Papers

We study the backward-Euler scheme for a class of stochastic functional differential equations (SFDEs) with memory driven by a fractional Brownian motion with Hurst parameter H > 1/2. We first establish the local consistency of the method, with a local truncation error of order H -$\rho$- $\beta$ + 1, and then, as the main result, prove the uniform global pathwise convergence of the numerical approximation, on a set of probability one, with order H -$\rho$ -$\beta$, for $\beta$ $\in$ (1-H, 1/2) and 0 < $\rho$ < H -$\beta$. A uniform high-probability formulation with deterministic constants follows as a corollary. In particular, the convergence order can be taken arbitrarily close to 2H -1, matching the order obtained for the explicit Euler scheme. The convergence proof relies on an exact integral representation of the scheme at mesh points, an a priori bound for the numerical solution, fractional calculus estimates, and a Gronwall inequality for weakly singular kernels.

For a $d$-dimensional probability vector $\mathbf{h}=(h_1,\dots, h_d)$, let $(S^{\mathbf{h}}_n)_{n\geq 0}$ be a nearest-neighbor random walk on $\mathbb{Z}^d$ such that at each step, it moves to one of the two nearest neighbors in the $i$-th dimension with probability $\frac{1}{2} h_i$ ($i=1,\dots, d$). Let $T^{\mathbf{h}}=\inf\{n\geq 1: S^{\mathbf{h}}_n=(0,\dots,0)\}$, the first return time to the origin. For two $d$-dimensional probability vectors $\mathbf{h}'$ and $\mathbf{h}''$ with the former majorizing the latter, we show that $T^{\mathbf{h}'}$ is stochastically smaller than $T^{\mathbf{h}''}$. In particular, the first return time for the $d$-dimensional simple random walk is stochastically larger than $T^{\mathbf{h}}$ for all $d$-dimensional probability vectors $\mathbf{h}$.

Let $\Omega\subset\mathbb{R}^2$ be a bounded planar set that is star-shaped with respect to the origin, and let $A,B,C,D$ be independent random points uniformly distributed on $\Omega$. We consider the random segments $S_{AB}$ and $S_{CD}$ and study the distribution of the smaller angle $\Theta\in[0,\pi/2]$ formed by them, conditional on the event that they intersect. Using the radial function of $\Omega$, together with a parametrization of each segment in terms of its supporting line and the positions of its endpoints along that line, we derive an integral representation for the conditional distribution \[ \Pr\{\Theta\leq\theta | S_{AB}\cap S_{CD}\neq\varnothing\}. \] The resulting expression makes explicit how the geometry of the boundary of $\Omega$ determines the angular distribution. The probability of intersection appears naturally as the normalizing constant and is related to the probabilistic version of Sylvester's four-point problem.

A central question in the prediction of stochastic processes is the extent to which past information can improve the probability of correctly predicting the next state. We introduce the Memory Prediction Excess (MPE) to address this question quantitatively. The MPE measures the average improvement in prediction accuracy obtained by using the entire observed history relative to using only the static marginal distribution, in discrete-time finite-state processes. It is defined as the difference between the expected optimal conditional prediction accuracy and the optimal static prediction accuracy. Its basic properties are examined: the MPE is always non-negative; it admits an upper bound depending on the static accuracy, attained if and only if the future is almost surely a deterministic function of the past; and degenerate cases in which the MPE vanishes are characterized. A normalized version, taking values in the unit interval, is introduced as a dimensionless measure of predictive efficiency. A lower bound is derived by comparing predictions based on histories of different lengths, showing that the expected optimal prediction accuracy is monotone with respect to the history length. The framework is extended to finite-length histories, where the finite-history MPE (FH-MPE) measures the predictive gain attainable when only the most recent observations are retained. This leads to the notion of a minimal memory length required to achieve the same predictive performance as the full history. For finite-order Markov chains, this minimal memory length is shown to be bounded by the Markov order. The MPE and its variants are formulated in terms of conditional probabilities and prediction accuracies, offering a probabilistic perspective on the predictive utility of memory that is complementary to classical information-theoretic approaches.

