Papers

In this paper we consider a two-dimensional generalized continuous-state branching process with mutually enhancing two-way interactions characterized by two stochastic differential equations driven by Brownian motions and spectrally positive $\alpha$-stable random measures. Such continuous-state population dynamics can also be identified as a stochastic Lotka-Volterra type population model. Infinite behavior such as explosion/nonexplosion for this population and staying infinite/coming down from infinity are derived under various conditions on the coefficients involved in the model and the conclusions are rather sharp

For even k, we derive a reflection identity for the basic count distributions in the problem of maximizing the probability that all n k-wise independent Bernoulli variables equal one. After separating the distinguished node n, the other support nodes reflect by s -> n-1-s while the Bernoulli parameter changes from p to 1-p. Corresponding basic weights differ by an explicit positive factor. Thus interior feasibility sets, and local changes of optimal support with their multiplicities, are reflected. This gives a proof of the exceptional-point reflection in Conjecture 5.11 of Berend, Ernst, Kontorovich and Kumar. The argument uses Lagrange interpolation and an elementary binomial reweighting identity; it does not require a conjectured ordering or connectedness of the feasibility sets.

In this paper, we study the diversification properties of convex combinations of iid infinitely divisible random variables. For Lévy processes with bounded variation sample paths, we characterize, in terms of subadditivity and concavity of the transformed Lévy tails, Lévy processes that exhibit the non-diversification phenomenon or the reverse diversification order with respect to the majorization order uniformly over all time horizons. For general symmetric Lévy processes without a Gaussian component, we show that the symmetric 1-stable Lévy process is the only nontrivial process exhibiting either phenomenon. We further investigate convex combinations of components of multivariate infinitely divisible distributions, allowing for dependent and heterogeneous components, and characterize the Lévy measures of multidimensional Lévy processes exhibiting the two adverse diversification phenomena uniformly over all time horizons. Explicit characterizations are obtained for the multidimensional symmetric Lévy processes, multidimensional $\alpha$-stable processes and multidimensional compound Poisson processes. Finally, we show that the non-diversification phenomenon extends beyond Lévy processes to running maxima and integrals of increasing convex functionals of Lévy processes, while both adverse diversification phenomena are preserved for Lévy-driven stochastic integrals with nonnegative deterministic kernels. Applications to ruin theory, storage processes and stochastic volatility are also discussed.

Neural surrogates can substantially accelerate computer-aided engineering (CAE) workflows, but their use in design requires uncertainty estimates that remain meaningful across varying geometries, spatial prediction fields, and engineering quantities of interest. We investigate how established uncertainty quantification (UQ) approaches behave when adapted to geometry-conditioned neural surrogates. We compare one closed-form and two sampling-based approaches-a Gaussian process (GP)-based method, concrete Monte Carlo (MC) dropout, and deep ensembles-and evaluate them on three large, industry-relevant CAE datasets for external aerodynamics and crash dynamics. We examine whether predicted uncertainties have credible magnitudes, identify locations with larger prediction errors, respond to unfamiliar inputs, and remain informative for derived engineering quantities. On the DrivAerStar dataset, where all three methods are compared, each generally assigns higher uncertainty to locations with larger prediction errors, and validation-based rescaling brings interval coverage close to nominal on a disjoint in-distribution test set. Results on AirFRANS and automotive crash also show useful error ranking and interval estimates, but the relative performance of the methods changes with the dataset and evaluation criterion. UQ methods and evaluation metrics should therefore be selected based on the intended downstream CAE decision.

We propose penalized nonreversible Langevin algorithms for sampling from $\pi(x)\propto e^{-f(x)}\mathbf 1_{\mathcal C}(x)$, where $\mathcal C\subset\mathbb R^d$ is a compact convex set. The algorithms combine a squared distance penalty with constant or compatible state dependent skew symmetric perturbations that preserve the penalized Gibbs distribution. For smooth, possibly nonconvex $f$, we derive nonasymptotic total variation bounds for the full gradient algorithm under a log Sobolev inequality. When unbiased stochastic gradients are available, we establish $2$-Wasserstein bounds under global contraction and Lipschitz conditions on the full drift in an adapted quadratic metric. For a fixed penalty parameter, the error relative to the penalized Gibbs distribution decays exponentially to an $\mathcal{O}(\sqrt{\eta})$ neighborhood, where $\eta$ is the stepsize. We also bound the discrepancy between the penalized Gibbs distribution and the constrained target. In a two dimensional quadratic model, we establish nonreversible acceleration by tuning the skew perturbation to the curvature imbalance induced by penalization. With the target accuracy and smaller curvature fixed and initial Wasserstein distances uniformly bounded, tuning the skew perturbation improves the sufficient Euler iteration bound from linear to logarithmic in the curvature ratio. Numerical experiments evaluate the algorithms on constrained Bayesian regression, classification, neural networks, and truncated sampling, and examine the acceleration mechanism in a stochastic quadratic model.

