Papers

New venture start-ups need to “survive” through multiple stages of reaching milestone targets. We investigate the strategies for start-ups in a milestone-oriented setting. We examine a model of an entrepreneurial start-up firm, where its state is captured by a diffusion process. The entrepreneur can choose between multiple activities (or controls), which incur different cost and determine the drift and the variance of the process. Depending on whether the process reaches a fixed upper boundary or a lower one, the start-up firm succeeds or fails. Continuous-time stochastic models with multiple ($\ge 3$) controls are typically very challenging to deal with. In this work, we are able to completely solve for the optimal policy and provide an explicit characterization of its structure. In particular, the optimal policy only uses controls from a set characterized by a so-called efficient frontier curve that orders the controls by two intuitive measures: riskiness (drift-to-volatility ratio) and cost-effectiveness (drift-to-cost ratio). A unique feature of our model is that depending on the model parameters, the efficient frontier curves can be of different types, resulting in qualitatively different structures of the optimal policy. As far as we know, this is the first study that analyzes a stochastic control model which admits efficient frontier curves of different types. Our work provides start-up firms with intuitive measures to evaluate their activities and offers valuable insights on how the optimal strategies in a milestone-oriented setting change qualitatively contingent upon the specific scenario. We believe the results provide a foundational block in the study of entrepreneurial decision-making.

We prove a universality theorem for random persistent homology over a class of triangulable spaces. More precisely, let $M \subset \mathbb{R}^D$ be a compact $C^2$-triangulable space satisfying a geometric quality condition and let $f: M \to \mathbb{R}$ be a probability density. Then the expected persistence ratio measure computed from the Čech or Vietoris-Rips complex of a Poisson point process with intensity $nf$ has a universal limit independent of $(M, f)$. Since smooth manifolds, algebraic varieties, semialgebraic sets and Whitney stratified spaces are all triangulable spaces, our theorem applies to a large class of non-Euclidean spaces. Beyond persistent homology, our proof covers a general class of scale-invariant functionals. It relies on a geometric transfer method that adapts constructions in Euclidean space to triangulable spaces through successive approximations by Freudenthal-Kuhn triangulations, and control of interference across singular strata.

In this article, we propose two other DLRA-type dynamics for stochastic differential equations (SDEs) than the one studied in arXiv:2308.11581, derived from a minimization of functionals and (informally) from a Stratonovich formulation, respectively. The former approach resembles the DLRA for SDE system proposed in arXiv:1803.00499. Providing the differentiability of the diffusion, the latter procedure registers an additional term in the drift. Indeed, its derivation exploits the Stratonovich formulation to write stochastic processes on manifold, and, hence, possesses a term that depends on the geometry of the manifold itself. These developments open the debate on which formalism is more suitable and what DLRA for SDEs really is.

We study first-order sensitivity of a scalar backward stochastic differential equation with a deterministic rough driver. The driver belongs to the nonlinear space of step-two weakly geometric $p$-rough paths, $2<p<3$, so an ordinary Banach-space difference quotient is not available. At a fixed rough path $x$, we use a weakly geometric, finite-$(p,p/2)$-variation tensor realization of the Qian-Tudor tangent structure, represented intrinsically by a first-level direction $h$ and a compatible second-level direction $\kappa$. Admissible rough-path secants are required to converge in a strong levelwise variation topology. Under bounded smooth rough vector fields, a bounded terminal condition, and a globally Lipschitz generator, we construct a continuous linear map $A_x:T_x^p\to S^\infty\times H^2_{\mathrm{BMO}}$. The proof first establishes a uniform four-jet expansion for reset rough flows along bounded realizations of full rough tangents. A Doss-Sussmann transformation transfers this expansion to quadratic generators. Uniform BMO and reverse-Holder estimates then yield a local difference-quotient theorem, which is propagated over a fixed deterministic partition and reconstructed in the original coordinates. The resulting derivative is independent of the joint lift, central decomposition, and radial realization. Consequently, solution difference quotients converge to $A_x(h,\kappa)$ for varying directions and arbitrary admissible secants with strong levelwise variation contact. This is intrinsic tangential Hadamard differentiability on that tensor-coordinate tangent class. Locally bounded ray-homogeneous selections give Frechet-differentiable chart pullbacks at the parameter origin. The latter statement is chartwise; it is not Frechet differentiability of the solution map in the homogeneous rough-path metric.