We study the short-maturity implied volatility and the skew stickiness ratio for baskets of assets with continuous, possibly rough, stochastic volatility. The fluctuation of the instantaneous basket variance has two sources: fluctuations of the constituent variances and fluctuations of the basket weights. We derive a near-the-money implied volatility expansion that separates these contributions. We then specialize the result to volatility models given by general functions of Gaussian Volterra factors and obtain an explicit basket skew coefficient in terms of the short-time kernel asymptotics, the factor sensitivities, and the return-factor correlations. A density expansion justifies differentiation of the near-the-money expansion at the money. Finally, using a Malliavin representation of the dynamics of total implied variance, we prove that the short-maturity skew stickiness ratio converges to the universal limit $H + 3/2$ for Gaussian factor basket models with $H \in (0, 1/2]$.

We prove that below the multicritical (Nishimori) point the ferromagnetic boundary of a biased Ising spin glass, argued by Nishimori (1986) to be exactly vertical, can be vertical and can also be reentrant, so that cooling at fixed disorder destroys order. It is vertical in the Sherrington-Kirkpatrick (SK) model with a Curie-Weiss bias $J_0$, and reentrant near the triple point of fully connected $p$-spin glasses and, at one disorder strength, on a decorated planar $\mathbb{Z}^2$-periodic lattice. For every $T<J$ the SK magnetization vanishes for $J_0\le J$, continuously at $J_0=J$, and is macroscopic for $J_0>J$. For every integer $p\ge3$, at each bias slightly larger than the triple-point bias, the $p$-spin magnetization is macroscopic at the Nishimori temperature and vanishes at an explicit lower temperature; this is Nishimori's recent replica prediction in two-temperature form. On the lattice ($\mathbb{Z}^2$ with each bond replaced by a fixed series-parallel gadget; maximum degree four), with iid $\pm J$ couplings at $\mathbb{P}(J_e=-1)=9/10000$, the Gibbs state is almost surely unique at high temperature, non-unique in a window containing the Nishimori temperature, and unique again at low temperature. As Dey and Kang observed, the SK boundary is set by the zero-field susceptibility together with an envelope bound on the free-energy gain in a field, which follows from the gauge inequality. We prove that this susceptibility is the curvature at the origin of the zero-field Parisi PDE solution, which Lopatto evaluated for every $\beta>1$ via his theorem that the support of the Parisi measure accumulates at the origin; we give a short alternative proof of that theorem. The $p$-spin and lattice results and the alternative proof are computer-assisted. The models are proxies for the nearest-neighbour $\pm J$ model on $\mathbb{Z}^2$ and say nothing about $\mathbb{Z}^d$.

We prove that the Benjamin-Bona-Mahony equation is almost surely globally well-posed with respect to random Gaussian initial data of negative Sobolev regularity in $H^{s}(\mathbb{T})$ for any $s>-\frac 14$. This result is sharp in view of the mild probabilistic ill-posedness due to Oh-Tzvetkov (2026). This also improves on a previous result of the author which established this only in a logarithmically negative regularity. To break through this logarithmic regularity barrier, we combine the $I$-method with the low-high argument from Bona-Tzvetkov (2007), which motivates a refined first-order expansion for solutions.

In spatial matching markets, a supply unit's flexibility is measured by its service radius, the maximum distance at which it can serve demand. In dimensions $k \geq 2$, we study how a platform should allocate service radii among the supply nodes subject to a budget on their sum. The platform makes this choice before observing supply and demand locations, with the objective of maximizing the expected fulfilled demand. We show that the shape of a preferred allocation depends on the total budget: under suitable conditions, large budgets favor allocations that are more uniform in the sense of majorization, while small budgets favor concentration. We also characterize a non-uniform allocation that is asymptotically optimal for a very-sparse regime, and show that the uniform allocation is suboptimal in this regime. Our results provide theoretical explanations for the radius allocation questions raised by the numerical experiments in [ASY26b].

We study one-sided, two-sided and reflected exit problems for a finite-state Markov-modulated additive-increase and multiplicative-decrease process. For upward passage, a first-jump decomposition and a spatial Laplace transform lead to a recursion indexed by finite words of phases and to a convergent Green series. For downward passage, the first-jump operator is a strict contraction and the known contribution from $[p_i b,b]$ becomes an explicit boundary forcing term in the same spatial recursion. These allow to derive the corresponding Green representation of the first passage matrix and its finite-dimensional compatibility condition. The derivation of the two-sided exit identity is based on the Markov property, which in this context is equivalent to the cocycle property of the Laplace transform, together with the recursive equation at the upper level. For reflection at the running infimum, a finite $\bar p$-geometric method of steps realizes a certain Picard map that allows to find the exit matrix. As a result, we construct a Doob martingale related to the first passage time of the reflected Markov-modulated AIMD.

