Papers

We study small-noise asymptotics for controlled functional differential equations driven by additive Gaussian noise. The Gaussian noise is modeled on an abstract Wiener space, covering both classical Brownian perturbations and non-Markovian perturbations such as fractional Brownian motion. The drift coefficient is assumed to be non-anticipative, Lipschitz continuous in the state path, and of linear growth. For bounded uniformly continuous cost functionals, we prove game-theoretic lower and upper bounds for the small-noise logarithmic value functions and identify their limit whenever the associated deterministic zero-sum game has a value. In the limiting game, one player chooses the drift control, while the other selects a Cameron–Martin shift of the Gaussian noise, penalized by the corresponding quadratic energy cost. We further provide sufficient Fan-type convexity and concavity conditions under which the game has a value, thereby obtaining a full small-noise Laplace principle. The proof combines the Boué–Dupuis variational representation on abstract Wiener spaces with pathwise stability of the controlled solution map and adapted finite-dimensional approximations of Cameron–Martin shifts.

We develop criteria for collision and non-collision of reversible infinitely many interacting diffusion processes in the real line. The approach is potential-theoretic and is based on capacity estimates for symmetric Dirichlet forms on the configuration space. Our main results are model-independent in the sense that no determinantal or Pfaffian structure, prescribed interaction potential, or explicit labelled stochastic differential equation is required. The non-collision criterion involves only the second and third correlation functions of the reversible measure, whereas the collision criterion is based on a local lower bound for a finite-volume conditional density of the reversible measure. As an application, we identify the sharp collision threshold for the diffusion associated with the $\mathsf{Sine}_\beta$-symmetric Dirichlet form: the collision set is polar if and only if $\beta\ge 1$. This provides an infinite-particle counterpart of the classical collision threshold for the finite-particle Dyson Brownian motion with inverse temperature $\beta$.

We introduce and analyze a fluctuating homogeneous Landau equation with regularised thermal noise. The model is motivated by the nonlocal gradient flow structure of the deterministic Landau equation, the fluctuation–dissipation principle, and the covariance of the martingale fluctuations of a Kac-like conservative Landau particle system. The noise is written in Landau-divergence form, is antisymmetric in the pair of velocities, and is interpreted in the Stratonovich sense after introducing a velocity correlation. To handle the vacuum singularity of the square-root mobility and the nonlocal Stratonovich-to-Itô correction, we replace the mobility by a regular coefficient. For moderately soft potentials, we prove the existence of probabilistic weak solutions to the regularised fluctuating homogeneous Landau equation. The proof is based on a three-level approximation scheme combining Galerkin approximations, coefficient regularisations, artificial diffusion, and compactness in both $L^2$ and $L^1$ frameworks. The solutions satisfy mass conservation, an energy inequality, and the entropy dissipation estimate. Finally, for a special class of admissible noise bases satisfying a tangential divergence-free condition, we obtain a refined entropy inequality in which the expected entropy is non-increasing relative to the initial entropy.

We introduce a stochastic model for the papal conclave in which $n$ cardinals vote repeatedly among themselves until one cardinal receives all the votes. In each round, the probability that a cardinal votes for a given candidate is proportional to the $\alpha$-th power of that candidate's vote count in the preceding round. For $\alpha=1$, the model reduces to the Wright-Fisher model and is dual to Kingman's n-coalescent. We reveal a sharp transition in the absorption time $\mathcal{T}$ at $\alpha=1$. It was known that when $\alpha=1$, $\mathcal{T}$ is typically of order $n$. We prove that for $\alpha>1$, it drops to order $\textit{loglog n.}$ In contrast, for $\alpha<1$, $\mathcal{T}$ is typically at least $\exp(\Omega(n))$. We also prove a sharp phase transition in the identity of the winner when $\alpha>1$. For every positive integer $k$, if $2^{1/k}<\alpha<2^{1/(k-1)}$ (where we write $2^{1/0} = +\infty$), with probability tending to 1 as $n\to\infty$, the eventual winner is the unique leader after round $k$. These results show that reinforced voting processes reach consensus remarkably quickly even for large electorates.

