Papers

This article develops a finite-horizon multiple-stopping framework and applies it to three American-style contracts on a geometric random walk in a Cox–Ross–Rubinstein market: an American put, a Russian option, and a floating-strike geometric-average Asian put. The general problem is represented by recursively defined Snell envelopes, with unused exercise rights encoded by a cemetery time; this yields an ordered optimal exercise vector without requiring all rights to be exercised. For the American put, a median representation of successive marginal values yields diminishing marginal values, nested exercise regions, and monotone exercise thresholds without relying on convexity of the marginal value. After suitable state reductions, analogous marginal-value arguments give threshold-type optimal exercise rules for the Russian and geometric-average Asian options. Independent random maturity is also incorporated, and its effect on the corresponding stopping regions is identified.

The Anderson tight-binding model is a fundamental model of quantum transport and localization in disordered media. Completing a problem left open by Bourgain and Kenig, this paper proves Anderson localization near the bottom of the spectrum for the lattice Anderson model with Bernoulli potential, in any dimension $d\ge 2$. The proof uses the multiscale framework of Fröhlich-Spencer and Bourgain-Kenig, and the main new ingredient is a probabilistic discrete unique continuation principle (PDUC) for the discrete Schrödinger equation. This PDUC is established via a bootstrap argument and a key probabilistic lemma proved by adaptively revealing the random potential.

A $d$-set mapping is a function acting on a domain $X$ equipped with a partition into $d$ disjoint subsets. While standard functions represent $1$-set mappings, generalizations to arbitrary $d$-partite structures appear naturally across discrete mathematics. In this paper, we develop an analytic combinatorial framework to quantify the functional graphs of these mappings. By leveraging generating functions and singularity analysis, we derive exact asymptotic expansions for macroscopic graph properties as the cardinality of $X$ tends to infinity, including the expected number of connected components, cyclic nodes, and tail lengths. We demonstrate the efficacy of this framework by recovering the classical bipartite mapping results of Hansen and Jaworski, and successfully generalize these mechanisms to arbitrary $d$-set mappings, providing the foundational architecture to establish their probabilistic limit laws.

We determine the Doob-Martin boundary of Hoffman's leaf-grafting graph on finite unlabelled non-plane rooted trees. Its full and minimal boundaries coincide and are parametrized by deterministic recursive paintboxes, identified when their finite sampling laws agree. Every central measure is a unique mixture of the corresponding extremal laws, and its limiting boundary point generates the completed tail field. We also show that the boundary is homeomorphic to the space of unordered root masses marked by child boundary classes. For the recursive Ewens family, we obtain the unique extremal decomposition from independent Poisson-Dirichlet splits, including the uniform recursive-tree and rooted-tree Plancherel cases.

We study the vanishing-noise limit of the rate function for the Donsker-Varadhan large deviation principle for one-dimensional diffusion processes on a circle. As is well known, the rate function can be represented either as the Legendre transform of the principal eigenvalue of the perturbed infinitesimal generator or by a variational formula. We analyze the asymptotic behavior of the principal eigenvalue as the parameter in front of the noise goes to zero and compare the Legendre transform of the limit with the expression obtained as the limit of the variational representation. We prove, in particular, that the resulting expressions do not always coincide, leading to continuity and discontinuity phenomena in infinite-dimensional functional spaces. Moreover, we use the previous analysis to pass to the limit in the LDP, using the notion of {\Gamma}-convergence.

Statistical model transformations based on randomly stopped sums, maxima and minima are widely used to extend statistical models. We characterize the complete set of stopping models for which randomly stopped sum and extreme model transformations function as statistically stable (idempotent) model extensions. Stability requires the underlying stopping model to be closed under pgf composition. We prove that any finite-dimensional, connected stopping model closed under pgf composition is necessarily a family of random variables whose pgfs commute. Using the corresponding Koenigs function, we establish that these models form a statistical manifold admitting a global, one-dimensional parametrization $\theta = \Pr(N=1) \in (0, \theta_*]$, where the probability mass at $i$ is a polynomial in $\theta$ of degree at most $i$. Finally, we establish a duality between stopping models closed and containing the identity variable (the ones yielding stable extensions) and the set of probability distributions supported on the positive integers. These findings disprove the long standing conjecture that statistical stability occurs only under geometric stopping.