Hedging a derivative by trading the underlying asset changes the payoff that the hedging intended to replicate. We study this phenomenon when trading generates price impact and execution costs. In a binomial model, we characterize replication through a fixed-point equation. In continuous time, we derive a nonlinear pricing PDE whose implicit terminal condition captures the nature of the moving target problem of the hedger. For monotone convex Lipschitz payoffs (such as calls and puts) we establish exact replication under midpoint execution costs. Numerical experiments illustrate: (i) how price impact shifts the effective strike, (ii) the non-linear dependence of the option price on the number of contracts, (iii) how execution costs smooth terminal holdings, (iv) the extent to which the hedger's own trading can bring an otherwise worthless option into the money, and (v) we explain the spread and the shape of the limit order book in the options market based on the price impact and the shape of the limit order book of the underlying.

When a free energy admits several local minimisers, the associated mean-field particle system exhibits a metastable behavior, undergoing random transitions between these minimisers. However, when the number of particles is large, these transitions are rare events and typically occur at a time-scale which is out of reach for simulations. In this work, we introduce a system of particles coupled with a slow macroscopic external variable which approximately follows a Langevin dynamics associated to a coarse-grained free energy. The empirical distribution of the particle system thus samples all local minimisers of the initial free energy, with transition times which are now independent from the number of particles. The construction is inspired by a theoretical tool: the two-scale approach for proving log-Sobolev inequalities for Gibbs measures.

We study feasibility correction for linear programs. Suppose an initial point satisfies most of the constraints. Can it be moved into the feasible region without changing the objective value? Our main deterministic result says that this is possible if the constraint matrix satisfies a restricted isometry property (RIP). We apply the correction theorem to obtain lower bounds for random linear programs with finite-moment or sub-Weibull entries. We complement these with matching-order upper bounds under a uniform $p$th-moment bound for some $p>2$ and a delocalization condition on the objective direction. As an application of our techniques, we also derive lower bounds for the Gaussian width of feasible regions under RIP assumptions.

In this paper, we consider Markov additive processes (MAPs) whose additive components are $\bR$-valued and have no positive jumps. Specifically, we assume that the modulators are Feller processes with only finitely many jumps on each bounded time interval and that the additive components also have only finitely many jumps on each bounded time interval. For such MAPs, we characterize the associated scale operators and use them to solve the two-sided exit problem, which concerns the first entrance time into one half-line on the event that it is reached before the other, and to characterize the potential measures of the processes killed when their additive components exit an interval. Our proofs rely on properties of analogues of local times, properties of the exit systems obtained by pairing them with suitable kernels, and results on the extension of $C_0$-semigroups to $C_0$-groups.

The lack of rigorous safety and performance certificates remains a key bottleneck to the deployment of modern learning-based methods. Sample compression has recently emerged as a powerful tool for deriving such certificates, with particularly sharp bounds available for algorithms satisfying a so-called preference property – also known as stability in the learning theory literature. These bounds find direct application across domains as different as the Scenario Approach, Pick-to-Learn, and Support Vector methods. However, whether they are tight has remained an open problem. In this paper we resolve this question affirmatively and show that the state-of-the-art bound for preferent compressions is provably tight. We establish this by exhibiting an explicit construction based on the uniform distribution and order statistics that attains the bound in the limit. Along the way, we also provide a considerably shorter and more accessible proof of this bound, requiring only elementary counting arguments and no infinite-dimensional duality.