We establish a general Poisson approximation for rare local patterns in critical Bienaymé–Galton–Watson trees with offspring distribution $\mu$ in the domain of attraction of a stable law, conditioned to have a large number of vertices. A pattern is specified by a sequence-dependent mark on fringe subtrees. If marked fringe subtrees remain microscopic and nearby marked occurrences have negligible clustering, then their count is asymptotically Poisson in total variation whenever its mean remains bounded; when the mean diverges, the count satisfies a law of large numbers. The main difficulty is the global dependence created by size conditioning. We overcome it by combining the cyclic-shift representation with a refined form of the Chen–Stein bound and a bridge-removal estimate controlling the interaction between a local mark and the remainder of the conditioned random walk. For non-fringe patterns, overlapping occurrences may form clusters and the raw count need not be asymptotically Poisson. We introduce declumped indicators which select boundary witnesses of these clusters and prove a general Poisson approximation for their count. As applications, we obtain sharp asymptotics for the maximum leaf-height, equivalently the maximum protection number, and for the height of the largest complete $r$-ary tree appearing as a non-fringe subtree. Unary-chain maxima, and the maximum leaf-height when $\mu_1>0$, exhibit lattice-modulated Gumbel behavior. Complete $r$-ary patterns for $r\ge2$, and the maximum leaf-height when $\mu_1=0$, are localized on one or two consecutive integers. The results require no exponential moment and include offspring distributions with infinite variance.

We introduce a two-factor stochastic volatility model in which the mean-reversion speed of a Cox–Ingersoll–Ross variance process is an observable functional of a market attention field rather than a latent factor. The field admits an exact Feynman–Kac representation and generates closed-form volatility forecasts. The model nests the Heston volatility model, provides measurement-stable dynamics under observable data aggregation, and preserves its structure under an equivalent martingale measure. A mean-field analysis establishes convergence of the underlying particle system, while simulations demonstrate volatility and option-pricing differences following attention shocks.

We introduce the concept of rough additive functions, which extends rough paths theory to line integrals of distributional 1-forms. In the context of gauge theory, we use this notion to define controlled gauge transformations and holonomies via RDEs. One of our main results is a rough version of Uhlenbeck compactness for distributional connections based on rough additive functions. The main ingredient is a singular elliptic PDE to obtain a Coulomb gauge. We solve this PDE using regularity structures and a method inspired by the implicit function theorem. Surprisingly, the model needed to solve the equation is determined entirely from rough additive functions, which is a simpler and geometrically more natural object. We apply our results to the Yang-Mills measure on the unit square, showing that it has a gauge-fixed representation with optimal regularity. Although we focus on the unit square in this article, we expect our results to apply to more general surfaces.

We study pathwise uniqueness for cyclic catalytic stochastic differential equations whose state-dependent square-root diffusion coefficients are non-Lipschitz and degenerate on the boundary. The approach is the direct construction of a strong solution using a Malliavin compactness criterion. The key is the development of a new family of boundary-sensitive weighted Malliavin estimates for the tangent processes of the smooth approximations. Pathwise uniqueness then follows from the dual Yamada-Watanabe argument together with the weak uniqueness available in the literature.

Regime-conditioned transition probabilities and moments are basic inputs for prediction and decision making in hybrid systems, but their short-time infinitesimal structure is not directly observable. We study nonparametric estimation of regime-indexed semigroup blocks for switching diffusions observed together with their regimes. A Dynkin–Taylor expansion identifies the first two block-generator coefficients, and localized probes recover switching intensities, drift, and diffusion. Under shrinking meshes, a common-design difference cancels the localized level and yields a martingale array; nonoverlapping second differences preserve within-block covariance and produce the variance factor two. The resulting first- and second-order estimators are asymptotically normal, admit feasible studentization, and support short-horizon approximation and specification checks for interactions among primitive coefficients. Fixed-mesh local-polynomial theory and a numerical study complement the recovery results.

