Papers

We construct a compact metric space with an atomless probability measure and a bounded Lipschitz cost for which the entropic optimal-transport minimisers have no zero-temperature weak limit. More precisely, $P_\varepsilon$ does not converge as $\varepsilon\downarrow0$. In the example, every unregularised minimiser is singular with respect to $\mu\otimes\mu$, so that the entropy on the optimal face is identically $+\infty$. We describe the cluster set by \[ \operatorname{Clust}(P_\varepsilon)=\{P_w:w\in\mathcal W\}, \] where $P_w$ is the mixture of the two zero-cost graph couplings with weight $w$, and where $\mathcal W\subset[0,1]$ is a non-degenerate compact interval. We then compute two explicit points $w^-<w^+$ in this interval. This shows that compactness, atomlessness, and Lipschitz regularity of the cost do not imply zero-temperature convergence. We also present a compactness theorem for the general problem. If $C\in L^1(\mu\otimes\nu)$ is continuous and bounded from below on Polish spaces, then the zero-temperature cluster set is a nonempty weakly compact connected subset of the optimal face. In the proof, we apply the cluster-point theorem of Bernton, Ghosal, and Nutz and the continuity of $\varepsilon\mapsto\pi_\varepsilon$. Finally, we give local and exterior first-order criteria for full convergence and cluster membership. We show that nonconvergence is possible, but only through a connected continuum of optimal plans.

We introduce Treasure Search Optimization (TSO), an interacting particle method for global optimization. Most swarm methods balance exploration and exploitation within a single population, and typically switch between the two by degenerating the noise, annealing a temperature, or tuning a parameter. TSO instead splits these tasks across two kinds of agents. A swarm of explorers stays in exploration mode and a single treasure hunter performs exploitation. The hunter drifts toward an objective-weighted average of the explorers and may teleport to it when the move lowers the objective. The swarm then re-centers around the hunter, creating a feedback loop between search and capture. We model the dynamics as coupled jump-diffusion stochastic differential equations (SDEs). The hunter's jumps are shared by all explorers and act as a common noise. The mean-field limit is therefore a conditional McKean-Vlasov jump-diffusion SDE, whose well-posedness we prove. We also characterize the steady state and prove, via Laplace approximation techniques, that the hunter settles near the global minimum with error of order $1/\alpha$, where $\alpha$ is the weight parameter. Linking the consensus drift to a smoothed free energy, we explain why the swarm ignores spurious local traps and demonstrate how to quantify uncertainty in inverse problems using post-processing Kalman steps after TSO iterations. Numerical experiments on ODE-constrained problems and a low dimensional Bayesian inverse problem demonstrate the effectiveness of the TSO method.

We study Poisson Galton–Watson trees on a standard Borel type space when the offspring kernel is multiplied by a scalar parameter. On finite trees, we identify the Radon–Nikodym derivative between two parameter values and show that it remains measurable after projection to the total progeny measure. Under a uniform bound on the offspring intensities, differentiation yields exact differential and integral equations for the projected laws without irreducibility, reversibility, or a positive eigenfunction. With an additional positive eigenfunction bounded above and away from zero, we relate these equations to an infinite spinal tree, uniform pruning, the Doob transform, and the Aldous–Pitman ascension process. For a uniformly bounded offspring kernel, we also prove uniform exponential integrability of the total progeny throughout the spectrally subcritical regime. As an application, under the graphical-kernel assumptions of Bollobas, Janson and Riordan, the number $K_n$ of connected components satisfies $K_n/n \to {\mathbb E}_{\pi}[1/T_u]$ in probability and in $L^1$, where $T_u$ is the total progeny of the associated branching process and $1/\infty=0$. If $q_u(x)$ is its extinction probability from type $x$, re-rooting and extinction duality give the explicit limit $$ \int_S q_u(x)\,\pi(dx) - {u\over 2}\int_{S\times S}\kappa(x,y)q_u(x)q_u(y)\,\pi(dx)\pi(dy). $$ This extends the finite-type and compact-continuous formulas to the full BJR graphical-kernel setting, allowing separable noncompact type spaces and kernels that may be unbounded or reducible.