We study Stein solution factors for indicator test functions by a common Mills-ratio method in discrete and continuous settings. In the discrete case, a general birth–death formulation gives exact solution envelopes and recovers the first-increment theory of Brown and Xia. For binomial and Poisson targets, and for negative binomial targets with shape parameter $r\ge1$, the additional algebraic structure of affine birth rates yields improved uniform bounds for the Stein solution. In the continuous case, we consider symmetric densities proportional to $e^{-V}$. Under natural structural assumptions on the potential $V$, Mills ratios yield explicit finite bounds for the indicator Stein solution and show that the uniform derivative and drift Stein factors have the sharp value $1$. Under an additional one-crossing condition, the solution-factor optimization can be carried out exactly and gives the optimal value $1/(4p(0))$. The even-power targets arising in statistical mechanics, including the quartic critical Curie–Weiss law, provide the original examples, and the calculation extends to the wider Subbotin family. For $1<\beta<2$ in the Subbotin family, the exact envelope has two off-center maximizers characterized by a unique incomplete-gamma equation, while the two first-order factors remain equal to $1$. The results exhibit a common discrete–continuous mechanism behind improved and, in the continuous sharp regime, optimal Stein solution factors.

We prove that the subcritical mass gap is comparable to the Lee-Yang gap over free-boundary square boxes raised to the power $8/15$. Random-current moment bounds, a correlation inequality and a finite block expansion give the upper bound. The lower bound follows from the Lee-Yang product, the susceptibility and the critical magnetization. We also obtain uniform factorial bounds on the field derivatives of the pressure.

The exact distribution of the sample variance for $n=3$ is known since Rietz (1931) for the uniform parent, and Royen (2007, 2008) gave a Fourier series for any bounded continuous parent. Neither settles which parents admit a finite closed form, nor which special functions it forces. In coordinates aligned with the cube diagonal the variance constraint becomes a cylinder and the cube cross-section a polygon with $S_3$ symmetry, hexagonal over the central band of the diagonal and triangular near either corner; the CDF is the volume of their intersection, the parent density entering as a weight. A closure hierarchy is then organized by the minimal function class containing the parent density: polynomial parents on any bounded interval always close in elementary terms, a theorem for the whole class; in the negative direction a single explicit parent already suffices, and for a rational one we prove the CDF is not elementary, the obstruction being an irreducible dilogarithmic part. Beyond those two theorems the hierarchy is a set of example-specific obstructions rather than a classification: for an algebraic parent the radial first-kind differential is shown non-elementary on a genus-two curve, and the exponential row is a conjecture supported by the Bessel structure of its radial integral. For the uniform parent we obtain a two-piece formula bifurcating at $Y=1/4$, where the variance disk first circumscribes the hexagonal cross-section at the cube center. For the singular arcsine parent we derive both endpoint laws in closed form and give a six-term approximation accurate to about $10^{-3}$, an accuracy Royen's universal series reaches at about a hundred terms.

In this paper, we analyze square-mean Weyl almost periodic solutions and square-mean Weyl double-measure pseudo almost periodic solutions for a class of semilinear evolution equations in separable Hilbert spaces driven by two-sided fractional Brownian motion with Hurst index $H<1/2$. Due to the non-integrability of covariance density for $H<1/2$, a Hölder-continuity condition on the diffusion coefficient is required. An illustrative example involving a stochastic parabolic equation demonstrates the applicability of obtained results.

We study determinantal planar Coulomb gases with a smooth radial potential, a smooth radial perturbation, jump-type singularities (including the bulk, semi-hard, hard, and soft-edge regimes), and a hard wall region given by a centered disk, a disk complement, or an annulus. We obtain explicit asymptotic expansions of the partition functions up to and including the term of order $1$. Our results unify and generalize many of the earlier works.