We prove that for every isotropic log-concave distribution $P$ on $\mathbb{R}^d$ and every even $m\ge2$, the polynomial $(Cm)^m\|v\|_2^m - \mathbb{E}_{X\sim P}\langle X,v\rangle^m$ is a sum of squares, where $C>0$ is a universal constant. This improves on the Poincaré-dependent bounds (Kothari and Steinhardt, 2017), recovering the optimal moment bounds for log-concave distributions. As an immediate corollary, we obtain computationally efficient algorithms with dimension-free error guarantees for a wide range of statistical estimation problems. Our proof proceeds by using stochastic localization to decompose $P$ as an average of strongly log-concave measures, whose centered moments admit the subgaussian certificates (Diakonikolas et al., 2025). With a covariance-adapted choice of localization, we show that a fourth-moment certificate derived from the variance inequality for quadratic forms (Letwin, 2026) suffices to control this averaging at every even degree.

We study the largest common subtree of two independent unlabeled uniform attachment trees (also known as random recursive trees). Our main result shows that, when the two trees have $n$ vertices each, their largest common subtree has at least $n^{0.83}$ vertices with high probability. This is obtained by starting with the common subtree induced by the Ulam–Harris labels in the two trees and improving using local optimization steps. We also give some upper bounds and bounds for general random tree growth models. We leave as an intriguing open question to understand the magnitude of the size of the largest common subtree.

We prove that the planar balanced excited random walk $M(2,1,2)$ is recurrent. This walk takes a horizontal simple random walk step on its first departure from each vertex and a planar simple random walk step on every later departure. Moreover, the number of distinct vertices visited before time $n$, multiplied by $(\log n)/n$, converges to $\pi$ almost surely and in every $L^p$, $1\le p<\infty$, the same limit as for the planar simple random walk. More generally, we prove recurrence of balanced excited random walks in spatially inhomogeneous cookie environments whenever the total positive and negative cookie strengths at each vertex are bounded by constants $A, B$ with $A+B<1+1/(2\pi +1)$.

We minimize $\mathbb E[h(Y-X)]$ over martingale couplings of $\text{Unif}[-1,1]$ and $\text{Unif}[-2,2]$ satisfying $Y\geq X-k$, where $h\in C^1([-3,3])$ has convex derivative. Feasibility holds exactly for $k\geq1$. For each such $k$, we construct a coupling that minimizes all costs in this class. For $1<k<3$, its support consists of two graphs, with $D(x)=x-k$ on $[k-2,1]$. The maps admit an explicit parametrization for $2\leq k<3$ and are determined by scalar equations with unique admissible roots for $1<k<2$. We prove optimality by a dual inequality, using an analytic estimate in the latter range. At $k=1$ the optimizer is $Y=X\pm1$ with equal probabilities; for $k\geq3$ it is the ordinary left-curtain coupling. In the unconstrained problem, left-monotonicity identifies the left-curtain coupling, which is optimal for this cost class. Under the constraint, a discrete example shows that the corresponding support condition, even together with every two-source comparison, does not imply optimality.

We consider a class of SPDEs driven by a standard real-valued Brownian motion, with drift and diffusion coefficients such that there exists a unique mild solution taking values in the interval $[-1,1]$ almost surely. To preserve this qualitative property of the exact solution, we propose a domain preserving Lie–Trotter splitting scheme: for any choice of the time-step size, the numerical solution takes values in the interval $[-1,1]$ almost surely. Furthermore, we prove mean-square convergence with rate $1/2-$ for the domain preserving Lie–Trotter scheme. These theoretical results are illustrated with numerical experiments.