We introduce ideal Gårding polynomials, a convexity-enhanced subclass of Gårding polynomials whose Gårding components are recursively convex under partial differentiation. This class strictly contains real stable polynomials and, after translation and homogenization, lies in the Lorentzian class. Our main result is that ideal Gårding polynomials still admit a robust structure theory despite this additional convexity: they are preserved under polarization, satisfy natural closure properties, and support a linear preserver theory. A key contribution of this paper is a universal model for univariate Gårding polynomials, described by monotone root sequences and equivalently by volume polynomials of Pitman–Stanley polytopes. We establish quotient concavity, and Newton–Maclaurin type inequalities, which leads to the polarization theorem, and suggests further connections with convex geometry and Lorentzian polynomials.

We consider the nearest-neighbor Edwards–Anderson Ising model on cubic boxes with random perturbations supported at the boundary. We prove that any perturbation admitting an energy envelope that is negligible compared with the volume, both in expectation and almost surely, leaves the limiting quenched specific free energy unchanged. For independent finite-variance bulk disorder in dimension at least two, an Efron–Stein estimate also yields almost-sure self-averaging along the full sequence of boxes. The hypotheses are verified for i.i.d. scalar surface fields, fixed random exterior spins, and periodic wrap-around bonds, with an $ O(L^{-1}) $ comparison of quenched means. The result concerns the specific free energy and does not assert convergence of finite-volume Gibbs measures.

Dropout and Random Gradient Masking (RaM) are two training techniques used to improve performance in deep learning. Both techniques inject randomness into the training dynamics, but in significantly different ways: dropout applies random masks to the activations in the forward pass, whereas RaM leaves the forward pass unchanged and instead masks the gradients. In particular, the noise induced by RaM in the parameter updates is unbiased, so standard explanations for the effectiveness of dropout, such as the penalization effect or the prevention of co-adaptation between neurons, do not apply to RaM. In this work, we show that the difference between the two methods disappears for ResNets in the large depth and width asymptotics: in the complete feature learning regime, they both converge to the same large-scale limiting dynamics. This asymptotic equivalence holds for several variants of dropout and RaM, including layerwise dropout as used in stochastic-depth ResNets, albeit at slower quantitative rates. In fact, we also show that several of these variants collapse to the same limit asymptotically.

We develop an algebraic transfer calculus for the oriented Schatten profiles of kernels underlying operator-valued Gaussian chaoses. A dimension-free link inequality propagates profile bounds through cut factorizations, tensor products, coefficient maps, and ordered contractions. Besides the constant-one strong Schatten theorem, we prove the sharp weak endpoint \[ \mathfrak{S}_{r,\infty}\times\mathfrak{S}_{r,\infty} \longrightarrow \mathfrak{S}_{r,\infty}(\log\mathfrak{S})^{-1/r}, \qquad 1<r<\infty. \] The estimate holds uniformly over all contracted Hilbert spaces, and the exponent $1/r$ cannot be decreased within the displayed $q=\infty$ Lorentz–Zygmund scale. A separate finite-complexity argument gives sharp effective-rank and finite-cut-rank logarithmic bridges from all-cut operator profiles to Gaussian operator norms. Combined with oriented-flattening Gaussian estimates, the calculus yields continuous multiplication on completed Wick chaoses with noncommuting coefficients, an associative algebra of factorially weighted analytic Wick series, and a local-to-global theorem for loop-free Peter–Weyl fusion trees. We then apply the method to singular Wick multipliers on groups of polynomial growth. For second-order multipliers we obtain sharp necessary and sufficient Schatten convergence thresholds; on $\mathbb{Z}^{D}$ we determine the full singular phase diagram at every order. Fourier transfer gives exact Sobolev, Schatten-class, compactness, and trace-class thresholds for sandwiched Wick multiplication operators on $\mathbb{T}^{D}$, together with sharp Fourier–Galerkin rates and approximation-number decay.

We consider a class of infinite critical tree-indexed random walks on $\mathbb Z$, where the motion of particles is subject to vertex reinforcement. We mainly focus on the strong reinforcement regime, where we expect the process to localize almost surely on two sites. Part of our analysis includes the study of a time-dependent generalized Pólya urn process, where the number of draws at each step is prescribed by a sequence $(\sigma_n)_{n\ge 1}$ of arbitrary positive integers, and the probability to pick a ball of a given color is proportional to a function of the number of balls of that color. In particular for bounded sequences $(\sigma_n)_{n\ge 1}$, we recover Rubin's characterization for the fixation of one color.