We ask whether transposition changes the stationary heavy tail of the co-moving recursion $\mathbf x_{t+1}=Q_tBQ_t^\top\mathbf x_t+Q_t\boldsymbol\eta_t$ in $d=3$. The model has independent Haar rotations, Gaussian body-frame noise with covariance $\Sigma$, and an explicit positive metric $\mathsf g$. The tail index $\alpha_\star$ depends only on singular values, so $\alpha_\star(B)=\alpha_\star(B^\top)$. The structure theorem and polar-twist mechanism are unconditional. In the $\mathsf g=I$ chart, write $B=S+\widehat{\boldsymbol\omega}$: $\Delta_6(B)=-16\det[\boldsymbol\omega,S\boldsymbol\omega,S^2\boldsymbol\omega]$. Then $B\not\sim_{O(3)}B^\top$, $\Delta_6(B)\ne0$, $\operatorname{rank}[\boldsymbol\omega,S\boldsymbol\omega,S^2\boldsymbol\omega]=3$, and controllability of $(S,\boldsymbol\omega)$ are equivalent; intrinsic operator chirality is a Kalman rank condition. Unlike the exponent, the tail level need not be transpose-invariant. The exact polar-twist identity rotates the anisotropic noise frame under mirroring, showing how level asymmetry can enter. A nonzero asymmetry is established only conditionally and numerically, not by a closed-form example: under Assumption R on the SPD chart, $\Delta\log C=\langle A,G\rangle+o(\|A\|)$ as $A\to0$, with $G=2H\circ[\Sigma,\partial_\Sigma\log C]$; finite-threshold simulations are consistent with this first-order law. Strict-Haar averaging makes every lagged second-order cross-moment transpose-blind; a fourth radial moment retains an odd channel. No finite scalar-weighted radial-moment combination uniformly cancels the even quadratic form while retaining the odd covector.

We introduce power variations of critical Gaussian multiplicative chaos (GMC) along refining partitions in arbitrary dimension. Under suitable renormalisation, we prove uniform fractional moment bounds via Laplace transform estimates as well as stable convergence to supercritical GMCs. Together, these results provide a unified approach to regularity and spectral questions for critical chaos, yielding: (i) the sharp exponent for the logarithmic modulus of continuity, (ii) the existence of non-empty essential spectrum for critical Liouville quantum gravity surfaces; and (iii) an almost-sure quantitative Fourier decay.

We study the edge scaling limit of the logarithms of the squared singular values of multiplicative Brownian motion on the general linear group $\mathrm{GL}_N(\mathbb{C})$ with general initial conditions. Building on earlier work [arXiv:2605.06429], we prove that, for every initial condition in the enhanced state space, the limiting dynamics are determinantal and derive an explicit double-contour formula for their extended multitime correlation kernel. The dependence on the initial state, including information not determined by the particle coordinates, is encoded by the associated Laguerre–Pólya entire function. We further extend the corresponding infinite-dimensional stochastic differential equation description to arbitrary initial configurations, including those with coinciding coordinates.

We study statistical limit theorems for time-series generated by multidimensional random fields that are finitary factors of iid random processes. Based on the decay of the tail of the coding radius, we establish the Central Limit Theorem with explicit rates of convergence, as well as the weak convergence to a Brownian sheet, for suitably regular functions. Our results can be applied to a variety of classical models in statistical mechanics, such as the Ising model, the hard-core model, and proper colourings.

This paper is a companion to one in which we prove there is a discontinuous phase transitions in the $k$-creation process with fast stirring in $d=1$ when $k \ge 2$. Dickman and Tomé (1991) introduced models on $Z$ in which $k$ consecutive occupied sites give birth at rate $\lambda$ and individual particles die at rate 1. Here, we show that without stirring the models with $k\ge 2$ have qualitative properties much like the contact process, which is the case $k=1$. The critical value can be charracterized by the speed of interface when the process starts from the initial configuration $(-\infty,0]$. The process dies out at the critical value. In the supercritical phase the complete convergence theorem holds which implies there is only one nontrivial stationary distribution, and convergence to the limit occurs exponentially rapidly. In the subcritical phase, the process dies out exponentially fast starting from any finite set.

For vertex-transitive Ramanujan graphs with logarithmic girth, a simple random walk bridge of length of order $\log N$, where $N$ is the size of the graph, has a maximum distance that changes from order $\sqrt{\log N}$ to order $\log N$ in a bounded critical window. We prove this by separating bridges whose lifts to the regular tree close from those whose lifts do not. A uniform two-term return estimate determines the probabilities of these two cases and the real-valued critical center. In the same $O(1)$ window, the normalized range has a two-point limiting law whose mixture weights vary nontrivially across the window.