Consider read-once polynomial functions over a finite field of order $k$, i.e., functions defined by expressions built from field addition, multiplication, and constants, in which every variable occurs at most once. Let $p=(p_1,\dots,p_k)$ be the distribution of the values of a read-once function on independent uniform inputs. We prove that for every $k\ge4$ all such distributions belong to a body $\mathcal{B}_k$, defined by the following relation on the sorted atoms $p_1^\downarrow\geq\cdots\geq p_k^\downarrow$ of the distribution: \[ \mathcal{B}_k=\left\{p:p_k^\downarrow\ge \frac{1-p_2^\downarrow-(1-p_2^\downarrow)^k}{k-1}\right\}, \] or equivalently, $\mathcal{B}_k = \{ p \colon 1-p_2^\downarrow-(k-1)p_k^\downarrow\le(1-p_2^\downarrow)^k\}$. Our main theorem is that this body is stable under the convolutions corresponding to both field operations. More generally, convolution for any quasigroup operation on $k$ points preserves $\mathcal{B}_k$; the multiplicative conclusion needs only an absorbing zero and a quasigroup operation on the nonzero elements. The body is full-dimensional and contains the uniform law and every point mass; its normalized volume is given by an exact one-dimensional integral, and the $k$th root of that volume tends to $0.2183305369\ldots$. The complete development, including the two stability theorems, the exact volume formula, and its sharp exponential rate, has been formalized in Lean 4.

In this paper, we study the sample complexity of the empirical plug-in estimator for the $2$-Gromov-Wasserstein distance $D_2$ between compactly supported probability measures on Euclidean spaces. Let $\mu$ and $\nu$ be supported on compact subsets of $\mathbb{R}^{d_x}$ and $\mathbb{R}^{d_y}$, respectively, and let $\widehat\mu_n$ and $\widehat\nu_n$ be their empirical measures based on independent samples of size $n$. We prove that \[ \mathbb{E}\left|D_2^2(\widehat\mu_n,\widehat\nu_n)-D_2^2(\mu,\nu)\right| \lesssim n^{-2/((d_x\wedge d_y)\vee 4)} (\log n)^{\mathbf 1_{\{d_x\wedge d_y=4\}}}. \] This rate is sharp up to the logarithmic factor in the critical dimension. The proof is based on a geometric representation of the Euclidean distance as a squared $L^2$-distance between half-space feature maps. This yields a variational dual formulation of the Gromov-Wasserstein functional in terms of a family of classical optimal transport problems indexed by an infinite-dimensional auxiliary parameter. Although the resulting cost functions need not be semiconcave in either argument, we introduce a marginal recentering of the costs that restores the concavity structure needed for sharp metric-entropy bounds. Combining this representation with empirical-process estimates gives a rate governed by the smaller of the two ambient dimensions.

We study stochastic, parabolic-parabolic Keller–Segel equations on the $d$-dimensional torus in scaling critical Besov spaces, for $d \geq 3$. Using stochastic maximal regularity estimates, we prove local well-posedness of the equation, i.e., that there exists a unique solution up to a maximal time of existence. We show that the time of existence can be made arbitrarily large with arbitrarily high probability, provided that the initial data is sufficiently small in these critical spaces. Contribution to the Oberwolfach Report for the Seminar 'Stochastic Partial Differential Equations in Critical Spaces' organized by Antonio Agresti and Mark Veraar.

