Papers

In this work, we are concerned with a class of multi-dimensional time-inhomogeneous stochastic differential equations (SDEs) on $\R^d$ driven by pure-jump Lévy processes, where the drift coefficient $b(t,x)$ satisfies $\lim_{t\to \infty}b(t,x) =0$ for every $x\in \R^d$. On account of three regimes associated with the index $\alpha$ of large jumps corresponding to the driven Lévy noise, we investigate the quantitative asymptotics of the corresponding rescaled processes. More precisely, for $\alpha\in (0,2)$, we prove that the rescaled processes, governed by time-inhomogeneous SDEs subject to additive processes, converge with respect to suitably chosen Wasserstein distances to time-homogeneous SDEs driven by symmetric $\alpha$-stable processes. Notably, the driven noise in the limiting SDEs depends only on large jumps of the underlying additive processes. In case of $\alpha\ge2$, a phase transition occurs and a diffusive phenomenon arises. In particular, in the setting $\alpha>2$, we establish the ergodicity of the rescaled process by means of asymptotic pseudotrajectories. The resulting time-homogeneous limiting SDEs are driven by Brownian motions, even though the transformed time-inhomogeneous SDEs are driven by (discontinuous) additive processes, and the effective noise intensity is determined by the entire Lévy measure of the original pure-jump process. As far as the critical case $\alpha=2 $ is concerned, we demonstrate that the noise intensity of the limiting SDEs driven by Brownian motions relies merely on the large-jump part of the Lévy measure.

We investigate the persistence of most probable paths through the Onsager–Machlup functional for multidimensional stochastic differential equations driven by fractional Brownian motion with time-dependent diffusion coefficients and Hurst parameter $H\in(1/4,1)$. Under suitable structural and variational conditions, deterministic trajectories remain most probable paths for sufficiently small noise in both the fixed-endpoint transition problem and the free-endpoint evolution problem, whereas sufficiently large noise destroys their local minimality. More generally, when exact persistence does not hold, global most probable paths converge to the corresponding trajectories of the noise-free system in both the uniform and Hölder topologies at the rate $O(\epsilon)$. We further analyze the second variation along periodic deterministic trajectories over time intervals of length $NT$. For $H>1/2$, positive definiteness, and hence local minimality, is lost on sufficiently long intervals. For $H\in(1/4,1/2]$, long-time positive definiteness holds for the fixed-endpoint problem, but this conclusion does not directly extend to the free-endpoint setting. We also establish the persistence of KAM tori in nearly integrable Hamiltonian systems in the sense of most probable evolution paths. Finally, a two-dimensional numerical example illustrates the persistence of deterministic trajectories under small noise and their pronounced deviation under large noise.

We prove a delocalisation bound for eigenvectors of Wigner matrices with the precise relationship between the size of the largest entry and the decay exponent of the probability. We also prove that the largest entry of an individual eigenvector and the largest entry of all eigenvectors are both Gumbel distributed. The proof is based on the inclusion-exclusion principle, the small probability comparison of Erdős–Xu, and partial diagonalisation of Gaussian divisible matrices.

We prove that the skew stickiness ratio converges to two at short maturity under local volatility models. This appears to be the first rigorous proof of this limit for a general time-dependent local volatility function. As a by-product, we strengthen the one-half rule of the implied volatility skew by removing uniform ellipticity and global bounds on spatial derivatives of order at least two. The proof uses a first-order Watanabe expansion.

The random-cluster model with cluster parameter \(0<q<1\) is conjectured to exhibit negative dependence, but even pairwise negative correlation between distinct edges remains open on general finite graphs. We first establish a reduction showing that the pairwise conjecture is equivalent to its restriction to regular graph sequences of diverging degrees $(d_n)$, even for edge pairs at distance \(o(d_n/\log d_n)\). We then prove strict pairwise negative correlation for a broad class of high-degree regular graph sequences satisfying a two-scale edge-isoperimetric condition. More precisely, for every fixed \(p\in(0,1)\), all sufficiently large graphs in the sequence have strictly negative covariance between any two distinct edge indicators at distance \(o(d_n/\log d_n)\), uniformly in \(q\in[0,1)\) and in the choice of the two edges. In particular, the result includes the \(q=0\) endpoint, corresponding to Bernoulli bond percolation conditioned to be connected. The isoperimetric hypothesis is satisfied by high-degree expander families and, with probability tending to one, by uniformly random regular graphs of diverging degree; its two-scale form also allows product geometries such as hypercubes and fixed-side high-dimensional discrete tori. The proof is based on a polymer representation and cluster expansion, together with a geometric identification of the leading contribution to the two-edge correlation.

