, with an absolute implied constant. The results of this paper have been formally verified in Lean.

We consider the largest eigenvalues of submatrices of the classical Gaussian random matrices with a finite rank spike in the critical BBP regime. We show that they converge to a multilevel interlacing particle system whose lowest level is the Airy$_\beta$ point process. Our construction and scaling limits extend in a natural way to non-classical $\beta >0$.

Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1}$ is absolutely continuous. The constant $C$ is independent of the dimension $n$, and the exponent $\frac{1}{k(d-1)+1}$ is sharp. This extends the scalar Carbery–Wright inequality and answers, in the log-concave setting, a question raised by Avni, Glazer, and Larsen. In addition, we show that the density of $\mu\circ f^{-1}$, whenever it exists, belongs to the Nikolskii–Besov space $B^{\frac{1}{k(d-1)+1}}_{1,\infty}(\mathbb R^k)$, with a dimension-free bound for the corresponding norm. A central difficulty in passing from scalar polynomials to vector-valued polynomial mappings is the lack of a suitable nondegeneracy parameter quantifying absolute continuity of $\mu\circ f^{-1}$, as the variance does in the scalar case. Natural candidates such as the covariance matrix or the Jacobian matrix either fail to characterize this property or do not lead to dimension-free estimates. We identify such a parameter and define it to be the covariance matrix of the vector formed by the monomials of degree up to $d^{k-1}$ in the normalized components of $f$. The dimension-free nature of our results allows us to extend Kusuoka's absolute continuity criterion for Gaussian polynomial random vectors to the log-concave setting. Moreover, in this setting, we obtain estimates relating convergence in distribution to convergence in total variation for polynomial random vectors.

Aumann (1974) showed that an atomless public randomization device makes the feasible- and equilibrium-payoff sets of a game with a fixed information structure convex, and asked whether they are closed. We show that, in every case the question leaves open, they need not be. One information structure drives all the examples: two sequences of fair signs whose coordinate correlations increase to a ceiling $\rho<1$ that no pair of separately measurable square-integrable rules attains. For every $0<\rho<1$ it yields a three-player game with a public randomization device whose equilibrium-payoff set is exactly the open interval $\{(0,0,t):-\rho<t<\rho\}$; a two-player game with a public randomization device whose equilibrium-payoff set is convex, full dimensional, and not closed; and, without any public device, nonclosed feasible- and equilibrium-payoff sets, the latter along equilibria with unique best replies modulo null events whose payoffs approach a vector that is not even feasible. With distinct but mutually absolutely continuous subjective priors, even the feasible-payoff set can fail to be closed in the presence of an objective public randomization device, together with every $\varepsilon$-equilibrium payoff set and the set of induced law tuples. Our construction also allows us to resolve a conjecture of Stinchcombe (2011). The main results and the lemmas supporting them are formalized in the Lean proof assistant; an appendix records the exact coverage of each statement, including the clauses for which only a paper proof is given.

We propose the VIX-derived volatility (VDV) model, a VIX-first framework for joint SPXVIX modeling. In the model, we define explicit dynamics for the VIX process to price VIX futures and options, yielding a VIX-side calibration that is independent of the SPX dynamics. Using the rolling-window definition of the VIX, we then derive a coupling function to obtain the SPX volatility process as a latent process consistent with the calibrated VIX dynamics. This produces a stochastic-volatility representation for SPX that can be further calibrated to SPX options without altering the VIX dynamics. Relative to existing joint-calibration approaches based on global optimization or highly flexible black-box dynamics, the method offers an interpretable and tractable decomposition of the joint problem by separating the calibration of the VIX from the calibration of the SPX. In a numerical experiment, we show that a VDV model with local volatility and mean-reverting dynamics for the VIX achieves a close model fit to the VIX futures market, VIX option market, and the SPX option market at the same time, providing a consistent joint framework.