This study investigates multiscale stochastic Burgers equations with reflection, wherein the slow component is modeled by a stochastic Burgers equation with reflection and the fast component by a stochastic reaction-diffusion equation with reflection. Using the weak convergence approach, we rigorously establish a large deviation principle for the slow component. Key technical tools include the penalization method, carefully constructed stopping times, and a refined adaptation of Khasminskii's classical time discretization scheme.

How deep does a graph neural network need to be on a sparse graph? We study its purest statistical form: node classification on the sparse contextual stochastic block model (CSBM) with average degree $\Delta=O(1)$, whose local weak limit is a broadcast-labelled Poisson Galton-Watson tree. Prior work derived a message-passing classifier $h_\ell$ that aggregates from each vertex at distance $k\le\ell$ the attenuated evidence $2\operatorname{artanh}(\gamma^k t(X_v))$, with $\gamma$ the edge signal and $t$ a bounded likelihood-ratio transform of the feature. We prove that the value of depth is governed by a single number, the Kesten-Stigum ratio $\kappa=\gamma^2\Delta$. Below the threshold ($\kappa<1$), the error sequence is Cauchy at a geometric rate, $|\mathcal{E}(\ell)-\mathcal{E}(\ell')|\le C\kappa^{(\ell+1)/3}$ for all $\ell'>\ell$, so all layers beyond depth $O(\log(1/\epsilon))$ change the error by less than $\epsilon$; conversely, under mild regularity each sufficiently deep layer still flips the decision with probability at least $c\kappa^{\ell/2}$, the empirically sharp exponent. Above the threshold ($\kappa>1$), depth is geometrically productive: $\mathcal{E}(\ell)$ is driven to a branching-process floor of order at most $1/(\kappa-1)$ at any geometric rate $\kappa^{-s\ell}$, $s<1$ (this bound has content only for $\kappa>17$). No local classifier of any depth beats the universal floor $e^{-\Delta}\Phi(-\zeta)$ set by isolated roots ($\zeta$ the feature signal-to-noise ratio), while the first layer provably helps by an explicit total-variation amount. Simulations with an exact belief-propagation baseline on the same trees show that the pairwise rule's error curve is mildly non-monotone in $\ell$, so an optimal finite depth exists (an exact instance is certified in the appendix), while BP saturates strictly faster, at an effective per-layer ratio below $\kappa$ that we identify.

This paper investigates second-order asymptotic expansions for the conditional higher moment (CoHM) coherent risk measure under a Farlie-Gumbel-Morgenstern (FGM) dependence structure, capturing a weak contagion between a primary loss risk and a reference risk. Assuming that the primary risk belongs to the Fréchet, Weibull, or Gumbel maximum domain of attraction, we systematically derive second-order asymptotic expansions using extreme value theory and second-order regular variation theory. Compared with existing first-order results, our refined approximations capture higher-order tail behavior and dependence effects more accurately. Numerical simulations confirm that the second-order asymptotics substantially reduce approximation errors, especially at extreme confidence levels. Empirical applications to insurance claim data further illustrate the practical superiority of the second-order approach.

The rapid digitalisation of elite sport has created new opportunities for integrating artificial intelligence (AI), performance analytics, and decision-support systems into athlete development and competition management. However, existing solutions remain fragmented, typically addressing isolated tasks such as performance analysis, athlete monitoring, or referee support. This paper presents \textbf{FST$\cdot$ai~2.5}, an explainable, uncertainty-aware, and secure AI framework for Olympic and Para-Taekwondo. \textbf{FST$\cdot$ai~2.5} introduces a unified digital ecosystem integrating athlete intelligence, competition analytics, federation-scale data management, AI-assisted decision support, athlete and event digital twins, explainable performance indicators, and adaptive training recommendations. The framework supports World Taekwondo (WT), Member National Associations (MNAs), coaches, referees, analysts, and athletes through transparent, secure, and federation-aware governance. By combining multi-source competition data, athlete-performance information, and contextual evidence, \textbf{FST$\cdot$ai~2.5} provides tactical diagnostics, longitudinal athlete monitoring, performance forecasting, personalised development planning, and federation-wide benchmarking using explainable and uncertainty-aware AI. Prototype deployments demonstrate the feasibility of the proposed framework. Although developed for Olympic and Para-Taekwondo, the methodology is broadly applicable to explainable AI, digital twins, and trustworthy decision support in combat sports and other high-performance sporting environments.