Robust Optimization Over Time (ROOT) is a recent branch of evolutionary dynamic optimization that seeks solutions capable of remaining effective across multiple consecutive environments. Unlike the traditional track-the-moving-optimum (TMO) paradigm, which reoptimizes after every environmental change, ROOT explicitly values persistence. Although the field has grown considerably, most contributions remain algorithmic and empirical, leaving several fundamental properties poorly understood from a theoretical perspective. One such property is survival time, defined as the number of future environments in which a deployed solution continues to satisfy a prescribed quality threshold. While survival time is widely used as a measure of temporal robustness, little is known about how its expected value depends on environmental dynamics, deployment quality, or problem characteristics. This paper studies expected survival time for a fixed deployed solution under isotropic Gaussian environmental dynamics. Modeling survival as a discrete first-exit problem, we derive a rigorous lower bound and a computable multi-step upper bound. The analysis shows that expected survival scales as ${\Theta}({\sigma}^-{2})$ in slowly varying environments and approaches its minimum value of one future change in high dimensions. A comprehensive Monte Carlo study validates the theoretical predictions, examines sensitivity to modeling assumptions and parameter uncertainty, and illustrates how the bounds can support deployment decisions after optimization. The resulting framework provides an analytical characterization of deployment lifetime and identifies when a required deployment horizon can be guaranteed, ruled out, or remains analytically unresolved.

We study the Sachdev-Ye-Kitaev (SYK) Hamiltonian $H_{\operatorname{SYK}}$ on $n$ Majorana modes with $k$-body interactions, and prove that $\mathbb{E}\|H_{\operatorname{SYK}}\|_{\operatorname{op}} = (1 - o(1))\cdot\sqrt{2n}/k$ for super-constant $k\leq o(\sqrt{n})$, where the expectation is over the disorder variables in the Hamiltonian. This confirms predictions due to Garcia-Garcia, Jia and Verbaarschot and answers a question posed by Feng, Tian, and Wei. Our results extend to the sparse SYK Hamiltonian. As a corollary, we obtain that a dissipative quantum algorithm due to Basso, Chen, and Dalzell provably computes the ground state energy of the SYK Hamiltonian up to an $O(1)$-multiplicative factor for all $k < \sqrt{n}/4$. Our key technical idea is identifying an explicit, deterministic linear operator $\mathsf{x}$ such that a fixed quadratic form of $\mathsf{x}^{2\ell}$ exactly equals the expected trace moments of the SYK Hamiltonian for every $n$ and $k$. This linear operator can be naturally viewed as a twisted model of bosons on the space of hyperedges of a hypergraph. The problem thus reduces to identifying the spectral edge of $\mathsf{x}$, which we show is dominated by the spectrum of a natural ${n \choose k}$-dimensional matrix from the Johnson scheme and is straightforward to compute using known results. To show that our bound is sharp, we construct a witness state with a large quadratic form on $\mathsf{x}$ and transform it into a certificate of a lower bound on the largest quadratic form on $H_{\operatorname{SYK}}$. Beyond the SYK Hamiltonian, our techniques also apply to the $k-$local quantum spin glass (QSG), yielding $\mathbb{E}\|H_{k-\operatorname{QSG}}\|_{\operatorname{op}}\leq O(\sqrt{n/k})$ for all $1\leq k\leq\sqrt{n}/2$, thus improving the standard $O(\sqrt{n})$ bound by a factor of $\sqrt{k}$.

Let $G=(V,E)$ be a possibly infinite, locally finite graph with non-negative Ollivier–Ricci curvature and degrees bounded by $d<\infty$. We prove that there exists a constant $C_d$ such that the continuous-time random walk displacement and log-volume growth satisfy \[ \mathbb{E}_x \mathrm{dist}(x,X_t)^2 \le t \exp\left[C_d \sqrt{\log t \log\log t}\right], \] \[ \log \mathrm{Vol}(B(x,r)) \le \exp\left[C_d \sqrt{\log r \log\log r}\right], \] for every $x\in V$ and all $r,t \ge e^e$.

These notes provide a self-consistent summary of the stochastic approach to anomalous diffusion based on Continuous Time Random Walks (CTRWs) and their scaling limits. An introduction to CTRWs and their relationship to the theory of semi-Markov processes is provided. A general technique to study scaling limits of CTRWs is then described, and the semi-Markov property of the limiting processes is discussed. With this at hand, the connection of limit processes with non-local (fractional-type) equations is introduced, and the most recent (and general) contributions, going far beyond fractional equations, are also described. Indeed, the theory presented here includes very general non-local evolution equations as abstract Cauchy problems as well as pointwise non-local fractional diffusion equations in bounded domains.