We initiate a geometric theory of $p$-Brownian motion, the nonlinear Markov process associated with the $p$-Laplacian introduced by Barbu-Rehmeier-Röckner. More precisely, we analyze its radial processes thoroughly, relative to an arbitrary center and in every dimension. On the one hand, we show explicit Tanaka–Meyer semimartingale formulas; our consequential characterization of nontriviality of the associated local times reveals notable differences to classical Brownian motion. In parallel, we establish sharp exit time estimates as well as scaling estimates for the self-similarly rescaled radial process. We also prove isometry of the corresponding marginal laws in each Wasserstein distance. Our contributions equally cover the Leibenson process - the nonlinear Markov process associated with the "porous medium equation" with $p$-Laplacian - recently introduced by Barbu-Grube-Rehmeier-Röckner.

We obtain multivariate martingale central limit theorems in $p$-Wasserstein distance with respect to the $\ell_r$ norm in $\mathbb{R}^d$ for $p\geq 1$ and $r\in [1,\infty]$, which generalize the results for $p=1$ and $r=2$ in the literature. As corollaries, we obtain the Yurinskii coupling and Cramér-type moderate deviation results. We also provide an illustrative application to the stochastic gradient descent algorithm. To prove our main results, we combine Lindeberg's swapping argument with a new Gaussian convolution inequality controlling the $p$-Wasserstein distance between a Gaussian convolved with a perturbation and the Gaussian with matching mean and covariance matrix. The latter is obtained by developing the recent line of research on $p$-Wasserstein bounds.

We compute the fourth-order correction to the full-collision asymptotics of the correlation functions of the $\mathrm{Sine}_\beta$ process. For $m \ge 2$ and $m\beta > 3$, the normalized correlation has an expansion through order $\epsilon^4$, with an explicit rational coefficient depending on the centered profile only through its fourth power sum and the square of its second power sum. The remainder is $o(\epsilon^4)$, locally uniformly in the collision profile. We evaluate the required fourth inverse moments of the Hua-Pickrell environment by finite-dimensional Ward identities and prove their convergence using characteristic-polynomial derivative bounds. A fourth-order expectation-Taylor lemma handles the full range $m\beta > 3$ without requiring fourth moments of every analytic derivative. For general unitary ensembles with a $C^4$ confining potential and a regular bulk point, we prove convergence of the finite-particle fusion coefficients through fourth order and a joint second-order limit, using complex kernel universality and divided differences. This unitary result imposes no fused-environment hypotheses. For arbitrary $\beta$, we retain a conditional quadratic transfer criterion.

We demonstrate explicit transformations of three independent uniforms with exact Gamma$(a,1)$, $a>0$, Poisson$(\lambda)$, $\lambda>0$, and Binomial$(N,p)$, $N\geq 1$, $0\leq p \leq 1$ laws using only elementary operations.

We develop a randomly-localized energy method to derive qualitative results on the support propagation of stochastic porous media equations with linear conservative noise. Unlike in previous works, where energies are localized by weighting them with a spatial bump function, we weight them with profiles which are solutions to stochastic transport equations. This modulates out the support propagation due to the conservative noise term, and energy arguments—which otherwise fail in this setting—are again applicable. As a result, finite speed of propagation along with sufficient and necessary conditions on the existence of waiting time phenomena (modulo stochastic transport) are proven. These methods and results demonstrate, on short time scales, that the support propagation may be disintegrated into two independent parts, one due to the evolution of the porous media equation and one due to stochastic transport.

The Derrida-Retaux continuous branching process, which is conjectured to be the scaling limit of the corresponding discrete model and of many other critical hierarchical renormalization models, is a process of cells evolving inhomogeneously on the time interval $[0, 1)$ via linear growth of each cell and independent splitting of cells at a specific rate. In this article, we show that the Derrida-Retaux continuous branching process is, on the one hand, encoded by a family of processes converging to a Brownian motion and on the other hand, fully obtained from the continuum Brownian tree encoded by the limiting Brownian path by time reversal and through length erasure of this tree. This direct representation explains the specific laws governing the significant quantities featuring the model (total mass, number of cells, law of large numbers) and provides a better understanding of its dynamics.

