Papers

We study Gaussian fluctuations of differential observables of centered stationary random fields on the discrete $d$-dimensional torus of mesh $1/N$. Our main result is a Central Limit Theorem for the fields $f\mapsto N^{-d/2-m}\sum_x\phi_x^N(Pf)(x/N)$, where $P$ is a scalar homogeneous constant-coefficient differential operator of order $m\geq0$ with Fourier symbol $p$, e.g., $P=\Delta$ and $m=2$. Under suitable assumptions on the spectral scaling limit $\gamma$ of the covariance and the normalized cumulants of order $r\geq3$, we prove convergence in law in a negative Sobolev space to a centered Gaussian random distribution with covariance multiplier $|p(k)|^2\gamma(k)$. The two factors separate the effects of the operator and the field and, notably, differentiation can compensate a low-frequency spectral singularity. For the lattice Gaussian Free Field, this gives a projected white-noise limit for the gradient. The proofs use Fourier analysis and the method of moments and cumulants, together with a tightness argument. Finally, we give two sufficient criteria for the cumulant assumption, namely a tree–graph bound and the Dobrushin uniqueness condition for finite-range Gibbs fields, and recover the white-noise limits of the centered Ising magnetization and Potts colour fluctuation fields at high temperature.

We study bootstrap percolation with independent vertex thresholds taking values one and two, with threshold-one probability $(1-c/d)/d$ for fixed $c>0$. The seed set is chosen uniformly among sets of a prescribed deterministic size, independently of the graph and thresholds. We prove threshold statements at fixed relative margins. For uniform simple graphs with prescribed nonnegative integer degrees of even sum and exact mean $d$, assume $\max_i|d_i-d|\le C\sqrt d$ and $c+1-v_n\ge\kappa>0$, where $v_n=\operatorname{Var}(D)/d$ and $C,\kappa$ are fixed. When $d\to\infty$ and $d=o(n^{1/7})$, the leading seed scale is $n(c+1-v_n)^2/(2d^4)$, without requiring a limit of $v_n$. At fixed relative margins below and above this scale, the final active set has size $O(n/d^3)$ and $n-o(n)$, respectively, with high probability. A separate result for $G(n,d/n)$ holds when $d\to\infty$ and $d^5/n\to0$, and gives scale $nc^2/(2d^4)$. Thus regular and independent-edge graphs have different coefficients at the same asymptotic mean degree. Local exploration estimates yield explicit inactive remainders and survive conditioning on simplicity in the prescribed model. We also quantify the precision obstruction to static inclusion transfer and compute deterministic response-barrier corrections, without identifying a shrinking random critical window.

We classify the linear solution spaces of the chordal Belavin–Polyakov–Zamolodchikov (BPZ) equations, which arise in Dubédat's commutation relations for multiple Schramm–Loewner evolutions (SLE). Without imposing growth conditions, we determine the subspaces selected by successive conformal Ward identities and their exact dimensions. For every $\kappa>0$ and BPZ spectral parameter $\lambda\in\mathbb{R}$, we determine the spectrum and Jordan structure of the translation operator. When $\lambda=0$ and $\kappa\in(0,8)$, we further determine all admissible scaling exponents and the dimensions of the corresponding translation-invariant solution spaces. A triangular change of derivative coordinates yields a first-order system of rational Knizhnik–Zamolodchikov type, allowing us to represent the Ward operators explicitly and reduce the classification to finite-dimensional linear algebra. For $2N$ boundary points and $\kappa\in(0,8)$, translation-invariant BPZ solutions with $\lambda=0$ and the homogeneity required by Möbius covariance automatically satisfy the third Ward identity and the usual power-law bound. This recovers, under weaker assumptions, the Catalan dimension formula previously established by Flores and Kleban. At $\kappa=8$, the third Ward identity imposes an additional constraint when $N\ge2$, but the full Möbius-covariant solution space still has Catalan dimension, establishing completeness of the partition functions constructed from uniform spanning trees. Finally, at $\kappa=4$ and $\lambda>0$, we construct an explicit basis of positive BPZ solutions and identify its elements as partition functions for level lines of a Gaussian free field with suitable harmonic means.