Let $\Lambda$ be a finitely generated abelian group and $\Gamma=\mathbb Z^{*d}$, we study the Cayley graph of the wreath product $G=\Lambda\wr\Gamma$ with natural set of generators and their inverse $S$. First, we establish a random matrix model $X_N=\sum_s X^{(s)}_N$ where the sum is indexed by the set $S$. As the size $N$ of the matrices goes to infinity, the traffic distribution of the $X^{(s)}_N$'s converges to that of the image of these generators in the reduced $C^*$-algebra of $G$. In particular, the spectral measure of $X_N$ converges toward that of the Cayley graph of $G$ with generators $S$. Moreover, in the case $\Gamma=\mathbb Z$, we establish a central limit theorem for linear statistics of this random matrix model. Then, we exhibit a formula for the asymptotic $R$-transform and derive the second-order distribution of the limit of $X_N$ in terms of its limiting first-order distribution.

We focus on a stochastic system that models a collection of $N$ identical oscillators with unidirectional coupling, perturbed by white noise. The unidirectional coupling breaks the symmetry of mutual dependence among the oscillators. It is shown that the associated random dynamical system $\phi$ admits a dynamical order, meaning that $\phi$ admits a simple asymptotic one-dimensional structure that is totally ordered with respect to the standard order in $\mathbb{R}^N$. Our approach takes a novel geometric perspective: we introduce a random dynamical system $\Phi$ on a smooth Riemannian manifold $M$, diffeomorphic to $\mathbb{S}^1 \times \mathbb{R}^{N-1}$, by “wrapping” the random system $\phi$ from $\mathbb{R}^N$ onto $M$. By choosing an appropriate random cone field $\mathcal{C}_M$ on $M$, we show that $\Phi$ is a differentially positive random system on $M$, which is a random counterpart of the differentially positive systems introduced by Forni and Sepulchre. We further demonstrate that the traditional well-known horizontal curves can be identified as the conal curves on $M$, thereby providing a crucial tool for establishing the dynamical order of the random system $\phi$.

HyperLogLog is a now classic probabilistic algorithm that provides an approximation of the number of distinct elements in a massive dataset, using only one pass over the data. In the original article, Flajolet, Fusy, Gandouet, Meunier (2007) provided a sharp analysis of the expectation and variance of the output, using explicit formulas analyzed using poissonization and Mellin transform. In this short article, we revisit the analysis of HyperLogLog with a more probabilistic viewpoint. This allows us to establish exponential deviation inequalities for the HyperLogLog estimator. The methods are elementary, but the estimates are non-asymptotic and totally explicit.

We establish a convex ordering result for count distributions arising from a parasite acquisition model with density-dependent deaths. If the ratio of two death-rate sequences is non-increasing, then the corresponding mean-parametrised distributions are ordered in the convex order. This yields, as a special case, the convex ordering of the mean-parametrised Conway-Maxwell-Poisson distribution in its dispersion parameter.

We develop a structural credit-risk model under incomplete information in which investors observe firm value only indirectly through noisy market signals and scheduled corporate disclosures. While disclosure dates are known in advance, their informational content is random, leading to stochastic discontinuities in the observation process. We derive the Kushner-Stratonovich equation for structural credit-risk models with endogenous default by applying the nonlinear filtering framework with predictable jumps. We then study the valuation and local risk-minimization hedging for default-sensitive securities under partial information. The interaction between predictable disclosure events and endogenous default produces discrete adjustments in the conditional default compensator, leading to announcement-driven distortions in credit spreads and hedge ratios that are absent from classical diffusion-based and inaccessible-jump models. Numerical experiments illustrate how scheduled disclosures affect filtered default probabilities, Credit Default Swaps (CDS) spreads, and hedging strategies, generating characteristic pre-announcement dynamics in credit spreads.