We study an interval-activation frog model on \(\mathbb Z\) with i.i.d.\ initial numbers of frogs \((\eta_x)_{x\in\mathbb Z}\), satisfying \(0<\mathbb{E}[\eta_0]<\infty\). Frogs at the origin are initially active and all others are sleeping. Each frog performs a symmetric integer-valued random walk and has a random lifetime \(L\) determined by an i.i.d.\ survival parameter \(\pi\in(0,1)\), with \(\mathbb{P}(L\ge k\mid \pi=p)=p^k\). Every jump activates all sleeping frogs at the integer sites between its endpoints. Let \(D^\to\) denote the maximal rightward displacement of a single frog before death. We derive survival and extinction criteria from the tail behavior of \(D^\to\). If \(\mathbb{P}(|\xi_1|\ge n)\sim n^{-\alpha}L_\xi(n)\), with \(L_\xi\) slowly varying, then survival holds with positive probability for \(0<\alpha<1\), while for \(\alpha=1\) both survival and almost sure extinction may occur. For \(1<\alpha<2\), assume \(\mathbb{P}(|\xi_1|>n)\sim c_\xi n^{-\alpha}\); in the finite-variance case assume \(\mathbb{E}[\xi_1]=0\) and \(\operatorname{Var}(\xi_1)=\sigma^2\in(0,\infty)\). Setting \(r=\alpha\) in the stable case and \(r=2\) in the finite-variance case, if the law of \(\pi\) has density \(f_\pi(u)\sim(1-u)^{\beta-1}\ell((1-u)^{-1})\) as \(u\uparrow1\), then, for \(0<\beta<1\), \(n\mathbb{P}(D^\to\ge n)\sim C_\beta n^{1-r\beta}\ell(n^r)\), with explicit \(C_\beta\). Hence the sharp off-critical threshold is \(\beta_c=1/r\): survival holds for \(\beta<1/r\), extinction holds almost surely for \(\beta>1/r\), and explicit sufficient conditions on the critical line leave a factor-four gap.

Our previous work established the multi-set allocation occupancy (MAO) distribution theory, including the norms, inequalities, and limit theory, under constraints on set sizes alone. By further conditioning on all pairwise intersections while leaving triple and higher-order membership patterns random, we develop the second-order extension. This extension naturally identifies the earlier marginal-size-only framework as the first-order MAO theory. Let $N$ be the population size, $T$ the number of labelled sets, and $M$ a feasible symmetric matrix whose diagonal and off-diagonal entries specify marginal sizes and pairwise intersections. Write $a=(a_B)_{B\subseteq[T]}$ for the atomic membership counts, where $a_B$ is the number of elements belonging to exactly the sets indexed by $B$, and let $A(N,M)$ be the resulting feasible atomic region. With labelled multiplicity $w(a)=N!/\left(\prod_{B\subseteq[T]}a_B!\right)$, $Z_{N,M}=\sum_{a\in A(N,M)}w(a)$, we define the second-order exact-$t$ and at-least-$t$ MAO norms by the direct atom formulas $\lVert t^r\rVert_T=\left(\sum_{a\in A(N,M)}\left(\sum_{|B|=t}a_B\right)_r w(a)\right)/\left((N)_r Z_{N,M}\right)$, $\lVert [t,T]^r\rVert_T=\left(\sum_{a\in A(N,M)}\left(\sum_{|B|\geq t}a_B\right)_r w(a)\right)/\left((N)_r Z_{N,M}\right)$. The moments can be calculated exactly based on the norms: If $X_{=t}$ and $X_{\geq t}$ denote the numbers of elements with membership degree exactly $t$ and at least $t$, respectively, then for every $\nu\geq1$, $\mathbb{E}_{N,M}[X_{=t}^{\nu}]=\sum_{i=1}^{\nu}S(\nu,i)\lVert t^i\rVert_T$, $\mathbb{E}_{N,M}[X_{\geq t}^{\nu}]=\sum_{i=1}^{\nu}S(\nu,i)\lVert [t,T]^i\rVert_T$. We further developed Poisson and normal limiting theory, verified by numerical approximation. The model provides a systematic route to higher-order MAO theories by prescribing intersections up to any chosen order.