This paper establishes a mesoscopic central limit theorem for linear statistics of $\beta$-ensembles or log-gas, as the dimension $N\to\infty$, in the temperature regime $1/N\ll\beta(N)\le 1$. For simplicity, we assume that the potential is one-cut regular and analytic. In this regime, the size of the fluctuations depends on $\beta(N)$ and the mesoscopic scale. We show that there is a transition at a critical $\eta\asymp 1/ N\beta(N)$ between a Random Matrix regime, where the limiting variance is given by the $\mathsf{H}^{1/2}$-norm and a Poisson regime where the limiting variance is given by the $L^2$-norm. We also describe the critical regime. The proof of the CLT relies on optimal local laws at intermediate temperatures and Stein's method for $\beta$-ensembles. In particular, in this regime, it is necessary to construct new correction terms to the classical equilibrium measure to obtain a suitable re-centring of linear statistics and describe their fluctuations. We also obtain a free energy expansion.

For systems of rough stochastic differential equations in the sense of Friz–Hocquet–Lê, with a fixed deterministic rough driver, we prove two results. The first is weak existence with unbounded coefficients. The Itô data is assumed continuous and of linear growth, with no Hölder or Lipschitz continuity and no ellipticity. The rough vector field is a controlled pair of linear growth with bounded derivatives, subject either to a norm-curvature decay or to diagonality with coordinatewise curvature decay; in the diagonal case a coordinatewise upper bound on the diffusion matrix is added. The existing results ask the Itô data and the rough field to be bounded. We remove that boundedness, and ask in exchange for a curvature hypothesis on the rough field, bounds on its derivatives, and a narrower index configuration. The conclusion is weaker on two counts: the bounds that define a solution are imposed in the mean rather than uniformly on the probability space, and the solution's moments stop at the order the initial law carries. The second is the Yamada–Watanabe implication for rough equations: weak existence, together with pathwise uniqueness, yields a strong solution and joint uniqueness in law. A solution here is not defined by a pathwise identity but by two bounds on a remainder, one of them conditional, and an arbitrary enlargement of the filtration need not preserve that one. The enlargements the Yamada–Watanabe construction produces are immersions, however, and along an immersion both bounds transfer unchanged.

We determine the most and least likely shapes for an instance of the critical beta-splitting tree via a connection to data compression and Huffman's minimum redundancy codes. This allows us to answer combinatorial questions about the distribution of clades posed by Aldous and Janson, stated as problem 7 in arXiv:2303.02529.

How do the zeros of a polynomial evolve under the holomorphic heat flow or repeated differentiation? In this work, we develop a unified probabilistic proof to three conjectures on the evolution of polynomial zeros under holomorphic heat flow and repeated differentiation. We study these two evolutions for random polynomials with i.i.d.~roots $z_1,\dots,z_n\sim\mu_0$, where $\mu_0$ on $\mathbb C$ satisfies suitable $2+\delta$ logarithmic moment conditions. For the heat flow of small time $t>0$ and Lipschitz continuous Stieltjes transform of $\mu_0$, we identify the limit distribution $\mu_t$ as explicit push-forward of $\mu_0$ under a transport map. For rotationally invariant $\mu_0$, we characterize the limit after $\lfloor tn\rfloor$ differentiations through its radial quantiles. For instance, the heat flow evolves the circular law into the elliptic and semicircle law, while differentiation moves surviving mass towards the origin. We also show that $o(n)$ derivatives preserve the initial distribution $\mu_0$. In fact, all convergences hold almost surely. The common proof strategy crucially relies on recursion identities from leaving out a root, and concentration inequalities, which lead to self-consistent equations for Stieltjes transforms. This brings a method familiar from random matrix theory to polynomial evolutions. Moreover, it allows for generalizations beyond rotational symmetric distributions and quantitative stability estimates for $o(n)$ differentiations and vanishing heat-flow times.

We study the finite-population approximation of a heterogeneous linear–quadratic mean-field control problem with common noise and random coefficients. Starting from the centralized social planner problem for $N$ non-exchangeable agents, we derive its exact finite-dimensional stochastic Riccati system and identify the scaling of its diagonal, off-diagonal, affine, and scalar components. We then compare this system with the Hilbert-space-valued Riccati system of the continuum model. Our main estimates give quantitative convergence of the complete backward system, including its common-noise martingale integrands. The error separates coefficient and kernel consistency, common-noise approximation, interface effects, exact diagonal corrections, and the mass of the limiting interaction kernel on the discrete diagonal band. We propagate these estimates to the feedback gains and, by a cellwise coupling, to the optimal states, controls, and initial social values. Finally, a cellwise projection of the limiting representative-agent feedback yields a decentralized finite-population strategy whose optimality gap vanishes. Under the stated regularity assumptions, the error bounds are expressed through explicit approximation and concentration moduli. Quantitative bounds on these moduli yield algebraic rates, including an $N^{-1/2}$ rate for the backward system in finite-type regimes; the rates for optimal trajectories additionally reflect the available conditional moment bounds.