Carathéodory's construction, also known as Method I, turns any weight on a family of sets into an outer measure. The difficulty is the extension problem: showing that the outer measure retains the prescribed weights. We record elementary properties of Method I, beginning with the fact that countable subadditivity on the covering family is exactly the criterion for faithful extension, and use them to prove standard results with only outer measures, $\sigma$-algebras, and measures, without premeasures on algebras. In topological spaces, a continuity principle reduces this criterion to finite subadditivity and finite approximation by open and compact sets. Applications include the Lebesgue integral as a measure, Tonelli's theorem, multidimensional Lebesgue–Stieltjes measures, Riesz representation for vector-valued functionals, Kolmogorov extension, mass distributions on nested partitions, and Frostman's lemma.

The speed v(lambda) of the lambda-biased random walk on a supercritical Galton-Watson tree without leaves is conjectured to be nonincreasing on [0,m), where m is the mean offspring. For non-constant offspring, monotonicity is known only for small bias: lambda <= 1/1160, lambda <= 1/2, and, when every vertex has at least m_1 >= 2 children, lambda <= m_1/(1+sqrt(1-1/m_1)). For offspring uniform on {2,3} (m=2.5) the last bound is 2/(1+sqrt(1/2)) = 1.17157... We prove, with computer assistance, that v is strictly decreasing on [0,1.755] for this law. The proof has three parts. Aidekon's speed formula gives v=(R-lambda)/(R+lambda) for an explicit functional R, so v decreases exactly when R/lambda does; we compare R/lambda at two biases directly, which avoids differentiating the conductance. A pathwise Lipschitz bound on the conductance in lambda turns that comparison into an inequality between expectations of explicit functions. A monotone sandwich of discretised laws gives two-sided bounds on the conductance law, and each lambda-cell is verified with exact rational arithmetic on top of bounded floating-point error; an independent interval-arithmetic implementation re-certifies five cells, including the one that sets the endpoint. The method stops where the crude Lipschitz bound becomes too weak; sharper control of the derivative of the conductance is what the full range needs. Code and certificates are public.

We identify the optimal constant and describe all maximizers for the Fourier endpoint extension inequality associated with the moment curve over finite fields. This confirms a conjecture proposed in the authors' earlier work. The proof proceeds by showing that the problem is equivalent to a purely probabilistic extremal statement: among all probability distributions on a finite set, the uniform distribution uniquely maximizes the expected number of distinct rearrangements of an i.i.d. sample.

For independent bond percolation on $\mathbb{Z}^d$ with parameter $p$, let $p_c^b(d)$ be the critical probability. We prove that for each $d \geq 9$, there is $\epsilon_d>0$ such that for each $p \in (p_c^b(d), p_c^b(d)+\epsilon_d)$, the complement of the infinite open cluster stochastically dominates a supercritical site percolation on $\mathbb{Z}^d$. This improves the previous results by Grimmett, Holroyd and Kozma 2014, and Bock, Damron, Newman and Sidoravicius 2020. Numerical estimates of $p_c^b(d)$ and $p_c^s(d)$ (the site critical probability) suggest that a similar stochastic domination result holds for all $d \geq 4$. For the Ising model on $\mathbb{Z}^d$ with inverse temperature $\beta$, let $\beta_c(d)$ be the critical inverse temperature. We prove that for each $d \geq 8$, there is $\epsilon_d>0$ such that for each $\beta \in [0,\beta_c(d)+\epsilon_d)$, the $+$ spins under the minus phase stochastically dominate a supercritical site percolation on $\mathbb{Z}^d$. This improves the previous result of Aizenman, Bricmont and Lebowitz 1987. Numerical estimates of $\beta_c(d)$ and $p_c^s(d)$ suggest that a similar stochastic domination result holds for all $d \geq 5$.