The global-in-time well-posedness and uniform large deviation principles (LDPs) are investigated for a wide class of Mckean-Vlasov stochastic non-local fractional $(\alpha,p)$-Laplacian equations with $\alpha \in (0,1)$ and $p>2$ driven by superlinear multiplicative noise defined on the whole space $\mathbb{R}^d$, where the non-local nonlinear fractional $(\alpha,p)$-Laplace operator is defined by a singular, symmetrical and translation invariant kernel function, the distribution-dependent drift terms have arbitrary polynomial growth and the distribution-dependent diffusion terms have superlinear growth. The global-in-time well-posedness is established under these conditions by using the monotone method and a domain expansion argument. Under additional conditions on the growth of diffusion terms, we establish the Freidlin-Wentzell and Dembo-Zeitouni uniform LDPs by using the generalized weak convergence method developed by Salins (Probab. Surv., 16:99-142, 2019). The idea of uniform tail-ends estimates, the pseudo monotone technique and the Arzelà-Ascoli theorem are combined to prove the weak-to-strong continuity of solution operators of the controlled equations in order to overcome many difficulties caused by the noncompactness of Sobolev embeddings on $\mathbb{R}^d$ and the nonlinearity of the fractional $(\alpha,p)$-Laplace operator. The superlinearly growing diffusion term is carefully controlled by using the dissipative drift terms and several algebraic inequalities.

Given an isotropic, exchangeable, and unconditional random vector $\mathbf X$, we consider the sample covariance matrix constructed from i.i.d. copies of several tensor models of $\mathbf X$, such as the tensor power $\mathbf{X}^{\otimes d}$. Under appropriate moment conditions on $\mathbf X$, we show that almost surely, the empirical spectral distribution converges weakly to the Marchenko-Pastur law. This extends previous results which required the coordinates of $\mathbf X$ to be independent. As we demonstrate, our extension applies to many new random vectors $\mathbf X$ of interest.

This paper investigates the optimal surplus management problem of an insurance company operating in a financial market with stochastic interest rates and jump-driven liabilities. The insurer dynamically allocates its surplus between a risky stock and a risk-free zero-coupon bond while facing insurance claims modeled by a compound Poisson process with exponentially distributed claim sizes. The short term interest rate follows a Cox-Ingersoll-Ross (CIR) process, which captures mean-reverting dynamics commonly observed in term structure models. The insurer maximizes the expected exponential utility of terminal surplus. Using stochastic control techniques, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation. Although the exponential utility structure suggests an exponential affine representation, the interaction between the interest rate hedge and the surplus state generates quadratic surplus terms in the HJB equation. To obtain a tractable formulation, we adopt a normalized surplus projection method, which provides an approximate reduction of the full three-dimensional problem to a nonlinear system of partial differential equations (which is subsequently numerically validated). The optimal investment policy admits an economically meaningful decomposition consisting of a myopic demand component and an interest rate hedging component. Numerical experiments illustrate how the optimal strategy and the surplus distribution depend on interest rate volatility, claim intensity, and risk aversion. The results highlight the importance of jointly modeling stochastic interest rates and insurance liability risk when designing optimal investment policies for insurance companies.

We study the problem of reconstructing the probability measure of a multi-group version of the Curie-Weiss or mean-field model of ferromagnetism from a sample of the voting behaviour the population. While originally used to study phase transitions in statistical mechanics, the Curie-Weiss or mean-field model has been applied to study phenomena where many agents interact with each other, in particular in case of a heterogeneous population with identifiable subpopulations. The degree of social cohesion within social groups manifests in the way the members of the group influence each others' decisions as well as how they behave under outside influence from voters belonging to another group. Contrary to single-group Curie-Weiss models, here we have a larger number of coupling parameters which have to be estimated. While the maximum likelihood estimator of the coupling parameters has desirable statistical properties in theory, computational challenges make applications to larger populations impractical. Therefore, we analyse an estimator based on asymptotic approximations to the behaviour of the Curie-Weiss model valid for large populations. Due to the wide applicability of models such as Curie-Weiss, the estimator is potentially useful in disciplines such as political science, sociology, automated voting, and preference aggregation.