How many distinct symbols should a password use? If its length is fixed at $n$ and an observer learns only which symbols appear, the number of compatible passwords is maximized asymptotically when $k/n\to1/(2\log2)$. We ask what changes when, in addition to the length, aggregate information about the repetition pattern is revealed. We model this by fixing a second additive profile $V_k=\sum_i v(J_i)$ at scale $V_k/n\approx\rho$. For $v(j)=\log(j!)$, the profile records the reduction in the logarithm of the number of compatible words caused by repetitions; we show that once the normalized profile $\rho$ exceeds $0.507834\ldots$, the limiting optimal fraction is pinned at $1/2$. For a typical multiplicity profile at the optimal support above this threshold, the excess in $V_k$ is carried by a vanishing fraction of used symbols. We interpret this as a form of non-equivalence of ensembles and extend the mechanism to other profiles and non-uniform allocation models.

We study the stochastic Landau–Lifshitz–Baryakhtar equation with multiplicative noise and homogeneous Neumann boundary conditions on smooth bounded domains in dimensions $d\le 3$. Relying on stochastic maximal regularity results, we establish global well-posedness in a variational strong, intermediate, and weak setting, as well as for initial data in the scaling-critical Besov spaces $B^{d/q-1}_{q,p}$ for every $q\in[2,\infty)$ and $p\in(2,\infty)$.

We study sticky Brownian motion on $\mathbb{R}$ with a set of sticky points $A$. In contrast to the classical case of a single sticky point, the set $A$ may have a more complicated structure and, in particular, may have accumulation points. The stickiness at the points of $A$ is described by a locally finite purely atomic measure $\nu$. We extend the classical time-change approach for sticky Brownian motion to this setting and use it to construct a weak solution of the corresponding stochastic differential equation with a local-time condition. We then prove uniqueness in law of the solution.

We show a large deviation principle for the weighted spectral measure of the Gaussian beta ensemble in the high temperature regime, when $n\beta_n\to 2c \in [0,\infty)$. The resulting rate function has a novel and interesting structure: it is finite only for discrete measures and moreover, in the very high-temperature regime (when $c=0$), additional terms may arise that penalize each support point beyond the first.

This article extends Stein's functional method to establish explicit rates of convergence in the Wasserstein-1 distance for Donsker's invariance principle under weak dependence. Our approach first reduces the problem of bounding distances between distributions on a function space to estimating distances between finite-dimensional marginals. We then employ the block technique, which is standard in Stein's methodology, to handle dependence. While our analysis focuses on $\phi$-mixing sequences, the method extends to other forms of weak dependence admitting suitable covariance inequalities.

Isoperimetric inequalities on the Boolean hypercube play a fundamental role in the analysis of Boolean functions. These inequalities have been established primarily for the uniform measure and for biased product measures, often through Fourier-analytic or inductive arguments. We develop a curvature-based semigroup framework to establish isoperimetric inequalities for measures with weakly dependent coordinates. In particular, we show that a Dobrushin-type condition, together with marginal boundedness of the distribution, implies the local Bobkov inequality, Talagrand's $L^1$–$L^2$ and variance–surface-area inequalities, the Kahn–Kalai–Linial inequality, and the Eldan–Gross inequality. Our results apply to zero-field Ising models with interaction matrix $J$ throughout the Dobrushin uniqueness regime $\|J\|_1<1$. The framework builds on discrete Bakry–Émery theory and gradient estimates and can also be extended to measures on Hamming slices or hypergrids.

We propose a kernel-based estimation framework for Multivariate Generalized Gamma convolutions (MGGC), a class of probability distributions on R d + . MGGC are useful for risk modeling e.g. Their densities are in general not explicit. The construction relies on a new Stein operator for the MGGC class, derived from a partial differential equation satisfied by the characteristic function and expressed as an integral against the underlying Thorin measure. This leads to construct a new kernel and an estimation procedure in the derived reproducing kernel Hilbert space (RKHS); with consistency results. Numerical experiments illustrate the practical behaviour of the proposed estimator.