Random basis expansion (RBE) search the span of a randomly sampled basis to find the best approximation of a target function. They are equivalent to single layer neural networks where the hidden layer weights are chosen randomly. Universal approximation properties have been established for RBEs using continuous weight sampling distributions and real-valued activation functions. Our results extend the universal approximation property to RBEs that use non-continuous weight distributions with dense support in the weight space and that use complex-valued activation functions. The result shows such random bases have the universal approximation property with arbitrarily high probability.

We study precise asymptotics for high-level exceedance probabilities of path functionals of continuous vector-valued Gaussian processes. The probabilities have the form $$ \mathbb{P}\{\Gamma_{[0,T]}(\check{\boldsymbol{u}}(\boldsymbol{X}-u\boldsymbol{b}))>L_u\}, \qquad u\to\infty, $$ where $\boldsymbol{X}$ is a centered $\mathbb R^d$-valued Gaussian process and $\Gamma$ belongs to a broad class satisfying natural monotonicity, scaling, no-atom, and continuity assumptions. The class covers classical sojourn times, Choquet-type sojourn integrals, area-under-the-curve functionals, and, through a local-footprint extension, shrinking-window Parisian persistence functionals. We obtain exact asymptotics in the stationary case and in the non-stationary case when the inverse generalized variance has a unique minimizer at the boundary point. The non-stationary theorem covers the regimes $\alpha<\beta$, $\alpha=\beta$, and $\alpha>\beta$, which lead respectively to Pickands-type, Piterbarg-type, and deterministic limiting constants. We also derive conditional limit laws for the first exceedance time. The main claims are stated in the body of the paper, while the proofs and auxiliary estimates are collected in the appendices.

We consider a supercritical two-type continuous-time linear birth-death process with mutation and selection, in which wild-type individuals give rise to mutant offspring with a larger net growth rate $\lambda_1$. In this setting, we investigate a component of the site frequency spectrum (SFS), describing the number of driver mutations present at any given frequency in the population. We establish asymptotic convergence results at large times and detection sizes. We examine three distinct regions of the SFS: (1) the small frequency region occupied by clones of size $\mathcal{O}(1)$, (2) the intermediate frequency region occupied by clones of size $1\ll j(t)\ll \exp(\lambda_1 t)$, and (3) the large frequency region occupied by clones of size at least $\exp(\lambda_1 t).$ We obtain strong laws of large numbers for the driver SFS in the small- and large-frequency regions by constructing suitable $L^2$-approximations. In the intermediate frequency region, we identify a cutoff frequency $j(t)$ growing with time at which there are an order-one number of mutant clones. At this frequency, we obtain a Poisson convergence theorem as $t\to\infty$. Overall, our results provide quantitative insights into how selection shapes the site frequency spectrum both at small and large frequencies, which can in principle be leveraged to construct estimators of relevant evolutionary parameters. As an example of this, we propose and show asymptotic consistency of a new estimator of the fitness increase.

For stochastic gradient descent (SGD) with a constant stepsize $\alpha$, the invariant law of the iterates, centered at a minimizer, describes the behavior of the algorithm over long time horizons. In the strongly convex case, this invariant law has the familiar $\sqrt{\alpha}$ scaling and a Gaussian limit as $\alpha\downarrow 0$. We show that this behavior changes fundamentally for convex objectives $H$ with flat minima and (sub)quadratic tails. More specifically, we study SGD with Markovian noise generated by a contractive driving chain. For every sufficiently small constant stepsize $\alpha$, we prove existence, uniqueness, and geometric convergence to an augmented invariant law in a Wasserstein distance induced by an $\alpha$-dependent metric. When the minimizer $x_\star$ has local flatness exponent $m\ge2$, meaning that $\nabla^2 H(x)\asymp \lVert x-x_\star\rVert^{m-2} I_d$ as $x\to x_\star$, we obtain a contraction bound with factor $1-c\alpha^{m-1}$, where $c>0$ is a constant. This recovers the factor $1-c\alpha$ in the quadratic case $m=2$. We then analyze the small-stepsize scaling limit. We show that the invariant law concentrates on the scale $\alpha^{1/m}$ and that the rescaled iterates converge weakly to the stationary distribution of the stochastic differential equation $$ dY_t=-h_0(Y_t)\,dt+\Sigma^{1/2}\,dB_t , $$ where $h_0$ is the limiting drift at the minimizer and $\Sigma$ denotes the asymptotic covariance. This recovers the Gaussian limit when $m=2$ and gives generally non-Gaussian stationary limits in the flat case $m>2$. Finally, we give corresponding results for coordinate-separable objectives with unequal flatness exponents.