We derive statistical limit theorems for sample-based approximations of infinite-horizon discounted stochastic optimal control problems in discrete time. Our first result is a functional central limit theorem for the sample-based value function under a uniqueness-type condition on population optimal policies. The limiting law is a mean-zero Gaussian process characterized by a linear fixed point equation that resembles a dynamic programming principle. We compare these asymptotics with those obtained from sample-based policy optimization and illustrate that their limiting variances can be different. We also derive a limit theorem for models with nonunique optimal policies, where the limiting law may be non-Gaussian. Applications to inventory control and renewable harvesting illustrate the theory.

KPZ-type extremal fluctuations have recently been proved for several models of random walks in space-time random environments (RWRE) in $1+1$ dimensions. A general moment criterion predicts the spatial scale at which this behavior should occur, but does not by itself guarantee non-trivial fluctuations at that scale. In this paper, we show that the averaging process provides an instance in which this criterion is not sharp: because of a degeneracy in the update mechanism, the actual KPZ limit appears only beyond the predicted scale. The relevant critical contribution arises from the interplay of two distinct fluctuation mechanisms, a phenomenon that appears to be rather special within the RWRE setting. Our proof builds on Dobrushin-type local-time limit theorems for zero-sum additive functionals, refined estimates for tilted $k$-point motions, and the recent moment-based axiomatic characterization of Cole–Hopf solutions to the one-dimensional KPZ equation. We also identify the behavior on the two sides of the critical scale, thereby sharply separating the subcritical, critical, and supercritical regimes of the model.

We study a network-archaeology problem for a dynamic graph whose latent substrate is a random recursive tree and whose observed topology is enriched by an independent homogeneous Erdős-Rényi shortcut layer. From a single unlabeled snapshot, the goal is to construct a confidence set of deterministic size for the first vertex. Since shortcut edges create cycles, the usual tree-based arguments using Jordan centrality do not apply directly. Our method uses auxiliary subcritical bond percolation to expose a tree-like renormalized structure: retained recursive-tree clusters form heavy-tailed blobs, retained shortcuts connect these blobs through a subcritical rank-one random graph, and large components are leading backbone blobs decorated by subcritical shortcut pieces. Applying Jordan centrality inside the largest auxiliary percolation components then gives a deterministic-size root confidence set for the cyclic observed network.

In this paper, we establish a moderate deviation bound for Gaussian maxima and the variance asymptotics of the Sherrington-Kirkpatrick free energy at criticality based on Gaussian convexity. First, let $(X_1,\ldots,X_N)$ be centered Gaussian vector with $\operatorname{Var}(X_i)\leq 1$. Suppose that, for fixed $\alpha\in(0,\sqrt 2)$ and $\kappa>0$, $\mathbb{E}\max_iX_i\geq\alpha\sqrt{\log N}$ and $\mathbb{E}\max_iX_i+\kappa\sqrt{\log N}\leq\sqrt{2\log N}$. We prove that $$ \mathbb{P}\left(\max_iX_i\geq \mathbb{E}\max_iX_i+\kappa\sqrt{\log N}\right) \leq N^{-\kappa^2/(2-\alpha^2)+o(1)}. $$ This answers a question of Ding, Eldan and Zhai. The exponent is sharp, as witnessed by an equicorrelated Gaussian field. Second, for the Sherrington–Kirkpatrick model at the critical inverse temperature $\beta_c=1/\sqrt2$, we prove $$ \operatorname{Var}\bigl(F_N(\beta_c)\bigr)=\frac16\log N+O(1). $$ Our argument provides the variance asymptotics at the critical temperature from an entropy perspective, via a route distinct from that of Du and Huang. For the upper bound, we express the variance as an entropy under exponential tilting and identify this entropy with the Kullback–Leibler divergence of a Gaussian synchronization model. Its derivative is then bounded using the I-MMSE formula, information percolation, and estimates for the susceptibility of the critical Erdős–R'enyi random graph. For the lower bound, we combine Gaussian convexity applied at the replica parameter with an estimate for inverse moments on the sphere and an identity relating GOE eigenvalue densities in consecutive dimensions.