We study randomized stopping times for a Markov process, described by a progressively measurable intensity $\alpha_t(\omega)$. We prove that every admissible intensity has a Markovian representative $\lambda(t,X_t)$, explicitly obtained from the observed measure and the surviving occupation measure. This representative preserves both the family of surviving mass measures and the joint distribution of the stopping time and the state at stopping. Whenever the relative entropy with respect to a reference intensity $r(t,x)$ is finite, we prove an exact decomposition showing that Markovization does not increase the entropy. As a consequence, variational problems in which a randomized stopping time enters only through its observed measure and an entropy penalty can be reduced to optimization over Markovian intensities $\lambda(t,x)$. For every observed measure admitting a finite-entropy representative, there is a unique minimum-entropy randomized stopping time, and it is Markovian. We discuss the connection with optimal Skorokhod embedding and prove a finite-constraint realization result for Brownian stopping. We also approximate arbitrary observed measures by those generated by bounded Markovian intensities.

We study rough differential equations driven by controlled rough paths with drift. By means of the canonical lift of the controlled driver, we establish pathwise well-posedness and continuity. Under compatible time-shift conditions, we construct differentiable solution cocycles for the associated dynamical system generated by the equations considered here. Furthermore, assuming the corresponding regularity, moment, and spectral hypotheses, we prove the invariance of the local stable, unstable, and center manifolds.

We study affine stochastic Volterra equations on the cone of symmetric positive semidefinite matrices. For scalar kernels acting entrywise on the matrix dynamics, we establish weak existence by exploiting stochastic invariance results for Volterra equations on convex domains and derive a conditional Fourier–Laplace transform formula characterized by matrix-valued Riccati–Volterra equations. As an application, we extend the Gibson–Schwartz commodity model by replacing its variance-covariance structure with a Volterra–Wishart process. The resulting model allows for memory in the variances and for stochastic instantaneous correlation, while retaining affine tractability. Its joint Fourier–Laplace transform admits an exponential-affine representation governed by a matrix Riccati–Volterra equation. While the existence theory considered here excludes kernels that are singular at the origin, shifted fractional kernels remain admissible and provide a tractable specification with power-law memory.

We consider $N \times N$ random band matrices $H = (H_{xy})$ with centered complex Gaussian entries, indexed by points $x,y$ on the two-dimensional discrete torus $(\mathbb{Z} / \sqrt{N} \mathbb{Z})^2$. The matrix entries $H_{xy}$ vanish whenever the distance between $x$ and $y$ exceeds the bandwidth parameter $W$. We prove that if $W \geq (\log N)^{7}$, then, with high probability, all bulk eigenvectors are delocalized. Equivalently, this yields a lower bound of order $\exp(W^{1/7})$ for the localization lengths of two-dimensional random band matrices, improving the superpolynomial lower bound $W^C$ for every fixed constant $C>0$ established in arXiv:2503.07606.

We study a stochastic Navier–Stokes system in which the spatial modes of the transport noise evolve dynamically and are coupled to the resolved velocity. The model couples a Navier–Stokes stochastic PDE to an infinite family of linearized Navier-Stokes equations for the noise correlation modes. These latter equations may contain a hyperviscosity. We prove existence of martingale solutions in two dimensions for hyperviscosity exponent $s\geq1$, and pathwise uniqueness for $s>1$ under additional regularity of the initial noise modes. In three dimensions, we establish existence of martingale solutions for $s>3/2$. The proof combines Galerkin approximations and compactness with estimates adapted to the coupled drift and stochastic terms.

We study the problem of nonparametric estimation of the interaction function in particle systems whose interaction enters through the diffusion coefficient. The particle system is observed discretely over a fixed time interval, and the interaction function is assumed to belong to a suitable nonparametric function class. We propose an estimation procedure based on empirical risk minimization over finite-dimensional sieve spaces, introducing a computable contrast function that exploits the local volatility of the observed trajectories. We derive a general non-asymptotic error bound under the intrinsic norm associated with the statistical problem and, focusing the analysis on multiresolution wavelet approximation spaces, we obtain explicit polynomial convergence rates in two distinct regimes, depending on whether the particle or discretization error dominates. Finally, we establish convergence rates for the classical $L^2$-norm.

We prove a decomposition theorem for canonical processes generated by independent symmetric random variables with log-concave tails, without the $\Delta_2$ condition. The proof uses the classical chaining method, based on admissible partitions and chopping maps. The argument also recovers the majorizing-measure characterization.

We consider a generalisation of the probabilistic reformulation of the supercooled Stefan problem from arXiv:1902.05174 allowing for non-integrable initial data. From an application of Ito's formula, we obtain a relationship between the Laplace transform of the initial data and a transform of the solution, together with its possibly discontinuous free boundary. Using this relationship we give a sharp existence result for global solutions, we extend previous results on the uniqueness of solutions, and we determine the asymptotic behaviour of solutions for a certain class of initial data.