This paper is concerned with a natural variant of the contact process that places a social cluster of individuals, rather than a single individual, at each lattice point, and distinguishes between symptomatic and asymptomatic individuals. In particular, the epidemic model depends on four infection parameters: the external (inter-cluster) and internal (intra-cluster) infection rates for both symptomatic and asymptomatic individuals. Newly infected individuals begin as asymptomatic before transitioning to the symptomatic state at a spontaneous mutation rate, and infected individuals recover at a baseline rate of one. We study the phase structure of the deterministic nonspatial mean-field model when each cluster reduces to a pair of individuals, and prove that the stochastic spatial model exhibits the same phase transitions with respect to the four infection rates for all cluster sizes. We also prove the existence of a phase transition driven by the cluster size itself for the spatial model.

We determine the leading logarithmic asymptotics of the fixed-maturity right tail of the SABR model with $\beta\in(0,1)$ and absorption at zero, for every correlation $\rho\in(-1,1)$. If $P(k)$ is the probability that the terminal forward is at least $f_0e^k$, then $-k^{-2}\ln P(k)\to(1-\beta)^2/(2\nu^2T(1-(\rho\wedge0)^2))$ as $k\to\infty$. For $\rho\ge0$ this is the rate predicted by the unrestricted hyperbolic geometry. For $\rho<0$ it is strictly larger: along the volatility excursion that produces the tail, survival forces the noise orthogonal to volatility to stay above a moving square-root barrier, at an additional cost of the same order $k^2$. Consequently the Black-Scholes implied volatility tends to $\nu\sqrt{1-(\rho\wedge0)^2}/(1-\beta)$: Henry-Labordère's conjectured wing limit $\nu/(1-\beta)$ holds for $\rho\ge0$ and fails for every $\rho<0$. For his second conjecture, whose unrestricted-distance Gaussian estimate does not hold for $\rho<0$, we also identify the correct replacement for this model: at the level of tail probabilities, the rate is given by the squared intrinsic Riemannian distance, within the survival domain, from the initial state to the terminal target set. The proofs are direct, combining a stopped Lamperti transform, a conditional time change, Gaussian moving-barrier bounds and change-of-measure constructions, and use no heat-kernel estimates.

Suppose $X=(X_t)_{t\ge0}$ is a supercritical superprocess with a non-local branching mechanism on a Lusin space $E$, whose mean semigroup admits a positive leading eigenvalue $\lambda_1$ with a bounded positive eigenfunction $\varphi$ and a finite eigenmeasure $\widetilde{\varphi}$. Let $Y_t:=\int_0^t X_s\mathrm{d}s$ be its occupation measure. We first prove a law of large numbers, that is, for every bounded measurable function $f$, $e^{-\lambda_1 t}\langle f,Y_t\rangle$ converges in $L^2$ to $\lambda_1^{-1}\langle f,\widetilde{\varphi}\rangle W^\varphi_\infty$, where $W^\varphi_\infty$ is the limit of the martingale $W^\varphi_t:=e^{-\lambda_1 t}\langle\varphi,X_t\rangle$. We then establish functional central limit theorems for the fluctuations of $\langle f,Y_t\rangle$ around this random centering. We show that the normalisation and the limit are determined by the spectral decay rate of $f$ compared with the critical value $\lambda_1/2$, and there are three possible regimes: after suitable normalisation, the fluctuation process converges in distribution in $\mathcal{C}[0,+\infty)$ either to a mixed Gaussian process $\sqrt{W^{\varphi}_{\infty}}G(f)$, or to a constant Gaussian process with random variance depending on $W^{\varphi}_{\infty}$, or to a non-Gaussian limit given by the limit of a martingale associated with the spectral projection of $f$. These results are the occupation-time counterparts of the fluctuation theorems for linear functionals of supercritical non-local branching superprocesses.

We study stochastic flows with interaction through an infinite-dimensional semigroup approach. By means of a kernel lifting procedure, we represent the solution to measure-valued stochastic equation as a Hilbert-valued Markov process. We identify its generator, construct the associated Markov semigroup, and establish a dense range condition using measure polynomial functionals. The results provide a framework for the analysis of stability of stochastic equations with interaction.

Per-target labeled random-finite-set filters, such as the state of the art $\delta$-generalized labeled multi-Bernoulli filter, cannot represent cooperating, swarm-like, motion between targets. In this work, we present a new filter that can. The new particle filter couples the joint particle filter with ensemble Gaussian mixture filter such that a measurement updates the density at every particle instead of merely reweighting. We prove that the new particle filter converges to the joint particle filter in the limit of particle number and thus to the true Bayesian posterior, under certain assumptions. Experimental results on two coupled systems, the Vicsek model and coupled Brownian motion provide evidence on the efficacy of the filter.