Optimization in non-convex neural network models is strongly influenced by the geometry of the solution space: sparse, isolated, point-like clusters are typically algorithmically inaccessible, whereas wide and flat regions can be found efficiently despite being relatively rare. At zero temperature this picture has been formalized in binary perceptrons through the overlap gap property (OGP), which limits algorithmic access to configurations with zero training error above a critical constraint density $\alpha_{\rm OGP}$. Here we extend this description to finite temperature, where a positive training error is allowed and statistically penalized. We first show that the frozen one-step replica-symmetry-breaking solution, dominating the zero temperature equilibrium measure, survives at any finite temperature. We furthermore derive a general criterion, based on the smoothness of the single-pattern Gibbs weight near the decision boundary, that determines when a finite-temperature relaxation of the loss removes freezing. We then extend the OGP construction to finite temperature and show that dense, algorithmically accessible regions of finite-energy configurations persist beyond $\alpha_{\rm OGP}$, up to a threshold $\alpha_{\rm OGP}(\epsilon)$ that grows with the allowed training error $\epsilon$. Finally, in the teacher-student setting, we show that these wide, finite-energy regions still retain good generalization. Using a finite energy message-passing algorithm, we demonstrate numerically that thermal noise enables effective generalization in the regime of constraint densities where both recovering the teacher and finding a zero temperature solution are computationally hard.

On a high-dimensional Poincaré ball, a Euclidean beta-type radial measure concentrates near a sphere, but hyperbolic distance amplifies the surviving radial spread, so the usual shell reduction loses part of the limiting metric. In each regime of the balance between this radial width and amplified angular separation, we determine the weak pyramid limit of the spaces rescaled at the order at which the transition between concentration and dissipation occurs. The four possibilities are a pyramid generated by finite star trees, the pyramid of spaces of diameter at most one, metric transforms of the Gaussian pyramid, and the Gaussian pyramid. Each star tree has branches from a common center, a shifted exponential distribution along them, and paths between different branches through the center. We also give a sharp criterion for convergence to the diameter-at-most-one pyramid.

We study the inradius and inverse inradius of the convex hull of standard planar Brownian motion. Our main goals are to improve the existing bounds on the expected values of these quantities and to establish variance bounds. The main contributions of the paper, however, are the following intermediate results we obtain in order to compute our bounds. We find closed-form expressions for the expected values of the minimum and maximum of the ranges of two independent one-dimensional Brownian motions. Besides improving the upper bound on the expected inradius, this result also gives an explicit expression for a quantity that has been investigated repeatedly in the literature and for which only a numerical evaluation has been known so far. Furthermore, we study the exit time of planar Brownian motion from an unbounded region that we call the hourglass domain and compute its first two moments. This result leads to a nearly fivefold improvement of the previous upper bound on the expected inverse inradius and plays an essential role in establishing the corresponding upper bound on the variance.

Many real-life systems can be found as examples of stochastic matching on hypergraphs, such as production lines or assemble-to-order systems. Two common features are the number of items required may vary between matchings, and there may intermediary items which exist as a combination of other items and not of external arrivals. Both of these phenomena can be modelled by considering the weighted variant of stochastic matching. In this work, we formalise the notion of stochastic weighted matching on hypergraphs. We also allow batch arrivals, meaning multiple items of multiple classes may arrive at the same time, and in particular, the arrivals can be correlated between classes. Unlike the classical setting where items arrive at discrete time $t \in \mathbb{N}$, we allow arrival processes to take place in continuous time $t \in \mathbb{R}_{\geq 0}$. We then extend the results from Nguyen and Bušić (2026) to overcome the intricacies brought up by this new setting. This allows us to derive necessary and sufficient criteria as direct generalisations of those in the unweighted setting, which depend only on the per-class arrival rates. As such, the correlation between classes bear no differences. The constructive proofs also give a maximally stable, periodic-review, size-based, arrival-rate agnostic policy.