We study a two-parameter family of quadratic BSDE systems of the form given by Jackson (2023) in his open questions on non-Markovian solvability. Fan, Hu, and Tang (2025) established well-posedness for every bounded terminal datum when $1/\alpha+1/\beta=1$. For every $\alpha,\beta>0$ and $M>0$, we establish existence and uniqueness in $\mathcal{S}^{\infty}\times\mathrm{BMO}$ for a dense $G_\delta$ subset of the space of terminal data satisfying $|\xi_i|\le M$ a.s., $i=1,2$, equipped with the topology of convergence in probability. We also obtain well-posedness for every terminal datum taking at most two values. Furthermore, we show that every bounded terminal problem can be reduced to one with a three-valued terminal datum while preserving its solution set. However, for every positive parameter pair with $1/\alpha+1/\beta\ne1$, we construct a three-valued terminal datum admitting at least two distinct solutions in $\mathcal{S}^{\infty}\times\mathrm{BMO}$. In particular, uniqueness can fail in Jackson's stochastic-game example $\alpha=\beta=1$.

How much information can a single neuron remember? How many memories can neural networks retrieve without creating false memories? These questions are related to a basic question: how many Boolean threshold functions $f(x)=\operatorname{sgn}(a_0+\langle a,x\rangle)$, $x\in\{-1,1\}^n$, are there? In this paper, we show that the number $T_n$ of distinct Boolean threshold functions is \[ T_n=2\binom{2^n-1}{n}\bigl(1+O(n^{-99})\bigr). \] Equivalently, the capacity of a single threshold neuron is $n^2-\log_2(n!)+1+O(n^{-99})$ bits, improving the $O(n)$ error term in the result of Kahn–Komlós–Szemerédi to $O(n^{-99})$. To prove this, we show that, for $1\le r\le n-1$, and $v_1,\ldots,v_r$ are chosen at random from $\{-1,1\}^n$, \[ \mathbb P\!\left\{ \langle v_1,\ldots,v_r\rangle\cap\{-1,1\}^n =\{\pm v_1,\ldots,\pm v_r\} \right\} =1-O(n^{-99}). \] In the context of the Kanter–Sompolinsky Hamiltonian for memory retrieval, this identifies $r=n-1$ as a sharp threshold, at which, for almost every collection of $r$ memories, the only ground states are these memories and their negatives, confirming a weaker form of the Kalai–Linial–Odlyzko conjecture. It also settles a recent open problem posed by M. Anthony on the specification number of Boolean threshold functions. In addition, we show that, for every $1\le r\le n-1$, \[ \mathbb P\{v_1,\ldots,v_r\text{ are linearly dependent}\} =2\binom r2\,2^{-n}+O\!\left(2^{-n}e^{-cn}\right), \] confirming a conjecture of Kahn–Komlós–Szemerédi.

We prove the existence of solutions to the stochastic magnetohydrodynamics (MHD) system, where randomness is introduced through random initial data and a stochastic integral appearing solely in the induction equation, while the fluid equations remain deterministic. We focus on the case where the adiabatic exponent $\gamma$ satisfies $\gamma > \frac{3}{2}$. The existence proof is carried out using the penalization method. We define the notion of a martingale solution and establish sufficient conditions for its existence. The proof then proceeds by means of the stochastic compactness method. Using an energy inequality, we derive a priori estimates in terms of expectation. Due to the stochastic nature of the problem, we demonstrate convergence in law of the penalized solutions and verify that the limiting object is a martingale solution.

We study a generalized contact process on \(\mathbb{Z}^d\) parameterized by an infection rate \(\lambda\) and a resetting probability \(p \in [0,1]\), modeling deterministic cure times. Once a vertex is infected, its recovery is scheduled exactly one time unit later. Incoming attempts to an already-infected vertex successfully reset its recovery clock with probability \(p\), and are ignored otherwise. This unifies spatial versions of classical Type I (\(p=0\), non-paralyzable) and Type II (\(p=1\), paralyzable) counters. Except in the fully resetting case, deterministic recovery deadlines destroy coordinatewise attractiveness and create a causal shielding effect. Using a first-moment bound on potential causal chains, we prove that the process dies out from finite configurations whenever \(\lambda<1/(2d)\), uniformly in \(p\in[0,1]\). The same causal-chain bound yields local convergence to the empty configuration from arbitrary initial states. In dimensions \(d \ge 2\), an oriented percolation exploration based on the first infective window establishes global survival for all \(p\in[0,1]\) when \(\lambda>-\log(1-p_c^{\mathrm{or}})\). Finally, in the purely non-resetting case \(p=0\), we derive a delayed identity for the one-site density and prove that this density remains strictly between zero and one at every finite time.