We study the proper hat guessing game on graphs, introduced by Adriaensen et al. in Hat guessing with proper colorings. In this game, the players are seated on the vertices of a graph $G$ and assigned hats from a set of $k$ colors such that the resulting assignment forms a proper coloring. The visibility of each vertex is limited to the hat colors of their neighborhood. Then they must simultaneously output a guess about the color of their own hat. The players win if at least one guess is correct. A parameter related to this problem is the proper hat guessing number $\operatorname{HG}_{P}(G)$ that is the maximum number of colors $m$ such that the players can guarantee a winning strategy. Motivated by the work of Shurman et al. \cite{shurman2026upper}, we establish the first upper bound that depends both on the number of vertices $n$ and the maximum degree $\Delta$ in the case where $\Delta \geq \frac{n}{e+1}$. This result leads us to show that the proper hat guessing number of the binomial random graph $G_{n,1/2}$ is bounded above by $cn$, where $c \approx 1.366$. Finally, we prove that graphs of maximum degree $(1-\gamma)n$ for some fixed $\gamma \in (0,1]$ cannot have $\operatorname{HG}_{P}(G) = (2-o(1))n$.

In his highly influential paper, On the maximal ergodic theorem for certain subsets of the integers, Bourgain introduced the study of pointwise convergence of ergodic averages along randomly generated subsets of the integers, developing a robust \(L^p\)-theory for \(p>1\). More recently, this theory was partially complemented at the \(L^1\) endpoint by work of LaVictoire, who treated averages along random sequences that are “slightly denser" than the set of squares, for which universal \(L^1\) pointwise convergence is known to fail. In this work, we prove that, almost surely, for every measure-preserving system and every integrable function, the ergodic averages formed by sampling along random sequences with quadratic growth converge almost everywhere. Our proof combines harmonic-analytic methods developed by M.~Christ with a “random sampling" of quadratic forms, thereby overcoming the main uniformity obstruction.

We study finite and countable point configurations in the trace of Brownian motion in $\mathbb{R}^d$, $d\geq2$. In the plane, the set of translations placing a prescribed finite configuration on the trace has Hausdorff dimension two in every nonempty open set; we prove analogous dimension-preservation results for radial and bilipschitz patterns, and every prescribed countable configuration has a dense translation set. For $d\geq3$ we prove a sharp occurrence criterion for regular scale-invariant families of configurations, in terms of the ranks of all labelled subconfigurations. It gives exact thresholds for arithmetic progressions and for homothetic and similar copies of finite sets. In particular we recover and extend a result of Benjamini and Kozma: almost surely the longest arithmetic progression in the trace has length five in $\mathbb{R}^3$, three in $\mathbb{R}^4$ and $\mathbb{R}^5$, and two in $\mathbb{R}^d$ for $d\geq6$.

In this paper we introduce a geometrically constrained optimal transport problem on Riemannian manifolds endowed with a Borel foliation, in which mass is transported only within individual leaves. We characterize the existence of foliated transport plans through the equality of the transverse projections of the marginals and show that the corresponding quadratic transport cost decomposes into a measurable family of leafwise optimal transport problems. We also formulate the problem as a classical Kantorovich problem with an extended-valued leafwise cost. Finally, we introduce a foliated continuity equation and establish a Benamou-Brenier formula relating the static and dynamical foliated formulations.

In this paper, we study the ferromagnetic nearest-neighbor Ising model on rooted spherically symmetric trees with one or \(q\) children per vertex, according to a prescribed sequence of branching levels. Unbounded runs of one-child levels force uniqueness at every nonzero homogeneous field and every finite inverse temperature. We quantify this effect through the locations of sufficiently long nonbranching stretches. Conversely, uniqueness need not entail any uniform rate: for each prescribed vanishing sequence, we construct a density-one tree whose finite-volume boundary influence decays more slowly along a subsequence. The tree retains branching number \(q\), although its nonzero-field coexistence region disappears. For eventually bounded gaps, we prove an optimal uniform comparison with the periodic tree having the largest allowed gap and obtain an explicit interval of coexistence fields. Thus bounded gaps are equivalent, within this family, to coexistence at some finite temperature and nonzero field. For eventually periodic gaps, Gibbs uniqueness is equivalent to fixed-point uniqueness of the full-period return map. A negative Schwarzian derivative yields a complete classification: in the supercritical regime the map has three fixed points inside a closed coexistence interval, two at its endpoints, and one outside. The nonzero endpoints are automatically nondegenerate folds, with square-root splitting of the merging branches.