We introduce a class of weighted Dirichlet spaces on the unit disk, called Douglas weighted Dirichlet spaces, characterized by the existence of a Douglas type boundary representation of the weighted Dirichlet integral. This representation naturally induces a nonlocal Dirichlet form on the unit circle in the sense of Beurling Deny and Fukushima. Our main result shows that the reproducing kernel of the weighted Dirichlet space associated with a regular Douglas weighted Dirichlet form is the image of the Szego kernel under the corresponding 1 resolvent. This resolvent representation is new for superharmonic weights. We also develop the potential theory associated with Douglas weighted Dirichlet spaces. We characterize the capacity in terms of reproducing kernels. Finally, as an application of the fact that every Douglas weight induces a regular nonlocal Dirichlet form, together with the general theory of Dirichlet forms, we obtain an associated symmetric pure jump Hunt process. In the classical Dirichlet case, this process is the wrapped Cauchy process.

We study a class of mean-field control (MFC) problems with singular controls over a finite horizon, allowing for general dependence of the cost functional on the measure argument. We derive an auxiliary mean-field game (MFG) with singular controls, which we refer to as a potential MFG, and show that, under suitable convexity assumptions, any solution to this potential MFG yields a solution to the original MFC problem. We apply this general result to a version of the classical Monotone Follower Problem by I. Karatzas and S. E. Shreve (SIAM Journal on Control and Optimization 22(6), pp. 856-877, 1984) with scalar mean-field interaction. The associated potential MFG with singular controls is solved by exploiting its connection with optimal stopping for the optimization step and by a suitable application of the Kakutani-Fan-Glicksberg fixed-point theorem. In the case of strategic complementarities, the mean-field equilibrium (and hence the optimal policy of the original MFC problem) is characterized by a continuous nonincreasing free boundary that uniquely solves a nonlinear integral equation. To the best of our knowledge, this is the first paper to provide a complete characterization of the optimal policy in a finite-horizon mean-field singular stochastic control problem.

This paper presents a strong error analysis of both semidiscrete and fully discrete approximations for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise and subject to nonsmooth initial data. The spatial discretization is based on a standard finite element method, coupled with the linearly implicit Euler scheme in time. Under low-regularity initial conditions, we derive sharp spatial and temporal regularity estimates that isolate the loss of initial regularity into an integrable temporal singularity, allowing us to establish optimal strong error estimates for positive times. Specifically, we prove strong convergence rates of order $O(h^\beta)$ for the spatially semidiscrete approximation and $O(h^\beta + k^{\beta/2})$ for the fully discrete scheme away from $t = 0$, where the parameter $\beta \in (0, 2]$ characterizes the spatial regularity of the noise process. Numerical experiments confirm the theoretical convergence rates.

We extend the Brenier-Strassen theorem about projections in convex order to non-flat spaces with curvature bounded from above. Precisely, for probability measures $\mu$, $\nu$ of finite second moment on a complete separable CAT(0) space, we prove that $\mu$ admits a unique W 2 -projection \bar{\mu} to the set of probability measures dominated by $\nu$ in convex order. Moreover, the unique optimal coupling from $\mu$ to \bar{\mu} is induced by a 1-Lipschitz map, without any absolute-continuity assumption on $\mu$. Our proof identifies the projection problem with a weak optimal transport problem whose cost is the squared distance to the set of convex means. We also establish a localized version on CAT(kappa) spaces with kappa \geq 0, where the optimal map is 1/2-Hölder continuous. Finally, we give a Strassen-type characterization of one-step barycentric martingales on proper CAT(0) spaces.

A pyramid is a generalisation of metric measure spaces (mm-spaces) introduced by M.~Gromov (Birkhäuser 1999) to establish a geometric framework for measure-concentration problems. An extended metric measure space (emm-space) is another generalisation of mm-spaces introduced by Ambrosio–Gigli–Savaré (Invent.~Math.~2014) to extend Sobolev calculus and optimal transport theory to a broader extent. We prove that every pyramid has a representation by an emm-space through $1$-Lipschitz order. Furthermore, if pyramids are concentrated, this correspondence is unique up to isomorphism. This shows, for the first time, that all pyramids can be realised by concrete geometric spaces. Based on this representation, we introduce a new notion, a concentrated emm-space. We then define the observable distance for concentrated emm-spaces and establish three equivalent characterisations of concentration in terms of pyramids, Lipschitz observables and the observable distance. Furthermore, by developing an fibration approach, we show the $\Gamma$-$\limsup$ inequality of Cheeger energies as well as the stability of the log-Sobolev inequality and the Poincaré inequality under the weak convergence of pyramids associated with emm-spaces. Our results provide a new geometric approach for studying both the convergence of emm-spaces and concentration-of-measure phenomena across a broad class of infinite-dimensional models that have so far remained largely beyond the reach of existing geometric methods. Many of the significant examples arise in probability theory, including the Wiener space, the configuration space, Gaussian fields such as massive Gaussian free fields, spatial white noise and massive bi-Laplacian fields.