Let $G$ be a higher-rank simple Lie group acting on a space $X$. A theorem of Nevo and Zimmer asserts that every ergodic stationary probability measure on $X$ is either $G$-invariant or admits a projective factor $G/Q$ for a proper parabolic subgroup $Q$. We develop a quantitative theory of stationary measures and prove an effective form of this dichotomy. We introduce notions of $\eps$-almost stationarity, $\delta$-almost invariance and $\delta$-almost projective factor, and show that every $\eps$-almost stationary measure is either $\delta$-almost invariant or carries a $\delta'$-almost projective factor, with $\delta,\delta'$ explicit in $\eps$ and depending only on $G$. No ergodicity, arithmeticity or Diophantine hypothesis is imposed, and the bounds are uniform over all $G$-spaces. Additionally, we prove a decomposition version of this theorem where each measure is decomposed into an almost invariant part and an almost projective part. This comes at the cost of worse constants. The proof introduces several tools: the entropigeonhole method, an entropy-based pigeonhole principle yielding a quantitative Mautner phenomenon; factor functions, quantitative analogues of functions on homogeneous factor spaces; and a fast generation dichotomy in the spirit of growth in groups. In a companion paper these are used to show, among other things, that a discrete subgroup of infinite covolume has injectivity radius at least $c\log^{(4)}r$ somewhere in the ball of radius $r$, which is an effective form of a theorem of Frączyk and Gelander. We also prove rates for Benjamini–Schramm convergence of quotients of higher-rank lattices.

We study a model of opinion dynamics / social learning / peer-review-based market economics on an evolving network, wherein i) each of the first $N$ agents adopts one of two available opinions arbitrarily, and ii) the $(n+1)$-st agent, for $n\geqslant N$, upon arrival, draws a sample of size $k_{n}$, with replacement, from the past agents, such that the $i$-th agent (for $i\leqslant n$) is included in the sample with probability proportional to the number of times they were previously sampled and agreed with. The $(n+1)$-st agent then decides which opinion to adopt i) based on the proportion of sampled agents conforming to each of the two opinions, and ii) according to a stochastic update rule that involves a memory parameter and a rather general reinforcement function. We study both i) the scenario where $k_{n}=k$ remains fixed with $n$, and ii) the scenario where $k_{n}$ grows at a suitable rate with $n$. This model can be represented as an evolving preferential attachment network wherein each vertex is endowed with one of two possible states, and all edges are directed. It can also be framed as a variant of the celebrated elephant random walk. We study the asymptotics of this stochastic process – in particular, the almost sure convergence, and in case of fixed sample sizes, second order fluctuations, of the relative dominance of each opinion, the influence capital and overall network activity.

Cover originally proposed a game in which Bob wrote two numbers on pieces of paper and allowed Alice to pick one and look at it. Cover then showed, as Blackwell is thought to have shown previously, that Alice has a strategy involving the selection of a random number which enables her to guess correctly with a probability greater than 0.5 which of the two numbers was larger. Central to Covers reasoning was that Alice could have picked either number to look at with equal probability. In this paper, we investigate the situation in which Bob obtains the two numbers as the result of an experiment based on a probability distribution, such as the spin of a roulette wheel. We then modify this to envision Alice and Bob conducting separate experiments based on probability distributions. Alice conducts one trial of her experiment and uses Covers strategy to guess whether the result from her experiment will be larger or smaller than the result from Bobs experiment whenever he chooses to conduct a single trial of it. The following two results are among those in the paper. (1) If the two numbers are observations of the same experiment, Alice merely has to observe one trial and apply the Cover strategy to have a probability greater than 0.5 of guessing the larger value, even if the second trial has not yet been conducted (2) There are many nontrivial examples where Alice and Bob conduct different experiments, and Alice can observe one trial of her experiment and apply the Cover strategy to have a probability greater than 0.5 of guessing the larger value, even if many of the specifics of the other experiment have not yet been determined and a trial of it has not yet been conducted.