The eigenvalues of a large normal random matrix form a two-dimensional Coulomb gas. At fixed weight, the spectral droplet grows with the matrix rank, and its semiclassical density increment is harmonic measure. We freeze an $N$-eigenvalue configuration and sample $M$ further eigenvalues using the projection onto polynomials of degree less than $N+M$ that vanish at the frozen points. Their conditional law is an exact polynomial ensemble in the potential of the frozen charges. For a regular analytic droplet, we prove that the spatial distribution of the new eigenvalues converges to harmonic measure on the old boundary when $M/\log N\to\infty$ and $M/N\to0$. This convergence holds conditionally on the frozen spectrum, outside a set of frozen configurations whose probability tends to zero. The equilibrium process formed from orthogonal polynomials of degrees $N,\ldots,N+M-1$ has the same limit. An exact finite-rank identity relates polynomial moments of the conditional mean to fluctuations of the frozen spectrum, with a remainder from polynomial modes of degree $N+M$ and above.

Probability-flow ordinary differential equations (PF-ODEs) are widely used as deterministic samplers for score-based diffusion models. Their usual justification is that the Fokker–Planck equation of a diffusion can be rewritten as a continuity equation driven by the score function of the forward diffusion. This identity does not, however, guarantee that the resulting velocity field generates a well-posed flow. We provide theoretical insights into the design of such deterministic samplers for generative models based on diffusions and reflected diffusions. We identify sufficient conditions for a regular Lagrangian PF-ODE flow to exist; reverse sampling and invertibility require two-sided divergence control. For learned scores, sampler stability is controlled by an unweighted velocity error, exposing a mismatch with density-weighted score matching and motivating architectural control of Jacobians, divergence, growth, and compression. Under the manifold hypothesis, positive-time regularization justifies an early-stopped PF-ODE while constants deteriorate near the data endpoint; an explicit sphere example shows that the exact deterministic flow becomes singular as the noise level vanishes even though the diffusion marginals remain well defined. These theoretical insights translate into concrete design principles for stable, invertible, and constraint-preserving diffusion samplers. We illustrate the practical relevance of these design principles using controlled numerical experiments.

We study metric graph sign clusters of the Gaussian free field on a large class of transient weighted graphs $G$ that are sufficiently low-dimensional, i.e. below the mean-field regime, and prove a macroscopic equivalence property: several natural ways to define macroscopic clusters - involving any of diameter, volume, or capacity functionals - are in fact the same. As a corollary, macroscopic clusters are characterized in low dimensions as those containing large loops in the corresponding loop soup picture, in contrast with the typical behavior in high dimensions. We then apply these results to the small-mesh scaling limit of critical clusters on $\varepsilon\mathbb{Z}^d$, $d=3,4,5$, as $\varepsilon \downarrow 0$. We prove that every subsequential scaling limit consists of sets with positive Brownian capacity whose Hausdorff and Minkowski dimensions are both equal to $1+\tfrac d2$, as conjectured by Werner. We also derive two-point crossing estimates in the limit, and show that each limiting loop soup cluster is the closure of the union of the scaling limits of the macroscopic loops it contains. Remarkably, unlike with planar Bernoulli percolation for instance, the determination of the Hausdorff dimension does not emerge from direct calculations involving a limiting object (such as $\text{SLE}_6$), but rather from the distinctive features of the discrete model itself.

Let $\mathcal{C}_\varepsilon$ denote the time it takes a Brownian motion to come within distance $\varepsilon$ of every point in the two dimensional torus $\mathbb{T}^2$ of unit area. We prove that for an explicit non-random centering $m_\varepsilon$, the variables $\{\sqrt{\mathcal{C}_{\varepsilon}} - m_\varepsilon \}$ converge in distribution as $\varepsilon \to 0$, to a randomly shifted Gumbel law.