We extend the notion of robust viscosity solutions to semilinear rough partial differential equations (RPDEs) of the form \[ \partial_t u + \mathcal{L}_t u + f(t,x,u,\sigma^{\top}\nabla_x u) + h(t,x,u)\,\diamond dW = 0, \quad u(T,x)=\xi(x), \] driven by a discontinuous path $W$ of finite $q$-variation for some $q<2$. This includes, in particular, typical paths from a broad class of Lévy processes, for instance from $\alpha$-stable Lévy processes with $\alpha<2$. Our notion of solution is formulated through a solution map that agrees with the classical viscosity solution for smooth drivers and is continuous in the terminal condition and the driver, with both the driver and the solution paths viewed in a suitable Skorokhod-type decorated-path metric \cite{chevyrev_superdiffusive_2024}. These requirements naturally imply that robust viscosity solutions must employ Marcus-type jumps, and we establish well-posedness, a flow property, and a stochastic representation in terms of the rough BSDEs with discontinuous Young drivers \cite{becherer_rough_2026}. For Lévy processes of finite $q$-variation with $q<2$, our notion of RPDE solutions provides a pathwise interpretation for corresponding SPDEs, which are seen to be Markov processes in an infinite-dimensional function space.

We prove the existence of a non-trivial continuous stationary reversible dynamics on the directed landscape as a subsequential scaling limit of Brownian last passage percolation under the Ornstein–Uhlenbeck dynamics. Strong passage-time stability estimates from the companion paper Bhatia '26, together with static endpoint regularity, give tightness at the critical dynamical scale $n^{-1/3}$. To establish non-triviality of the subsequential limiting dynamics, we show that, in dynamical BLPP, the fractional overlap between geodesics at times zero and $a n^{-1/3}$ tends to one as first $n\to\infty$ and then $a\downarrow0$. The proof refines the excursion argument of Ganguly-Hammond '24, using strong passage-time stability and the fact that a fixed dynamical time is increasingly subcritical at finer spatial scales.

We consider Brownian last passage percolation (BLPP) in which the Brownian increment process on each unit horizontal interval is independently resampled at rate one. By combining strong passage-time stability estimates with a multiscale analysis of static near-optimal paths, we show that, for every $\varepsilon>0$, the union of all geodesics between two KPZ-scale rectangles of transverse width of order $n^{2/3}$ and longitudinal length of order $n$, separated by a distance of order $n$, visits at most $n^{1+\varepsilon}$ unit horizontal cells in the bulk during the critical time interval $[0,n^{-1/3}]$, both in expectation and with stretched-exponentially high probability. We also obtain the quantitative bound $n\exp\{C(\log\log n)^2\}$ on the expected hitset size, with a corresponding failure probability at most $Ce^{-c(\log n)^2}$. Using this, we establish that the set of times admitting a non-trivial bigeodesic has almost surely zero Hausdorff measure for the subpolynomially decaying gauge $H(r)=\exp\{-L(r)^2\log L(r)\}$, where $L(r)=\log\log(1/r)$, as $r\downarrow0$. In particular, this set almost surely has Hausdorff dimension zero. For each fixed deterministic non-axial direction, we further show that almost surely no time admits a bigeodesic in that direction.

We solve the perpetual liquidation problem for a geometric multi-skew Brownian motion carrying local-time pushes upward at a support level and downward at a resistance level, a model of technical analysis that is Markov in the price alone. Three geometries arise, separated by a closed-form criterion: the continuation band lies below resistance, straddles it, or pins its upper edge there. Selling into resistance is a conclusion rather than an assumption. Identification is asymmetric: exercise behaviour determines the support parameters exactly, while the strength of resistance is recoverable only on an interval with a closed-form endpoint.

We consider the Sachdev-Ye-Kitaev model with $N$ Majorana fermions and interaction order $q$. For even $q\in \{2,4,...,N-2\}$, we give an elementary proof and show that the eigenvectors of the model are completely delocalized in all deterministic directions.