We consider stochastic heat equation (SHE) defined on 1-d torus $\mathbb{T}$ of the form $$\partial_t u=\Delta u+g(u)\dot{W},$$where $\dot{W}$ is a space-time white noise and $g$ is a real-valued function which is uniformly elliptic (i.e., $|g|$ is uniformly bounded away from 0), and is globally $\beta$-Holder continuous for some $\beta\in(0,1)$. We prove that weak uniqueness holds as long as $\beta>\frac{2}{3}$. The same uniqueness holds for vector-valued solutions where the coefficient $G$ has the same dimension as the white noise. Previously, uniqueness of solutions to the SHE with Holder diffusion coefficient was only established for $\beta>\frac{3}{4}$ via a Yamada Watanabe argument by Mytnik and Perkins (arxiv:0809.0248) without assuming $g$ is nonzero. And when $\beta<\frac{3}{4}$, Mueller, Mytnik and Perkins (arXiv:1201.2767) constructed a non-unique SPDE example satisfying $g(0)=0$. A later generalized coupling argument for nondegenerate $g$ also stopped at the same threshold $\frac{3}{4}$. Our result shows that uniform ellipticity of $g$ restores uniqueness to SHEs in the Holder regime where the same SHE with non-elliptic $g$ and the same Holder regularity are often non-unique in law. This constitutes the first general class of SHE weak uniqueness results in the $\beta\in(\frac{2}{3},\frac{3}{4}]$ regime.

The pseudo-quantum representation re-encodes a finite, irreducible, reversible continuous-time Markov chain with $M$ states as a complex-orthogonal flow on a doubled space of dimension $2M$. After uniformisation, a square-root gauge, a doubling of the state space and a diagonal unitary twist, the chain generates an entire one-parameter group $W_z = e^{zK}$ with $z$ complex, and this single group is the object of the paper. Its real slice, $z$ real, reproduces the stochastic semigroup exactly through an explicit decoding whose probabilistic meaning is the parity of the number of ticks of the uniformised chain. Its imaginary slice, $z = i\theta$, is a finite-dimensional unitary quantum system. The chain and the quantum system are therefore not two analogous models but two restrictions of one entire representation. The setup also produces, with no further input, a pseudo-Schroedinger equation, a bilinear von Neumann equation for a complex-symmetric pseudo-density, and a pseudo-Bloch vector equation on a non-compact quadric. We develop the construction from first principles and work out one model completely, the usual symmetric Ehrenfest urn. Its gauged generator equals $(2/n) J_x$, the spin-$n/2$ operator, its relaxation modes are Lorentz boosts, and its imaginary slice is a depth-one quantum circuit whose Born distribution is the classical urn law under the clock $\theta(t) = \arccos(e^{-2t})$. The same clock identifies the Fisher-Rao lift of the urn trajectory with a rigid spin-coherent-state orbit. This is a preliminary simplified version of a longer paper. It treats only finite state spaces and only the symmetric Ehrenfest urn, and it previews without proofs the asymmetric urn, two queues, the symmetric simple exclusion process and the stochastic Ising model. Appendix primers make the quantum, geometric and information-theoretic language self-contained.

We study the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) generated by the nonsmooth energy kernels $K(z)=-|z|^q$, $0<q<2$. In dimensions $d\ge2$, the corresponding energies are not displacement semiconvex, so standard Wasserstein-gradient-flow theory does not apply. When $d+q-2>0$, we prove global well-posedness on $\mathbb{R}^d$ for probability densities in subcritical $L^p$ spaces, with targets in the same integrability class and with finite moments. We also include the one-dimensional Coulomb endpoint $d=q=1$. For the associated $N$-particle system, we prove global noncollision and fixed-$N$ convergence to the collision-free critical set, a particle-to-continuum criticality principle, and a modulated-energy mean-field estimate that yields convergence of the particle dynamics to the continuum flow as $N\to\infty$ on every finite time interval. We also construct collision-free saddle equilibria, showing that deterministic particle trajectories need not approach global empirical minimizers. For $1\le q<2$, every continuum solution in our class has a narrowly relatively compact orbit, every $\omega$-limit point is Lagrangian critical, and the orbit approaches the Lagrangian critical set. For $0<q<1$, the same conclusions hold under uniform-in-time moment and subcritical $L^p$ bounds. We prove that an absolutely continuous Lagrangian critical point equals the target when the source and target have finite moments of order $q$, except when $0<q<1$ and $d\in\{1,3\}$. Under the preceding uniform bounds, rigidity gives convergence of the continuum flow to the target throughout the rigid part of the well-posedness range. Finally, we show that no initial-data-independent multiplicative MMD decay modulus exists on $\mathbb{R}^d$, and that global Polyak–Łojasiewicz inequalities fail in several whole-space and periodic Riesz/Coulomb regimes.