We study an infinite-dimensional Skorokhod problem on the positive cone of $L_1([0,1])$, where a regulator process constrains a free càdlàg path to remain nonnegative through an oblique reflection mechanism determined by a positive linear integral operator $\boldsymbol{F}$ on $L_1([0,1])$ with spectral radius less than $1$. This constrained process $Z$ remains in the positive cone, and the regulator $Y$ is nondecreasing with respect to the cone order and satisfies a pointwise complementarity condition, increasing only on those space-time regions where the corresponding component of $Z$ vanishes. This formulation extends the classical multidimensional Skorokhod problem on the nonnegative orthant to an infinite-dimensional setting. Under suitable assumptions on the operator $\boldsymbol{F}$ that guarantee a dominant positive eigenfunction, we proceed through a diagonalization argument to give an alternative representation of the system in which the $L_1$ norm of the linear operator is less than 1. From this characterization, we establish existence and uniqueness of solutions for arbitrary input paths $X\in D([0,T],L_1([0,1]))$. We further show that the associated Skorokhod map is well defined and Lipschitz continuous with respect to both the uniform and Skorokhod $J_1$ topologies. As a consequence, solutions may be obtained as limits of multidimensional Skorokhod problems, providing a constructive characterization of the infinite-dimensional reflection mechanism. These results provide a framework for the study of constrained stochastic systems with infinitely many interacting components and provide a foundation for reflected stochastic processes evolving in function spaces.

In this article, we give a global construction of the Virasoro conformal blocks on Teichmüller space of Riemann surfaces of genus $g$ with $m$ marked points, when the central charge is $c>25$. We prove that they are global holomorphic sections of a holomorphic line bundle and they satisfy the Ward identity, which encodes their conformal invariance. For the sphere with $3$ points, the space of conformal blocks is one-dimensional. For a given marked Riemann surface and a marked pair of pants decomposition, it is in general an infinite dimensional Hilbert space, isomorphic to $L^2(\R_+^{3g-3+m})$ with respect to some measure involving the DOZZ structure constants. We show that it carries a projective unitary representation of the mapping class group. More generally, there are unitary isomorphisms for each Moore-Seiberg move, changing a marked pair of pants decomposition into another one.

This paper is the second part of the proof of the conformal bootstrap for Liouville conformal field theory on compact surfaces with boundary. It is devoted to the spectral theory of the half-annulus semigroup and to the resulting bootstrap formula. Building on the Segal axioms and gluing properties established in Part~1~\cite{GRW1}, we identify the generator of the half-annulus semigroup with the boundary Hamiltonian and establish its spectral decomposition by scattering methods. This yields a direct-integral decomposition of the boundary state space into irreducible Virasoro representations, governed by the boundary spectral measure. Combining this decomposition with Ward identities for boundary deformations, we express general correlation functions as integrals over spectral parameters attached to cutting curves, with integrands given by products of bulk and boundary structure constants and the corresponding conformal blocks. These results also underpin the analyticity and symmetry properties of conformal blocks~\cite{GRSSbloc}, and contribute to the construction of mapping class group representations on spaces of conformal blocks~\cite{PapierBlocs}.

We study gradient-type dynamics for weak stochastic control through the laws of controlled diffusions on path space. Although the objective is generally nonconvex in the drift, a uniformly convex running cost induces a strictly convex functional of the corresponding path law and hence a natural Bregman geometry. We construct the associated implicit proximal scheme and prove its well-posedness and convergence to a continuous-time mirror flow. When the mirror cost coincides with the running cost and the law-dependent potential is linear, the flow becomes affine in centered entropy coordinates. This yields a variational construction of the resulting Newton flow, uniform stability estimates, and exponential convergence to the optimizer. For a general uniformly convex mirror cost and a nonlinear convex potential of the law, we construct the natural-gradient flow globally in learning time using BMO estimates for quadratic BSDEs, and establish quantitative convergence and a first-order discretization error. We further analyze interacting finite-particle approximations, proving exponential convergence to the finite-particle optimum and an O(1/N) normalized discrepancy from the mean-field optimum. In the Markovian setting, reflection coupling yields uniform spatial regularity and exponential convergence of the feedback controls on Rd, including for nonquadratic state-dependent Hamiltonians.