Let $G_{2n}$ be a complex symmetric random matrix whose entries above the diagonal are independent standard circular complex Gaussians, and let $H_n=\operatorname{haf}(G_{2n})$. We prove the uniform shifted anticoncentration bound $$ \Pr\!\left( \left| \frac{H_n}{\sqrt{(2n-1)!!}}-z \right| \le \varepsilon \right) \le 2\sqrt{\frac n\pi}\,\varepsilon^2 $$ for every $z\in\mathbb C$ and $\varepsilon>0$. This establishes a local anticoncentration property that supports hardness arguments for quantum advantage in Gaussian boson sampling.

Neural Stochastic Differential Equations (Neural SDEs) provide flexible continuous-time generative models, but generic neural drift and diffusion networks are costly to simulate on long horizons and can give unstable gradients when the training signal is a path functional rather than a pointwise observation. We introduce SLiSDE, a family of Neural SDE models built from structured linear stochastic layers. Parallel-in-time simulation is obtained at the layer level, while expressivity is recovered by gated in-flow stacking: previous-layer paths modulate the next layer's latent flow through learned gates. For functional calibration tasks in which rare paths dominate the loss, we add an optional Girsanov tilt that acts as a learned importance sampler with an exact likelihood-ratio correction. We prove well-posedness, a discretisation error bound, validity of the change of measure, and a universality result: the terminal laws of the gated stack are dense in the space of square-integrable laws. Experiments on functional calibration benchmarks show that the structured model outperforms fully neural SDE baselines while retaining parallel-time simulation and stable importance weights.

For the fractional heat equation $\partial_t u + (-\Delta)^{\beta/2}u=0$ on $\mathbb{R}^d$ with $\beta\in(0,2)$, Weber and Zacher introduced the optimal constant $C_{\rm LY}(\beta,d)$ in the Li–Yau type estimate $$(-\Delta)^{\beta/2}\log u(t,\cdot)\leq \frac {C_{\rm LY}(\beta,d)}{t},\quad t>0.$$ They asked whether $C_{\rm LY}(\beta,d)$ converges to the classical Li–Yau constant $d/2$ as $\beta\uparrow2$. In this note we prove that the answer is affirmative. More precisely, for every fixed $d\geq1$ there exists a constant $K_d>0$ such that $$\frac{d}{\beta}\leq C_{\rm LY}(\beta,d)\leq\frac{d}{\beta}+K_d(2-\beta)\log\frac{e}{2-\beta},\quad 1\leq\beta<2.$$ Consequently, $\lim_{\beta\uparrow2}C_{\rm LY}(\beta,d)=d/2$. The proof has two main ingredients. First, the Bochner subordination formula, together with a concavity argument, shows that the supremum in Weber–Zacher's formula for $C_{\rm LY}(\beta,d)$ is attained at the origin. Second, the factor $2-\beta$ is derived from a lower bound for the fractional heat kernel profile in the large-distance regime, where the Lévy–Itô decomposition plays an important role.

In this work, different from previous results, the explicit expression of numerical Onsager-Machlup action functional for Euler discretized SDEs driven by fractional Brownian motion is derived provided the drift coefficient and numerical reference path satisfy some suitable conditions. Then numerical fractional Euler-Lagrange equations for numerical Onsager-Machlup action functional are also obtained. Finally, numerical experiments to illustrate and support the theoretical findings.

Flow matching (FM) has recently emerged as a promising framework for generative modeling due to its conceptual simplicity and strong empirical performance. In FM, samples are transported along a vector field parameterized by a neural network, inducing a probability path that evolves from a simple noise distribution to the target data distribution, governed by an ordinary differential equation (ODE). However, existing FM approaches predominantly rely on probability paths derived from optimal transport (OT) between Gaussian distributions, which may be suboptimal for capturing complex data with inhomogeneous structures such as heavy tail or sharp contrast. In this work, we generalize FM to the broader class of location-scale families for handling data inhomogeneity and introduce a novel class of probability paths defined as geodesics on the manifold of probability distributions. We name this approach probabilistic geodesic flow matching to distinguish it from prior geodesic (Riemannian) FM methods defined in input space. We argue that Euclidean OT-based paths are not necessarily optimal in probability space and may limit modeling flexibility. Through synthetic benchmarks and scientific datasets at different scales, we demonstrate that the proposed method more effectively captures complex distributions, leading to improved or comparable performance compared with SOTA geometry-motivated generative models.