The Ball-Nayar-Tkocz conjecture asserts that, for independent real random variables $X,Y$ with a common log-concave density, the function $t\mapsto h(\sqrt tX+\sqrt{1-t}Y)$ is concave on $[0,1]$. Gaussian distributions have this property. We construct strongly log-concave counterexamples arbitrarily close to the Gaussian and study their persistence under Gaussian smoothing. Let $X$ have mean zero and variance one, let $G$ be an independent standard Gaussian, and let $q\in[0,1]$ denote the signal variance proportion in the smoothed variable $\sqrt qX+\sqrt{1-q}G$. In the asymmetric case, for every fixed $0<q\le1$, there are counterexamples for which the entropy of the weighted sum of two independent copies of the smoothed variable has strictly positive second derivative near the endpoints. In the symmetric case, for every $4/7<q\le1$, there are counterexamples for which an unequally weighted sum has strictly greater entropy than the equally weighted sum. Both families have smooth, strictly positive densities and are strongly log-concave before and after smoothing. The density ratios $f/\varphi$ converge uniformly to one, and each derivative of fixed positive order converges uniformly to zero, where $\varphi$ is the standard Gaussian density. The symmetric counterexamples can also match any prescribed finite number of Gaussian moments exactly. The asymmetric construction uses an endpoint expansion of entropy, whereas the symmetric construction exploits the different decay rates of high-order Hermite perturbations under different weights. Thus, densities violating concavity approach the Gaussian density in the strong sense described above. The question of uniform concavity in the symmetric class for $0<q\le4/7$ remains open.

We study when the image of a measure under a random Hölder map attains full packing dimension. Our main tool is a family of packing intermediate dimension profiles $\dim_{P,\theta}^{s}\mu$, indexed by $\theta\in(0,1]$ and $s>0$, which refine the packing dimension profiles of Falconer and Howroyd and reduce to them at $\theta=1$. For a large family of random $\alpha$-Hölder maps $f_\omega:\mathbb{R}^n\to\mathbb{R}^m$, which includes index-$\alpha$ fractional Brownian motion as a particular case, we prove that for every compactly supported Borel probability measure $\mu$ on $\mathbb{R}^n$, $$\dim_P \mu_{f_\omega}=m \quad\text{almost surely} \quad\Longleftrightarrow\quad \alpha m\leq \lim_{\theta\to 0}\dim_{P,\theta}^n \mu.$$ We further study the profiles $\dim_P^s\mu$ themselves, obtaining a quantitative lower bound and a Marstrand-type identity for their limiting behavior as $\theta\to 0$. Finally, we obtain the analogous characterization for analytic sets: $$\dim_P f_\omega(E)=m \quad\text{almost surely} \quad\Longleftrightarrow\quad \alpha m\leq \lim_{\theta\to 0}\dim_{P,\theta}^n E,$$ where $\dim_{P,\theta}^n E=\sup\{\dim_{P,\theta}^n \mu :\, \mu\in \mathcal{M}_c^+(E) \}$.

We introduce a family of discrete hyperbolic secant distributions on $\mathbb{Z}$, whose normalizing constants arise from series values calculated by Ramanujan and are expressed in terms of Gauß' constant $G=\varpi/\pi$, where $\varpi=\Gamma^2(1/4)/(2\sqrt{2\pi})$ is the lemniscate constant. Using the elliptic lambda-star function $\lambda^*$, we construct scaled versions of these distributions parametrized by $\sqrt{r}$ and $1/\sqrt{r}$ for $r \in \mathbb{N}$. For the first such distribution, we compute the moments up to the eighth degree in closed form via Poisson summation, exploiting that the hyperbolic secant is a fixed point of the $-2\pi i$-Fourier transform. As a byproduct, we obtain closed-form values for series of odd powers of the hyperbolic secant up to the ninth degree, e.\,g.\ $\sum_{k \in \mathbb{Z}} \operatorname{sech}^3(\pi k) = (G^3+G)/\sqrt{2}$, and reinterpret several classical Ramanujan series probabilistically. Finally, we apply our results to evaluate a contour integral, on the critical line, of the product of Dirichlet's beta, gamma, and Riemann zeta functions.