The analysis of large data panels is important in econometrics and beyond. Prediction and inference methods for such data typically rely on simplifying model assumptions for the covariance structure of errors. One convenient assumption is what we call groupwise sphericity: that errors are uncorrelated across individuals and have constant variance within certain groups. While theoretically useful, in large panels groupwise sphericity is often too restrictive to apply in practice. We therefore develop new quantitative inference tools to test whether deviations from this model assumption are practically relevant. Our approach covers both large-dimensional data matrices and regression panels, in a regime where the cross-sectional dimension is proportional to the sample size. The theory is based on the analysis of extreme eigenvalues of the empirical covariance matrix and uses recent advances in random matrix theory. Numerical experiments demonstrate accurate size control and good power in finite samples.

This paper establishes uniform local large-deviation asymptotics for the entrance law $\underline n_t(dx)$ of excursions of a Lévy process with negative drift and locally regularly varying Lévy measure $\nu$, reflected at its running infimum. In the fixed-time regime, we prove that as $x\to\infty$, $$ \underline n_t\big((x,x+\delta]\big) \sim \underline n(\zeta\wedge t)\, \nu\big((x,x+\delta]\big), $$ uniformly in $\delta\in[\delta_0,\infty]$ for every $\delta_0>0$. Our principal result concerns the large-time regime and shows that, as $t\to\infty$, $$ \underline n_t\big((x,x+\delta]\big)\sim \int_x^{x+\delta} V(z) \,\nu(\beta t+dz), $$ uniformly in $x\geq0$ and $\delta\in[\delta_0,\infty]$, where $V$ denotes the renewal function. Using a fluctuation-theoretic decomposition of the killed semigroup, we further derive corresponding uniform local large-deviation asymptotics for the Lévy process killed upon its first passage below zero.

The exact limiting distribution of a volatility target index has recently been established under a log-normal assumption on the risky asset. In this paper, we show that the structural limit remains valid under a broad class of stochastic volatility processes. In addition, we show that, when the limit of the joint distribution of the volatility target index and the driving Brownian motions is considered, an extra independent Brownian motion must be involved. Besides the structural limit theorem, we derive a correction formula for the expected quadratic variation of the discrete rebalancing volatility target index, under the assumption that the volatility of the underlying risky asset is a deterministic function of time. As a consequence of the discrete correction formula, it is shown that, as the rebalancing time step $\Delta t\to 0$ and the observation window parameter $\lambda \to 1$ simultaneously along a path, the expected quadratic variation converges to the target variance if and only if $(1-\lambda)^{-1}\Delta t \to 0$ along the same path. Numerical results are provided to support the structural limit, using the local volatility model and the Heston stochastic volatility model as examples, as well as the effectiveness of the discrete correction formula.

We consider a continuous-time particle system with exclusion interaction on the integer lattice. Each particle is assigned intrinsic left and right jump rates by an independent draw from a common random environment. We identify the decomposition of the system into maximal stable subsystems, which we call clouds. We show different qualitative behaviour for the cloud decomposition corresponding to different regimes for the random environment law that parallel the classical Sinai and Kesten–Kozlov–Spitzer regimes from one-dimensional random walk in random environment. Our main result in the Sinai regime is that the system is decomposed into an infinite number of finite clouds which all go to minus infinity with rapidly decreasing speed magnitudes, and we obtain explicit associated distributional limits. We analyse the model through the potential function associated with the random environment. In the Sinai regime, deep potential wells trap particles, producing stable clouds whose speeds are exponentially small in the well depth. Since successive wells are deeper, the resulting clouds are successively slower. Thus the localization mechanism of Sinai's random walk manifests itself here as a jamming phenomenon, in which a potential well traps not one walker but an entire block of mutually excluding particles.

We present a simple proof of the $\varepsilon$-Dvoretzky conjecture, which asserts that the dependence on the approximation parameter $\varepsilon$ in Dvoretzky's theorem is polynomial in $1/\varepsilon$. In particular, if $n \geq (C/\varepsilon)^{\ell/2+1}$, then any $n$-dimensional convex body has, through any given interior point, an $\ell$-dimensional section that is $\varepsilon$-close to a Euclidean ball. Here, $C > 0$ is a universal constant. We in fact obtain a sharper dependence on $\varepsilon$. The proof is probabilistic, but uses a different probabilistic model from those employed previously. We also prove a simultaneous version of our theorem for a finite family of convex bodies with the origin in their interior, yielding a common linear subspace on which all of the bodies have nearly-Euclidean sections. Finally, we compare the radius of the approximating Euclidean ball with familiar geometric parameters, such as the mean widths of the body and its dual.