We prove the logarithmic fractional-moment exponent for the mass of a shrinking ball under the critical two-dimensional stochastic heat flow. The proof builds on the geometric block decomposition of Gu and Tsai and introduces a fractional-product change of measure that preserves block independence and allows the fractional moments to be analyzed.

Let $X$ be a centered isotropic log-concave random vector in $\mathbb{R}^n$. For $\theta\in S^{n-1}$, let $\mu_\theta$ be the law of $\langle X,\theta\rangle$, and let $\Theta$ be uniformly distributed on $S^{n-1}$, independently of $X$. We prove the sharp estimate \[ \textsf{E} W_1(\mu_\Theta,\gamma_1) \le \frac{C}{n}. \] Here the Wasserstein distance is computed after the direction is fixed and is then averaged over the sphere. No symmetry assumption is imposed. A product measure with centered exponential coordinates gives a matching lower bound of order $n^{-1}$. The proof separates the averaged-direction law from the fluctuation among fixed directions. For the first part, a Taylor expansion in the random radius retains a mean-zero cancellation and yields an $O(n^{-1})$ error. For the second, a weighted $L^2$ distance between distribution functions is converted into an exact spherical kernel depending only on $|x|^2$, $|y|^2$, and $\langle x,y\rangle$. Expanding this kernel in $\langle x,y\rangle$, we control its linear, quadratic, and cubic terms using the quadratic variance inequality $\operatorname{Var}(X^\top M X) \le 8\operatorname{Tr}(M^2)$, while fixed-order moment estimates control the remainder.

Bandit algorithms generate data for downstream inference, but adaptive sampling biases post-bandit sample means. We analyze this bias for stable index algorithms, including UCB1 and its generalizations, and derive sharp leading-order expressions for the sample-mean bias and expected $Z$-statistic, in bandit experiments of fixed horizon $T$. Our characterization reveals the algorithmic origin of bias through a key index-function-dependent quantity, which we term effective exploration rate. For example, under UCB1, the effective exploration rate is of order $\sqrt{\log T}$, and the standardized bias of any arm (that is not uniquely optimal) decays at the extremely slow rate $1/\sqrt{\log T}$. We also show how the choice of the index function affects both regret and bias, which reveals a regret-bias trade-off: more exploratory algorithm reduces bias but increases regret. We further show how bias most severely distorts confidence intervals and hypothesis tests when the tested arm is one of the tied-optimal arms. Our sharp characterization for bias uses a novel empirical fluid approximation of the algorithm's sampling dynamics, which may be of independent interest.

We consider the two-dimensional incompressible Euler equation on $\mathbb T^2$ with Gaussian random initial data having independent Fourier coefficients. For every $\sigma>0$, we prove that such a Gaussian measure on $H^\sigma(\mathbb T^2)$ is invariant under the Euler flow if and only if it is supported either on shear flows or on cellular flows. This settles the invariant-measure classification conjecture posed by Bedrossian and Latocca (Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2026). The proof relies on closure of the Fourier support under non-degenerate interactions and an affine relation among the inverse variances along Euler triples.