A random sequence sampled from a known continuous distribution is observed with the objective to choose an item with the overall rank one. A rejected item cannot be recalled and is immediately erased from the memory. Under this memory constraint, the choice problem is not amenable to recursive methods of optimal stopping and becomes a global optimisation task. We focus on a heavy-traffic form of the problem with infinitely many choice opportunities, which we state in terms of a planar Poisson process (PPP). Symmetries of the PPP are used to derive basic structural properties of the optimal stopping rule, including the balance at the boundary equation, and two key integral identities. Throughout, we make throrough comparison to the classic full-information counterpart of the problem, revisiting both discrete- and continuous-time models. The optimal value, stopping rule and other characteristics of the problem are determined analytically and approximated numerically with high precision.

We study randomly advected incompressible Navier-Stokes equations, where the advecting field is a mean-zero, divergence-free, space-time stationary velocity field with smooth order-one correlations. We introduce a two-parameter family of models in which the advection is accelerated on a fast temporal scale $\varepsilon^2$ and has spatial correlation length $\delta$; the critical regime $\varepsilon = \delta$ corresponds to the natural parabolic scaling of the Navier-Stokes equation. In the full subcritical regime $\varepsilon = o (\delta)$, we prove a law of large numbers in dimensions $d = 2, 3$: the solutions converge to a deterministic Navier–Stokes system with an enhanced diffusion coefficient given by a Green-Kubo formula. In two space dimensions, under the slightly stronger assumption $\varepsilon = o (\delta^{1 + \iota})$ for some $\iota > 0$, we identify the leading-order fluctuations: after subtracting deterministic macroscopic corrections satisfying a nonlinear system of Navier-Stokes type, the rescaled fluctuations converge to a Gaussian field solving a linearized Navier-Stokes equation driven by multiplicative space-time white noise.

We provide a proof that the empirical spatial distribution estimator in $\mathbb R^d$ as well as the corresponding plug-in estimator of the spatial depth are uniformly $L^1$-consistent. The consistency rate only depends on the sample size $n$, not on the dimension $d$ or any tuning or regularization parameters. This is a rare property. The result of this note originates from a conversation with ChatGPT 5.4 Pro as part of some of our own earlier experiments on its mathematical reasoning capabilities.

Let $d \geq 3$, \[ D \subsetneq \{ 0,1,\dots,d-1\}, \qquad |D| \geq 2, \ 0 \in D \] be a finite alphabet, and define the integer Cantor set \begin{align} \mathcal{C} := \mathcal{C}_{D} := \bigcup_{J \geq 0} \Big\{ \sum_{j =0}^J a_j d^j : a_j \in D \Big\}. \end{align} We prove that for any $\sigma$-finite measure-preserving system, $(X,\mu,T)$, and any $f \in L^p(X)$, $2\leq p<\infty$, the ergodic averages \begin{align} \frac{1}{|\mathcal{C}_N|} \sum_{n \in \mathcal{C}_N } f(T^n x), \qquad \mathcal{C}_N := \mathcal{C} \cap \{1,2,\dots,N \} \end{align} converge $\mu$-almost everywhere. By rescaling, this allows us to resolve the question of lacunary differentiation of Cantor measures at self-similar scales: if \begin{align} \mathcal{C}' := \Big\{ \sum_{j \geq 1} a_j d^{-j} : a_j \in D \Big\} \subset [0,1] \end{align} is a real-variable Cantor set, and $\nu$ denotes its natural measure, then we prove that \begin{align} \lim_{k \to \infty} \int f(x-d^{-k} t) \ d\nu(t) = f(x) \end{align} Lebesgue almost-everywhere for any $f \in L^2_{\text{loc}}(\mathbb{R})$.

A foundational principle in nonlinear dynamics is that the structure of a dynamical system can be recovered from a small number of generic measurements or coordinates. We develop an analogous principle for the identification of dynamic models for time series with noise, which builds on previous identification results for noiseless dynamical systems. The noise is allowed to be non-iid, non-Gaussian, and dependent on the state. Our results cover noisily observed differential equations and discrete-time dynamical systems, as well as stochastic models with process noise. We illustrate the utility of this identification principle using a Lorenz-63 model and a Hénon map model, both with observational noise.