In this paper, we study the problem of sampling from a probability distribution using Langevin-type dynamics. We consider overdamped Langevin dynamics perturbed by adding a Stratonovich perturbation that preserves reversibility. We prove an optimal scaling for the Stratonovich perturbation applied to Gaussian target distributions and derive an algorithm to construct such perturbation. Our theoretical results are supplemented by numerical experiments in which we compare our proposed method with nonreversible samplers and overdamped Langevin dynamics. Our numerical experiments demonstrate the efficiency of the approach beyond the quadratic case.

We study the scaling limit of the collision measure of two i.i.d. discrete-time random walks on $\mathbb Z^2$, which records their collision sites and times. This is a critical regime for collisions: the walks collide infinitely often, whereas the limiting Brownian motions do not collide at positive times. Hence, existing general results on the convergence of collision measures do not apply. Assuming that the jump distribution has mean zero and finite second moment, we prove that, under a logarithmic scaling, the collision measure converges in distribution to a non-trivial random measure. We also give an explicit representation of the limiting measure in terms of a Poisson point process. To prove convergence, we introduce a new cluster decomposition of the Laplace transform of the collision measure, revealing the cluster structure of collisions.

We develop a cancellation mechanism for linear second-order kinetic equations in non-divergence form with measurable uniformly elliptic coefficients. We decompose the dynamics into spatial waves and follow two families of velocity paths along which the waves acquire nearly opposite phases. The parabolic Krylov–Safonov theory gives a common lower bound for the two velocity marginals. The corresponding contributions cancel up to a small phase error, producing a contraction \textità la Döblin}. Iterating this contraction yields exponential Fourier decay estimates with enhanced dissipation. We then present two applications. The first contribution is a kinetic Aleksandrov estimate: a maximum principle in which the source is measured in an $L^p$ norm. For rough coefficients $A(t,v)$ independent of position, we obtain the estimate for every $p>2n+1$, where $n$ is the dimension of position and velocity. The Fourier decay also gives spatial smoothness, and parabolic regularity gives Hölder continuity in time and velocity. For autonomous coefficients $a(x,v)$ in dimension one, position serves as time away from zero velocity. A Harnack comparison controls returns to small velocity intervals and yields the estimate for every $p>4$. Both thresholds are optimal among those valid for all ellipticity ratios. The second contribution concerns position on the torus and velocity on the sphere, with measurable uniformly elliptic diffusion coefficients $A(t,v)$ independent of position. We prove enhanced dissipation with the optimal square-root frequency power and Gevrey regularity in position. For autonomous coefficients $A=A(v)$, we also obtain exponential convergence to equilibrium in $L^2$ weighted by the invariant velocity measure, and a quantitative spectral gap. To our knowledge, these are the first results on decay established for laws of kinetic equations with rough coefficients.

Neural network-based approaches have emerged as efficient alternatives to traditional optimization-based procedures for the calibration of stochastic volatility models. However, existing work has focused primarily on predictive accuracy, with comparatively little attention devoted to understanding the structure of the learned inverse calibration mappings. In this work, we analyze neural calibration mappings for the Heston and rough Heston models across multilayer perceptron, highway, and softmax-parametrized highway architectures, using complementary Shapley-based methods from explainable AI. Specifically, we consider SHAP and $\nu$SHAP explanations, which capture distinct, complementary notions of feature relevance, corresponding to sensitivity and sufficiency of feature subsets, respectively. Short maturities and smile wings consistently dominate parameter inference, and the dominant attribution structure remains qualitatively stable across architectures despite differences in predictive accuracy and parameter count. Parameter-specific differences between SHAP and $\nu$SHAP further reveal how distinct regions of the implied volatility surface contribute to parameter recovery and expose substantial redundancy in the calibration input. Building on this redundancy, we show that $\nu$SHAP explanations can guide a significant reduction in input dimensionality for the rough Heston model while matching calibration accuracy relative to the full implied volatility surface. These findings demonstrate that complementary Shapley-based methods provide structural insight into learned inverse calibration mappings beyond predictive error metrics, and offer a practical route to feature selection in neural calibration problems.