We present self-contained martingale proofs of Talagrand's convolution conjecture in the Gaussian and Boolean settings with explicit dimension-free constants. The argument is based on the analysis of the stopped density martingale and a weighted Itô isometry estimate. The Gaussian case was settled by Eldan and Lee, and Lehec; the Boolean case was proved up to a $\log\log$ factor by Chen and recently settled by Lu, Guo and Fang.

This paper investigates finite-horizon tail and ruin asymptotics for models with random-size clusters of dependent main and delayed claims under Levy investments. Under long-tailed, dominatedly varying claim distributions, a conditional, uniformly weighted Ko-Tang condition within clusters, and moment conditions, univariate asymptotics are derived for discounted aggregate claims and ruin probabilities. For two business lines sharing renewal arrivals but having independent claim marks and Levy processes, marginal and joint tail asymptotics are established as thresholds diverge, without restricting their ratio. Results provide asymptotic formulas for ruin in both lines, simultaneous ruin, and ruin in at least one line.

We consider a pinned harmonic chain perturbed by random velocity flips, which conserve the total energy. Previous results have established the diffusive hydrodynamic limit for the energy profile at the level of expectations. In this article, under suitable fourth-moment bounds on the initial data, we prove for the first time a law of large numbers for the empirical energy distribution. Our approach is based on the Wigner distribution and shows that its random fluctuations vanish in the macroscopic limit. Consequently, the empirical energy profile converges in probability to the deterministic solution of the corresponding heat equation.

In this paper, we consider a spiked Wigner type matrix with a heteroscedastic and unknown variance profile. It is well known that in the supercritical regime of the BBP transition, strong spikes can create outliers in the spectrum. Unfortunately, in the heteroscedastic case, in general it is not possible to estimate the spike strength from these observed outlier consistently, as the latter is a solution to a Dyson equation with unknown parameters from the variance profile. In this paper, inspired by the work on sparse matrix completion \citep{BordenaveCosteNadakuditi2023}, we introduce an asymmetrized model by randomly splitting the spiked matrix into two parts, which transforms the noisy Wigner type matrix into a non Hermitian random matrix, while preserving the Hermitian spikes at the cost of a dilution. We establish a BBP type transition for the asymmetrized model, from which we can estimate the strength of the spikes precisely, even without knowing the variance profile of the noise part. We then further apply our approach to study the correlation between two correlated spiked models, where the spike/signal parts of the two models are correlated, and the noise parts are independent but may both be heteroscedastic. By applying our asymmetrization approach to the two models separately and also jointly, we are able to obtain a precise estimate of the correlation between the signal parts of the two models.

Menshikov and Volkov [Electron. J. Probab. 13 (2008)] studied recurrence and transience of a class of Markovian random walks on $\mathbb R_+$ whose conditional drift depends on both time and position and is of order $\rho x^\alpha t^{-\beta}$ with $\rho>0$. The case on the critical line $2\beta-\alpha=1$, with $\alpha\in(-1,1)\setminus\{0\}$, remained open. We prove recurrence in this remaining case. Furthermore, we establish recurrence and transience criteria that complete the classification for $-1<\alpha<1$ and $\beta\ge 0$, without assuming the Markov property and under weaker assumptions on the increments than those imposed by Menshikov and Volkov.

We study a continuous-time opinion model in which agents interact in pairs and each agent has a fixed anchor. At each interaction, both opinions are updated according to a possibly nonlinear rule with random inputs. Under suitable stability and moment assumptions, we establish uniform-in-time propagation of chaos: as the population grows, any fixed number of agents become asymptotically independent, with a common law solving a nonlinear Boltzmann equation. The proof combines a graphical approximation on finite time intervals with exponential convergence of the finite system and its nonlinear limit to their respective stationary laws. We examine two applications. For an anchored variant of the Friedkin-Johnsen model with random coefficients, we give conditions for sub-Gaussian or power-law tails of stationary deviations from the anchors. Occasional overreaction can produce power-law tails with Gaussian anchors and noise, even as the nonlinear law converges exponentially fast to stationarity. We also study an anchored model of biased assimilation, determining when a nearby opinion moves toward or away from neutrality with the anchor and incoming evidence held at neutral values. Simulations illustrate the tail predictions and show how a single peak in the smoothed empirical opinion profile can split into two.