A two-level analytical framework is presented for modeling random walk transport of messenger ribonucleic acid (mRNA) molecules along neuronal microtubules from soma to synapses. Motivated by empirical observations of mRNA cargo motion, the transport within a dendrite is modeled by a persistent telegraph process with pauses. Theoretical expressions for the probability of traversing the dendrite and the mean time for such travel are derived for different and equal probabilities of persistence. These results are used for the construction of a semi-Markov model of motion of mRNA cargo within the whole neuron. The semi-Markov model provides the probabilities of absorption at a given synapse and corresponding mean first-passage times (MFPTs) from the soma, where mRNA is transcribed. The theoretical expressions, together with experimentally obtained parameter values, are used to calculate MFPTs for neurons with empirically reconstructed morphology. The model predicts that when retrograde persistence is stronger, the MFPT to each synapse is effectively the same. Otherwise, when the persistence is more pronounced in the anterograde direction, the transport in the neuron resembles the motion along a single dendrite – nearly linear dependence of MFPT on the distance between soma and synapse. These findings are theoretically justified when the lengths of dendrites are considerably longer than the distance traversed during a typical run.

We study long-memory continuous-time moving-average processes driven by a Levy process and observed at random renewal times. The sampling scheme introduces an additional source of randomness through irregular observation times. We establish the asymptotic behaviour of normalized partial sums under both finite- and infinite-mean renewal sampling.

We study a spectral regularization of the Dean–Kawasaki equation and quantify how the failure of positivity preservation affects its weak approximation of the empirical measure of independent Brownian particles. For initial densities bounded away from zero, we prove a uniform-in-time weak error measured through the Laplace transform and prove a superpolynomial convergence rate for smooth test functions. When the initial density is allowed to vanish, the negative part of the regularized solution leads to weaker upper bounds, and a one-dimensional example gives a lower error bound ruling out superpolynomial convergence. We also present numerical experiments that confirm the theoretical results and illustrate main observations.

We establish a convex order comparison for products of averages of exchangeable nonnegative random variables. As a consequence, we solve a conjecture of Lamkin and Tkocz (2022) on the log-convexity of moments of sample means, as well as its pointwise version, in a more general form involving arbitrary integer partitions.

In earlier works, we studied natural examples of stable-like random walks on finitely generated nilpotent groups and, more generally, on groups of polynomial volume growth. These walks are driven by measures of radial type, coordinate-wise type, or convex combinations of such measures. In the present work, we explore when two such random walks have comparable behavior, as measured by the equivalence of their associated Dirichlet forms. We develop criteria for this equivalence in terms of both geometric and algebraic features of the driving measures and the group.

An ordered positive bidiagonal factorization (PBF) is used to develop a spectral and probabilistic theory for Markov transition matrices of arbitrary finite bandwidth. The factors determine two families of mixed-type multiple orthogonal polynomials, an entrywise positive $q\times p$ matrix of measures, and a sequence of elementary death-or-stay and birth-or-stay transitions. This yields Karlin-McGregor formulas for transition probabilities, Green kernels, resolvents, potentials, and first-passage transforms without reversibility or block symmetrizability. Rational stochastic PBFs are characterized by ordered finite-urn experiments. Cyclic reorderings give Darboux intertwinings, while a factor-resolved continued fraction gives the first-return law. Grouping states produces a finite-phase quasi-birth-and-death process and a matrix continued fraction for return time and phase. For bounded continuous-time generators, uniformization preserves the spectral data. For unbounded rates, a conservative generator whose shifted leading truncations all admit scalar PBFs must be tridiagonal; nevertheless, $Q=V(T-I)$ preserves arbitrary fixed bandwidth and separates the embedded chain from the holding rates. The theory is explicit for mixed Piñeiro and Jacobi-like systems. For Piñeiro, the exact PBF region and larger structural-band positivity regions are obtained. Jacobi-like beta-convolution weights admit Gamma-factor cancellations, complete cancellation recovering Piñeiro. For $q\in\{2,3,4\}$, the strict ordering conditions give the only open PBF region, with further lower-dimensional cancellation strata. Rational $(3,2)$ examples provide all factors and the resulting Markov models.