This work studies the mean-field limit of large networks of interacting stochastic leaky integrate-and-fire (LIF) neurons subject to short-term synaptic depression (STD). The macroscopic dynamics of this system is governed by a two-dimensional, non-linear McKean-Vlasov equation that couples the evolution of the neurons' membrane potentials with a synaptic depression variable. We investigate the long-time behavior of this limit system. To this end, we introduce an auxiliary linearized Markov process by freezing the interaction non-linearity to a constant. By exploiting the regeneration of the membrane potential at spike times, we are able to explicitly compute the conditional expectation of the synaptic depression variable, conditionally on the potential value, under the invariant measure of this two-dimensional linear process. This is a crucial ingredient to study time-dependent local perturbations thereof. As a consequence we are able to identify an analytic criterion guaranteeing the local stability of any invariant probability measure of the fully non-linear system. This stability criterion is formulated in terms of the zeros of the Laplace transform of a specific linear response function. Finally, we provide numerical examples demonstrating that the two-dimensional framework induces a richer spectrum of long-time dynamics than purely one-dimensional models. For example, synaptic depression can lead to low-frequency oscillations around a unique, unstable invariant measure where the oscillations are much slower than the neurons' firing rates.

We settle the Rajchman problem for canonical scalar dyadic Mandelbrot cascades at the minimal Kahane–Peyrière integrability threshold. If $\mu$ is the cascade on $[0,1]$, then $\widehat{\mu}(\xi)\to 0$ as $|\xi|\to\infty$, almost surely on non-extinction. For every fixed nondegenerate $C^2$ embedded arc $\gamma:[0,1]\to\mathbb{R}^2$, the pushforward $\gamma_\#\mu$ is likewise Rajchman almost surely on non-extinction. The analogous conclusion holds for the scalar cascade on the parameter circle pushed forward by any fixed nondegenerate $C^2$ Jordan curve. No moment condition of order strictly greater than one is imposed; in particular, the results include the regime $\mathbb{E}[W^q]=\infty$ for every $q>1$. The proof combines a spine-based lower-deviation principle, adaptive terminal approximation, and predictable capping to obtain almost-sure estimates uniform over large frequency annuli without higher moments. For curved pushforwards, an endpoint-safe phase decomposition controls direction-dependent stationary regions, including those meeting the endpoints of an arc, and couples the geometric and probabilistic arguments through a common dyadic kernel. Combined with the exact Fourier-dimension formulas for the corresponding models, the theorems show that Rajchman decay persists at zero Fourier dimension.

This work is concerned with the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with non-diagonal diffusion terms of type \begin{equation} u_{t}-\sum_{i,j=1}^{d} D^{2}_{ij}(a_{ij}(x)\beta(x,u))+ \text{div}(b(x,u)u)=0 \quad \text{in}\; (0, \infty) \times \mathbb{R}^{d} ,\notag \end{equation} with initial condition $u(0,x)\equiv u_{0}(x)$, where $a_{ij}$, $\beta$, and $b$ are suitable functions. Under suitable assumptions, this equation generates a continuous contraction semigroup $S(t): L^{1}(\mathbb{R}^{d}) \rightarrow L^{1}(\mathbb{R}^{d})$, and $u(t)=S(t)u_{0}$ is a mild solution to the equation. Our main contribution is to prove that this mild solution is unique in the much larger class of distributional solutions. This extends previous uniqueness results for the diagonal (also called isotropic) diffusion case $a_{ij} \equiv \delta_{ij}$. Another key analytical result of this paper is the uniqueness for distributional solutions of the associated linearized equation. As a main application, we prove weak uniqueness for the corresponding McKean-Vlasov SDEs. Moreover, we prove that, the probabilistically weak solution to the McKean-Vlasov SDEs is also the unique probabilistically strong solution. Furthermore, we establish a new $L^{\infty}$ estimate for mild solutions starting from data in $L^{1}\cap L^{\infty}$ and this estimate is used in the construction of nonlinear Markov processes. Finally, we prove that the path laws of the solutions to the McKean-Vlasov SDEs form a nonlinear Markov process in the sense of McKean.