We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

In this paper, we study averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection. First, we derive a general averaging principle applicable to such equations under minimal assumptions. Subsequently, since the coefficients of the obtained averaged equation still depend on the small scaling parameter $\e$, we impose either periodic or asymptotic conditions on the coefficients, thereby obtain two distinct averaged equations whose coefficients are independent of $\e$ and establish two averaging principles. Stopping times and Khasminskii's time discretization schemes play an important role. Finally, a concrete example is provided to illustrate the applicability and validity of the theoretical results.

We perform a numerical study of uniform meandric systems via their encoding by pairs of simple random walks and their interpretation as a model of critical percolation on random planar maps. We investigate certain natural box-crossing events and provide numerical evidence for a strong negative association between opposite open crossings and formulate conjectures on their asymptotic behavior. Our second contribution is a simple law-preserving dynamics on meandric systems, by local peak/valley flips in the walk encoding. Our simulations suggest that the percolation structure decorrelates much faster than the underlying random planar map.

Recent work of the author investigates certain random processes, valued in abelian p-groups, that naturally arise in the study of Haar random matrices over $\mathbb{Z}_p$. It was found, somewhat surprisingly, that these processes are reversible Markov chains. In this short note, we give a simple alternative derivation of this fact. The new derivation also proves that this phenomenon is not specific to $\mathbb{Z}_p$, but generalizes to Haar random matrices over any profinite local ring.

We prove that the Parisi measure of the Sherrington-Kirkpatrick model has connected support at every positive temperature and every positive external field. When the model breaks replica symmetry, its support is an interval contained in $(0,1)$, its restriction to the interior has a smooth density that remains strictly positive up to both endpoints, and both endpoints carry atoms. The proof rests on a strict inequality for moments of spatial derivatives of the Parisi PDE, which both excludes gaps in the support and forces positivity of the density. The same estimates recover the zero-field support description and give a lower bound for the mass of its upper atom.

Local anticoncentration controls the conversion of additive output probability estimates to relative estimates in Gaussian boson sampling. We prove a center-uniform quadratic small-ball bound for hafnians of complex Gaussian transpose Gram matrices for $2n$ selected modes, with an explicit finite coefficient that is polynomial when the number of Gaussian rows is at least of order $n^2/\log n$. Letting the Gaussian row count tend to infinity yields anticoncentration for the independent complex symmetric Gaussian ensemble. For the transpose Gram hafnian bound, the proof compresses dependent hafnian cofactors by Fourier interpolation and controls their self-coupled variance through a conditional Wishart identity that preserves the transpose Gram coordinate. Independent perturbations satisfy related bounds, and a positive mixture density shows that the local quadratic power is exact. Together with the uniform hiding theorem in the companion paper, the result gives local anticoncentration for finite Haar interferometers.

Using estimates established in a previous paper, we prove quantitative universality results for cokernels of products of i.i.d. random matrices and for flags associated to k-tuples of random i.i.d. matrices, over finite local rings. In the case when the ring is a quotient of $\mathbb{Z}_p$, this gives a quantitative analogue of some previous results of Huang, Nguyen and Van Peski.

We consider the cokernel corners process for an i.i.d. matrix with entries in a finite local ring. When the distribution of the entries is uniform, this process is a Markov chain, and hence the ergodic theorem for Markov chains can be applied. This implies, in particular, that for uniformly distributed p-adic random matrices, the cokernels of the corners are distributed according to the Cohen-Lenstra measure, almost surely. The purpose of this note is to show that the conclusion of the ergodic theorem also holds for i.i.d matrices, provided that the distribution of the entries is not concentrated on the translate of a subring, or the translate of an ideal. This will follow from the bounds proved in a previous paper of the author.

The signatures of $p$-rough paths form a subgroup of sufficiently high-level truncated tensor algebras, whose inverse limit is a subgroup of the full tensor algebra. For $p \geq 1$, we provide a top-down description of the signature group as the inverse limit of finite-dimensional Carnot–Carathéodory geometries in the $p$-variation setting. We show that every compatible choice of metrics induces a topological tree structure on the inverse-limit group, under which the signature group is not a topological group. This extends the results of Enrico Le Donne and Roland Züst from bounded variation to rough paths. We also characterise the dependence of the inverse-limit groups and their metric completions on the choice of metric, identifying them with the tree-reduced path group of Horatio Boedihardjo, Xiang Geng, Terry Lyons, and Danyu Yang.