We study the long-term behaviour of refracted Lévy processes within a half-line, with killing on exiting the domain and at state-dependent rate inside it. We identify the decay rate of the killed semigroup together with the associated invariant function and measure, and we obtain convergence at exponential rate to a Yaglom limit. The proofs make use of R-theory and Lyapunov function techniques.

We investigate a variant of the deep BSDE method introduced by E et al. (2017, 2018). The key novelty is that we establish an a-posteriori convergence result for the approximation of strongly coupled forward-backward stochastic differential equations (FBSDEs), i.e., our result holds without any assumptions on small time horizons, monotonicity or weak coupling that are typically imposed in the literature on the deep BSDE method. Instead, we rely on smoothness assumptions on the coefficients and cover FBSDEs in which the coupling of the BSDE into the SDE depends on both the backward component $Y$ and the control component $Z$. Numerical experiments illustrate the theoretical results and demonstrate the applicability of the proposed approach.

The Markov kernel based metric $D_1$ was introduced in 2011 in order to construct the scale-invariant dependence measure $\zeta_1$, which assign each bivariate copula $C$ a dependence value in $[0,1]$, with $0$ exclusively for the case of independence, and $1$ exclusively for complete/functional dependence. In the original paper it has been shown that the resulting metric space $(\mathcal{C},D_1)$ is separable and complete, however, no further topological properties were studied. Considering that $D_1$ has proved useful in a variety of contexts, using tools from infinite-dimensional topology, we here close this gap, show that $(\mathcal{C},D_1)$ is homeomorphic to the Hilbert space $(\ell_2,\Vert \cdot \Vert_2)$, and prove that several subfamilies are either homeomorphic to $(\ell_2,\Vert \cdot \Vert_2)$ or to the Hilbert cube $(\mathcal{H},\rho)$. Moreover, allowing for a better assessment of relative sizes, we show that various subfamilies are so-called $Z$-sets in $(\mathcal{C},D_1)$, implying that they are topologically negligible in the full space.

We study the Smoluchowski-Kramers approximation for a stochastic wave equation with state-dependent damping on a bounded domain, driven by both a $Q$-Wiener process and a Lévy process with finite second moment. As $\varepsilon\to0$, we prove that $u^\varepsilon$ converges in distribution, in the Meyer-Zheng topology, to the unique weak solution of an overdamped stochastic parabolic equation. The proof relies on a nonlinear transformation associated with the damping coefficient, uniform energy estimates, and compactness arguments in the pseudo-path topology. We identify the limiting equation, which contains both the classical Gaussian noise-induced drift caused by state-dependent damping and an explicit jump correction generated by the Lévy noise. We show that the latter coincides exactly with the Marcus-to-Itô correction associated with the canonical jump flow induced by the nonlinear damping transformation.

For $0<\kappa\leq4$, we construct a closed response calculus for $\mathrm{SLE}_\kappa$ weldings and a canonical Dirichlet form. The divergence covariance combines the Weil–Petersson and Velling–Kirillov forms. The integrated response gives exact changes of measure and canonical Liouville–capacity increments on conformally removable weldings.

We develop a novel pathwise approach to study the convergence of the law of killed diffusion processes conditioned on non-absorption, towards a quasi-stationary distribution (QSD) as time goes to infinity. We start from the general observation that the dynamics of an absorbed Markov process conditioned upon survival up to time $T>0$ is the minimizer of the pathwise relative entropy with respect to its unconditioned dynamics, under a simple distributional constraint at that time; in other words, a Föllmer process. We then show how this result applies to a Brownian diffusion process softly-killed at a state-dependent regular rate, and characterize the associated drift change. In the case when the diffusion process is moreover reversible, we leverage this idea and recent results on the propagation of weak log-concavity of HJB semigroups to prove that, under strict asymptotic convexity of the potential, the conditioned dynamics satisfy a contractivity property in $1$-Wasserstein distance, uniformly in $T>0$. Under a general ergodicity condition on the associated Feynman-Kac semigroup, we then establish the existence of a QSD with a large domain of attraction, and the exponentially fast convergence to it of the conditioned semigroup in the $1$-Wasserstein distance as $T$ goes to infinity. Finally, we deduce the exponentially fast convergence, also in $1$-Wasserstein distance, of the law of the corresponding Q-process towards its equilibrium.