We consider a fractional Brownian motion $B$ with Hurst index $0<H<1/2$, and its maximiser $\tau$ on $[0,1]$. We show that the rescaled process $a^H(B_{\tau+\,\cdot\,/a}-B_\tau)$ converges in $C_{\mathrm{loc}}(\mathbb R)$ to a limiting tangent law that is $H$-self-similar, supported on nonpositive paths pinned at zero, and rerooting-rescaling invariant: rerooting the limit process at its maximum on any fixed compact interval separated from zero and rescaling again asymptotically reproduces the same law. We also identify the tangent law as the limit of two-sided finite-grid hard-wall laws as the mesh vanishes and both horizons diverge, which can informally be interpreted as conditioning fractional Brownian motion on a nonpositive path. As an application, we consider persistence probabilities for fractional Brownian motion: a tilted variant of $B$ yields a different tangent law with a finite left horizon and an infinite right horizon and we show that \[ \mathbb P(B_t\leq1\text{ for all }0\leq t\leq T) = \big(C+o(1)\big)\,T^{-(1-H)},\quad \text{as }T\to\infty, \] where the leading order coefficient $C\in(0,\infty)$ has an explicit representation in terms of the expected maximum and the tilted tangent law.

Let \(\cL_D^\theta\) be a Brownian loop soup of intensity \(0<\theta<1/2\) in a bounded smooth planar domain. We estimate the probability that one cluster of intersecting loop traces meets a fixed number of separated shrinking bulk discs. For every fixed \(q\ge3\), this probability is, up to an arbitrary loss in the logarithmic exponents, the product of the corresponding one-arm probabilities. The upper bound is not obtained by multiplying pairwise estimates or by a formal application of BK. We condition on the loops not confined to one target collar, charge a local arm down to their penetration radius at every target, and control the joint penetration depths by a multi-target Brownian loop-measure estimate. The latter follows from a marked Schur-complement expansion on finite killed networks, uniform planar capacity estimates, and a random-walk-loop-soup coupling. The lower bound uses winding separators, one macroscopic bridge loop, and Poisson FKG. The one-window exponent and the two-window estimate are due to Jego, Lupu and Qian.

In the paper we introduce vector-valued dynamic extended graphons in order to obtain the mean-filed limit for certain class of problems with adaptive weights. We impose fairly general assumptions on the matrix of initial connections, which allows us to cover the case of sparse graphs.

We propose a novel probabilistic formulation for optimal stopping mean field games (OS-MFGs) with randomized strategies. We characterize mean field equilibria through a new class of coupled forward-backward systems, termed coupled reflected forward-backward McKean–Vlasov stochastic differential equations (MKV-RFBSDEs). An equilibrium is represented by a quintuple $(X,Y,Z,A,L)$, where $L$ is an adapted, $[0,1]$-valued, non-increasing càdlàg process representing the randomized stopping strategy. The optimality of randomized stopping strategies is characterized through two novel Skorokhod-type conditions involving $L$. This characterization is new even for classical optimal stopping problems without mean field interactions. We rigorously prove an equivalence between solutions of the MKV-RFBSDE system and OS-MFG equilibria in randomized strategies. We establish the existence of equilibria by applying the Kakutani–Fan–Glicksberg fixed-point theorem to a set-valued best-response correspondence, relying on new stability, compactness, and continuity results for the coupled MKV-RFBSDE system. We also prove uniqueness under suitable conditions. Under alternative monotonicity assumptions, we develop a new order-theoretic approach based on Tarski's fixed-point theorem, yielding the existence of extremal equilibria and constructive schemes for the minimal and maximal solutions. We further show that a mean field equilibrium induces an approximate Nash equilibrium for the associated $N$-player stopping game. Finally, we connect our probabilistic formulation with the analytical approach characterized by a coupled system of constrained partial differential equations.