We establish the well-posedness of stochastic scalar conservation laws on moving hypersurfaces driven by Brownian motion. To handle the interaction between stochastic forcing and evolving geometry, we derive an Itô formula on moving surfaces and introduce the notion of generalized entropy solutions incorporating the relevant stochastic interaction terms. A martingale entropy solution is constructed via the vanishing-viscosity method, based on a uniform $L^\infty$-bound in space and time, an $L^1$-estimate for the spatial gradient, an $L^1$-continuity estimate in time, and a suitable tightness argument. Pathwise uniqueness is established by adapting Kruzhkov's doubling-of-variables method to moving hypersurfaces, yielding an $L^1$-contraction property. Finally, together with the Yamada-Watanabe theorem, these results yield the well-posedness of the problem.

Remarkable breakthroughs [25,43] established the so-called strong asymptotic freeness between classical Gaussian random ensembles and their semicircular free counterparts. Along the same lines, the Lehner formula [31], associated with the free counterpart, precisely determines the spectral edges of Kronecker-Gaussian matrices. We here revisit and study this formula without the utilization of random matrix theory and spectral methods. In particular, relying on concepts utilized within Random Duality Theory (RDT) [45,46,51], we reconfirm Lehner's formula and effectively reprove the key asymptotic freeness results obtained via spectral methods in [25,43].

Let $D$ be a bounded analytic Jordan domain and let $\gamma$ be chordal $\mathrm{SLE}_2$ in $D$, equipped with its $5/4$-dimensional natural-content measure $\mu_\gamma$. We retain the root in the standard integrated Brownian-bridge representation of Brownian loop measure and let $\mathcal M_\varepsilon^\gamma$ be the root-intensity measure of loops with duration in $[\varepsilon^2,t_0]$ whose traces hit $\gamma$. For every $f\in C_c(D)$, we prove that $\varepsilon^{5/4}\mathcal M_\varepsilon^\gamma(f)$ converges in $L^1$ to $\frac{4}{5\pi}\bar v_{\mathrm{BB}}\mu_\gamma(f)$, where $\bar v_{\mathrm{BB}}\in(0,\infty)$ is the mean specific area swept out by the Brownian-bridge offset of a natural-time two-sided whole-plane $\mathrm{SLE}_2$. In particular, the positive random measures converge vaguely in probability. The proof uses a duration-octave identity, a deterministic finite-$R$ reference coefficient obtained from a stopped two-arm Markov skeleton, a marked physical Palm tangent, an annular remote-return estimate, and a legal mesoscopic diagonal. As an application, the uniformly time-marked roots of an independent Brownian loop soup satisfy the corresponding vague law of large numbers. The analytic-boundary hypothesis enters only through a global finite-domain uniform-integrability estimate; its analogue for an arbitrary bounded Jordan domain remains open.

Let $\mu$ be a probability measure on the unit circle with Verblunsky coefficients $\alpha$. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of $\alpha$ into components localized at those points. We give a counterexample for two critical points of multiplicity three, found by GPT-5.6. The construction uses two phase modes with common power-decay exponent $3/20$. The sequence admits the Lukic's decomposition conditions, but its weighted entropy for the corresponding two-point weight equals $-\infty$.