In this article, we consider the stochastic nonlinear heat equation driven by Lévy space-time white noise in dimension one. For the spatial average of the solution, we prove quantitative and functional central limit theorems under $m_1+m_{2p}<\infty$ for some $p\in(1,\frac{3}{2})$. These results extend the Gaussian fluctuation theory for the parabolic Anderson model to the nonlinear setting. The main new feature is a minimum-type term in the second Malliavin derivative estimate caused by the nonlinear coefficient.

In this paper, we study the asymptotic behavior of the extremal process of a standard one-dimensional branching Brownian motion (BBM) with drift $-\rho>-\sqrt2$ and absorbing barrier at level $-x$. We prove that the two-dimensional point process, with first component being the extremal process of the BBM and the second component being the running minimum of the BBM with drift, converges weakly to a decorated Poisson point process (DPPP) on $\mathbb{R} \times [0, \infty)$. This framework allows us to supply a common limit coupling of the extremal processes for different barrier levels and to explicitly derive the limit, as $t\to\infty$, of the extremal process of the killed BBM killed at level $-x$, demonstrating that the double limit, when $t\to\infty$ first and then $x\to\infty$, of the extremal process of the BBM killed at level $-x$ coincides with the limit of the extremal process of (un-killed) BBM.

We study a multidimensional nonlinear filtering model whose coefficients depend on a given observation-adapted predictable process and whose observation drift may grow linearly in both the state and the random input. Due to the unboundedness of the observation drift, a global reference measure is not available. To overcome this hurdle, a localized entropy argument is adapted to prove the stopped likelihood to be a uniformly integrable martingale at each control-energy stopping level. The stopped Zakai equation, and hence, the stopped filtering equation is derived. The global filtering equation is then established by de-localization. The uniqueness of the solution to the stopped Zakai equation is obtained by a duality backward stochastic partial differential equation. This uniqueness then propagates to that of the global filtering equation through the stopped ones. Finally, a stability result is established in $W_1$-distance of measures.

We establish an averaging principle for a class of multiscale mean-field forward-backward stochastic differential equations and identify several novel phenomena that are absent from classical fast-slow systems. In contrast with classical fast-slow systems, the effective dynamics cannot in general be obtained by simply freezing deterministic slow parameters and averaging against the invariant measure of the resulting fast equation. The appropriate averaging object is instead provided by a frozen fast dynamics in a random environment and its associated conditional invariant measures, which retain the coupling between the slow state and its distribution. The forward-backward structure creates a further obstruction: local averaging estimates need not remain stable when propagated over an arbitrary time horizon. We identify a uniform restart stability condition for the averaged system under which this obstruction can be overcome. Using a joint lifted semigroup for the state-law dynamics, together with a two-scale discretization and a Gordin-type decomposition, we prove strong averaging for both the forward and backward components with optimal convergence rate $O(\varepsilon^{1/2})$. As an application, we apply the general theory to a class of mean-field stochastic control problems and develop an efficient algorithm for solving such mean-field control problems.

We establish moment and mixed-tail inequalities for fixed-order decoupled homogeneous chaoses generated by independent, centered, sparse bounded random variables. Our bounds apply to arbitrary real rectangular coefficient tensors and describe the fluctuation scales through weighted slice and partition norms, with a Bennett-type logarithmic improvement in the largest-entry term. As an application, we derive guarantees for sparse Khatri–Rao embeddings that explicitly account for sparsity and input geometry.