Let $D$ be an oriented graph (a digraph with no directed 2-cycles) with maximum degree $\Delta\ge 1$, equipped with nonnegative arc weights of total weight $w(D)$, and let $\mathrm{fas}_w(D)$ denote the minimum weight of a feedback arc set of $D$. Alon (2002) proved $\mathrm{fas}_w(D)\le(\frac{1}{2}-\frac{1}{16\sqrt{2\Delta}})w(D)$. We determine the optimal constant: \[\mathrm{fas}_w(D)\le(\frac{1}{2}-\frac{\sqrt{2}}{6\sqrt{\Delta}})w(D).\] In fact, we show a stronger result: $\mathrm{fas}_w(D)\le\frac{1}{2}w(D)-\frac{\sqrt{2}}{12}\sum_v w_2(v)$, where $w_2(v)$ is the $\ell_2$-norm of the weights of the arcs incident with $v$. Both bounds are attained by the unit-weight directed triangle, so the constant $\sqrt{2}/6$ is best possible (already among unweighted oriented graphs). The proof combines the vertex-peeling scheme of Berger and Shor with a continuous random-ordering analysis: realizing the random order by independent uniform labels renders the expected local imbalance at each vertex exactly an integrated Khintchine-type functional, and the theorem reduces to the sharp evaluation \[\inf_{\|a\|_2=1}\int_0^1 \mathbb{E}|\sum_j a_j B_j(q)|\,dq = \frac{\sqrt{2}}{6},\] where the $B_j(q)$ are i.i.d. Bernoulli$(q)$ random variables, which we prove via Fourier analysis. The proof also yields a randomized, near-linear-time algorithm attaining the bounds in expectation.

Exact uniform and local moduli of continuity for anisotropic Gaussian random fields are established under a general framework based on correlation bounds for pairwise increments. This framework does not necessarily require strong local nondeterminism (SLND), stationarity of increments, or spectral-type representations. As an application, we solve an open problem about sharp moduli of continuity for a class of linear stochastic PDEs in a particular case where the solution is $C^{1-}$ in space and SLND is not available. In this case, we establish decorrelation of the spatial increments via new localization estimates, which may be of independent interest. We also briefly discuss an example about Gaussian Volterra processes.

We prove constant lower bounds for the spectral gap of the following random walks on unitary groups $\mathsf{SU}(2^n)$ on $n$ qubits. (i) Random Pauli Rotation: choose an $n$-qubit Pauli operator $P$ and an angle $\theta \in [0,2\pi)$, both uniformly at random, and apply $e^{\mathrm i \theta P}$. (ii) Brickwork Random Unitary Circuit: choose $n-1$ unitaries $U_{i}$ uniformly at random from $\mathsf{SU}(4)$ independently, and apply $U_{2j-1}$ on two qubits $2j-1, 2j$ and then $U_{2j}$ on two qubits $2j, 2j+1$. Importantly, the spectral gap lower bounds are independent of $n$ and apply for all finite dimensional unitary representations of $\mathsf{SU}(2^n)$ uniformly, including those that appear in unitary $t$-designs. We also prove analogous constant-gap results for orthogonal and Clifford groups; the latter is indispensable for our result on the Brickwork Random Unitary Circuit.

We prove an Eyring–Kramers law for metastable transitions of the hypoelliptic third-order Langevin diffusion in the low-temperature limit. This diffusion is a three-level Markovian lifting of Langevin dynamics: the Brownian noise acts only on the highest auxiliary variable and reaches the position variable through a third-order H"ormander chain. For a double-well potential with a unique index-one transition saddle, we determine both the Arrhenius exponential scale and the sharp prefactor of the mean transition time. The prefactor is governed by the unique positive unstable rate of the deterministic linearization at the saddle, equivalently the positive root of a cubic polynomial. Our proof combines a weak-capacity framework with a saddle-adapted boundary layer, an explicit Gaussian current calculation, committor localization, and intrawell flatness. Under matched kinetic normalizations, the resulting metastable prefactor is strictly smaller than its underdamped counterpart. A numerical experiment for a one-dimensional double well illustrates the Arrhenius scaling and the predicted prefactor comparison.