We study the joint diffusive-mean field limit for a system of weakly interacting kinetic Langevin dynamics, extending the results of Delgadino, Gvalani and Pavliotis (Arch. Ration. Mech. Anal., 2021) to the hypoelliptic/hypocoercive case. We show that, in the absence of phase transitions, the two limits commute, and we calculate the covariance matrix of the limiting Brownian motion using the Green-Kubo/Kipnis-Varadhan formula. However, at low temperatures, and in the presence of phase transitions, the two limits may not commute. We demonstrate our findings by providing a detailed analysis of the diffusive-mean field limit for the $O(2)$ model in a magnetic field. Our analysis is based on the systematic use of recently developed hypocoercivity techniques, together with an appropriate linearization of the mean field McKean-Vlasov-Fokker-Planck partial differential equation.

The class of $q$-Bass martingales provides a natural answer to a central question in martingale optimal transport: how to construct martingales with prescribed initial and terminal marginals whose transition kernel remains as close as possible to a given reference measure $q$. We prove the existence of $q$-Bass martingales when the initial marginal is supported on finitely many atoms, and establish uniqueness, up to an additive translation constant, of the associated Bass measure. Our approach is geometric and relies on the analysis of a suitable parametrization of convex polygonal chains.

We prove that every connected simple graph of bounded degree satisfying the classical dimension-free Bakry–Émery condition $\mathrm{CD}(0,\infty)$ for the unnormalised Laplacian is volume doubling and supports, at all integer graph scales, a scale-invariant $L^2$-Poincaré inequality with dilation two, with constants depending only on the maximum degree. This settles the polynomial-growth conjecture of Cushing, Liu, and Peyerimhoff in a stronger form. The main novelty is a dimension-free adaptation of the graph-theoretic modified nonlinear heat-flow method introduced by Münch and extended to infinite weighted graphs by Pajot and Russ: point-mass consequences of $\Gamma_2\geq0$ and positive-resolvent smoothing replace any global $\mathrm{CD}(0,n)$ reduction, while diffusive exit-time control and finite-volume localisation yield the Poincaré inequality.

We study the vector Dyson equation $$-\frac{1}{m(z)}=z\mathbf{1}-\mathbf{a}+Sm(z),$$ with parameter $z$ in the complex upper half-plane $\mathbb{C}_+$, where $\mathbf{a}\in\mathbb R^d$ and $S$ is a nonnegative matrix, not necessarily symmetric. This equation has a unique vector solution $m(z)\in\mathbb{C}_+^d$, for which we establish a complete measure decomposition and prove regularity. We then develop a graph-theoretic approach to the singularity and stability problem for non-symmetric matrices $S$. The graph structure of $S$ identifies the possible degeneracies of the stability operator as $z$ approaches the real axis. In particular, for non-backtracking matrices, we prove square-root growth at regular edges, cubic-root growth at regular cusps, and complete stability estimates. We also obtain the corresponding estimates for symmetric matrices in the periodic setting.

Score-based generative models exhibit a puzzling behavior: they often appear to cover all modes of a target multimodal distribution and yet may fail to learn the correct relative mode amplitudes, which can be interpreted as mixture weights. We resolve this apparent paradox by relating the diffusion score matching (DSM) loss to the error in estimating mixture weights from generated samples. We show that, even when the target score is insensitive to mixture weights, generated samples can recover the weights accurately if the scores at intermediate noise levels are informative about the weights. Accordingly, we define the diffusion score sensitivity index (DSSI) as the variation in the DSM loss relative to changes in a parameter. We then show that the DSSI governs the accuracy with which the parameter of the target distribution can be estimated from generated samples. For Gaussian mixtures in arbitrary dimensions, we prove that the mixture weight estimation errors are on the same order as the DSM loss under mild conditions. Empirically, we show the emergence of sensitivity during the noising process of benchmark data distributions under typical noise schedules, and that these sensitivity values predict how well a well-trained model recovers mixture weights. Furthermore, we show that the choice of noise schedule can reduce diffusion sensitivity, leading to mode amplification. Although we focus on mixture weights, the proposed sensitivity framework governs the recovery of any qualitative parameter of the target distribution.

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same is true for equilibria obtained as limits of finite approximations. Techniques developed in this paper show that infinite games long treated as intractable become amenable to direct equilibrium analysis.