We classify signatures of bounded-variation paths in finite-dimensional real normed spaces whose logarithms are entire, in the sense of superexponential homogeneous decay. The zero-first-level fibre is trivial; if the first level is $v\neq0$, the signature is $a\e^v a^{-1}$ with $a$ a bounded-variation signature, and every such conjugate is entire. For a given path, $a$ may be chosen from a prefix. For a tree-reduced representative, a gate-selected prefix gives weak path conjugacy to a line. The proof uses a finite-dimensional spectral-growth statement: if $q\geq1$, $A:\C\to M_q(\C)$ is entire, and $\norm{\e^{A(z)}}\leq C\e^{\tau\abs{z}}$, then the $k$th characteristic-polynomial coefficient of $A(z)$ has degree at most $k$ for $1\leq k\leq q$. Universal matrix isospectrality, resonant developments, metric-tree fixed points and compactness, and Stieltjes–Fourier reconstruction establish the bounded-variation modified Lyons–Sidorova conjecture under a hypothesis imposed only on the whole path.

In this paper we study autoencoders, a special class of deep neural nets (DNNs) whose performance can be characterized via their fixed points. This perspective naturally raises questions of existence, stability, and basins of attraction of these fixed points. These questions are addressed via the contractive properties of autoencoders, and are closely related to the notion of edge-of-chaos. Edge-of-chaos (EoC) is an important notion in the theory of DNNs. It describes the critical regime separating ordered and chaotic signal propagation through a randomly initialized network. Initialization at or near this critical regime offers several theoretical and practical advantages, including stability of the network w.r.t. perturbations of the input. EoC was previously introduced for broad classes of neural networks using mean-field averaging methods. In this paper we modify the notion of EoC for the study of autoencoders. Specifically, we introduce local and global EoC for autoencoders that control local (small) and global (arbitrary) perturbations of the input respectively. The study of stability of autoencoders falls within the scope of nonlinear problems in Random Matrix Theory (RMT). Our analysis of local EoC is based on spectral techniques of RMT, whereas global EoC is studied by employing Sudakov-Fernique inequality for Gaussian processes.

We develop a refined asymptotic theory for renewal processes with regularly varying inter-arrival distributions in the finite-variance regime. Assuming that $1-F(t)=t^{-\alpha}L(t)$ with $\alpha\in(2,3)$ and $L$ slowly varying, we establish second-order asymptotic expansions for both the renewal convolution and the renewal function. The results reveal a two-scale structure: the equilibrium tail $Q_F(t)$ governs the local renewal correction, while the integrated tail $\rho(t)=\int_t^\infty Q_F(u)\,du$ determines the global deviation from equilibrium. A particular emphasis is placed on the critical threshold $\alpha=2$, where the asymptotic behaviour undergoes a structural transition. In this borderline case, the power-law hierarchy collapses and the dominant correction is governed by an integrated slowly varying tail. Nevertheless, the local renewal correction mechanism persists, and the correction term remains asymptotically proportional to the equilibrium tail. The analysis is based on a systematic use of Laplace-transform techniques and Tauberian transfer principles, which allow us to relate singular behaviour in the Laplace domain to precise time-domain asymptotics. The results provide a unified description of second-order renewal asymptotics across the entire range $2\leq\alpha<3$, and point to possible extensions to renewal equations and to age-dependent branching processes.

Let $q$ be an odd prime power, and let $A_n=(a_{ij})\in\mathbb F_q^{n\times n}$ be a random matrix whose entries are independent and uniformly distributed on $\mathbb F_q$. The permanent of $A_n$ is defined by $\operatorname{per}(A_n)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}$, where $S_n$ denotes the symmetric group on $[n]$. Ghasemi, Gross, and Kopparty conjectured the zero-mass asymptotic $\Pr[\operatorname{per}(A_n)=0]=1/q+o(1)$ for every fixed odd prime power $q$, and Hunter, Kwan, and Sauermann subsequently stated its equivalent full-distribution formulation: for every fixed $q$ and every $x\in\mathbb F_q$, \[ \lim_{n\to\infty}\Pr[\operatorname{per}(A_n)=x]=\frac1q. \] In this paper, we prove this conjecture. More precisely, we prove that there is an absolute constant $C>0$ such that \[\frac12\sum_{x\in\mathbb F_q}\left|\Pr[\operatorname{per}(A_n)=x]-\frac1q\right|\le C\frac{\log n}{n}\] for every odd prime power $q$ and every $n\ge 7$. The estimate is uniform in $q$, so the conclusion remains valid for every sequence $q=q(n)$ of odd prime powers.