Papers

We consider the three-dimensional stochastic wave equation (SWE) driven by a multiplicative Gaussian noise that is white in time and colored in space: \[ \frac{\partial^2 u}{\partial t^2} = \Delta u + b\bigl(u\bigr) + \sigma\bigl(u\bigr)\,\dot{W}, \] where the drift function $ b $ and diffusion coefficient $\sigma$ are assumed to be locally Lipschitz and exhibit logarithmic superlinear growth at infinity. We establish the existence and uniqueness of a global mild solution on any fixed time interval $[0,T]$ under suitable assumptions on the spatial covariance function $ f $ of the noise $\dot W(t,x)$. Our results apply, for example, to the case \[ b(u) = u (\log_+ u)^{\theta_1} \quad \text{and} \quad \sigma(u) = u (\log_+ u)^{\theta_2}, \] with parameters $\theta_1 \in (0,2)$ and $\theta_2 \in \bigl(0, \tfrac{\bar{\nu}+1}{2}\bigr)$, and $\log_+(z)=\log(z\vee e)$, where $\bar{\nu}$ is determined by the assumptions on $ f $.

In 1999, two papers of Benjamini, Lyons, Peres, and Schramm showed that for Bernoulli percolation on unimodular nonamenable quasi-transitive graphs, there are no infinite clusters at criticality. Using the theory of countable Borel equivalence relations and Gaboriau's cluster graphing construction, we give a short proof of this result. We also point out other uses of the cluster graphing construction in the literature, for instance in showing nonuniqueness at $p_u$ in arXiv:1509.00247 [math.GR]. The purpose of this short note is to make folklore proofs known to experts more accessible to the wider community of probabilists and measured group theorists.

Lewis Carroll's Pillow Problem asks for the probability that three points chosen at random in the plane form an obtuse triangle. The question has no answer until the sampling scheme is pinned down, and the cleanest way to pin it down is to put the points on a circle, which gives 3/4. This paper solves the version in which the three vertices lie on three concentric circles of radii $r_1$, $r_2$, and $r_3$. The answer is a sum of three terms, each the probability that a weighted sum of two independent arcsine variables exceeds a threshold, and it collapses to 3/4 when the radii are equal. Reading the formula gives closed forms in two subfamilies, one an arcsine and one the Legendre chi function; a Pythagorean condition deciding which vertex can carry the obtuse angle; and the bound $1/2 \leq P < 1$, with the minimum attained only when one vertex sits at the common center and the other two radii are equal. Conditioning on the radii shows that the same formula is the kernel for every independent rotationally symmetric sampling scheme, so the bound applies to all of them at once and averaging over Rayleigh radii recovers the known Gaussian value of 3/4 in closed form, supplying a link between the circular and Gaussian conventions that the literature records as missing. Simulation confirms the results throughout.

We prove universality of limiting local eigenvalue statistics for random matrices over $\mathbb{Z}_p$. In previous work of the author and Van Peski (arXiv:2601.06283), the limiting eigenvalue correlation functions of additive Haar random matrices were studied in arbitrary finite extensions of $\mathbb{Q}_p$. The same Haar random matrix model plays a central role in the Ellenberg-Jain-Venkatesh heuristic for zeros of $p$-adic $L$-functions. We show that its limiting local eigenvalue statistics are unchanged for a broad class of random matrices with independent entries satisfying a mild non-concentration condition. Thus the random matrix predictions underlying the Ellenberg-Jain-Venkatesh heuristic are not artifacts of the particular Haar ensemble, but instead reflect universal limiting eigenvalue statistics. In this sense, our results provide additional theoretical support for the robustness of their random matrix heuristic. Our proof is based on a new framework, which we call the resultant distribution method. The method recovers limiting laws and root statistics of $p$-adic polynomials from the distributions of their resultant valuations against fixed test polynomials, together with suitable degree estimates. As a second application, we consider random $p$-adic polynomials with independent coefficients satisfying a mild non-concentration condition. Caruso (arXiv:2110.03942) determined the joint root correlation functions of the Haar coefficient model over finite extensions of $\mathbb{Q}_p$. We prove that, for roots of absolute value one, these limiting correlation functions are universal and persist for a broad class of independent coefficient distributions.

We consider the ruin problem for an insurance company investing its whole reserve in a risky asset whose parameters depend on a Markov random environment. Using the Green's function method we prove $C^2$-smoothness of the survival probability for annuity payments under minimal assumptions-the jump distribution need only be a probability measure on the positive half-line. For two-sided jumps smoothness holds whenever, in each regime, the jump distribution has no atoms on the negative half-line or the survival probability vanishes at the origin. This condition is sharp: an isolated atom combined with a positive value at the origin makes the second derivative discontinuous, and the size of the discontinuity is computed explicitly.

In previous work, we identified, for every $p\geq3$, a unique first critical temperature $\beta_1^p$ and proved that the Parisi measure is replica symmetric (RS) for $0<\beta\leq\beta_1^p$ and one-step replica symmetry breaking (1-RSB) on a nonempty interval immediately above $\beta_1^p$. In this sequel, we determine the rest of the phase diagram. We prove that there is a unique second critical temperature $\beta_2^p>\beta_1^p$ such that the Parisi measure is 1-RSB for $\beta_1^p<\beta\leq\beta_2^p$, while for $\beta>\beta_2^p$, $\operatorname{supp}\mu_\beta=\{0\}\cup[q_\beta,q'_\beta],$ for some $0<q_\beta<q'_\beta<1$, and is therefore full replica symmetry breaking (FRSB). Combined with our earlier results, this establishes the two transitions predicted by Gardner for the Ising pure $p$-spin glass.

A sequence of sparse weighted graphs $(G_N:N\ge 1)$ indexed by the number of vertices $N$ is said to be left-convergent if all (suitably weighted) subgraph counts converge to a limit as $N\to\infty$. This notion generalizes in a natural way Benjamini-Schramm's definition of local weak convergence. A broad research agenda aims at determining which `global' graph properties are determined by left or local weak convergence (in other words, which of these properties are in fact local). We prove that, under a spectral condition on the weighted adjacency matrix ${\boldsymbol A}_N$ of graph $G_N$, the free energy density of the Ising model on this weighted graph is continuous in the left convergence topology (and hence is a local function). Our proof uses the decomposition of the Ising measure as a log-concave combination of product measures, and of the rapid mixing of Langevin dynamics for log-concave measures. As applications, we derive new limit theorems for the free energy density of spin glasses, antiferromagnets, and magnetization constrained ferromagnetic models, on locally tree-like graphs.

For a convex body $K \subset \mathbb R^d$ let $\Delta(K)$ be the expected distance between two independent uniform points of $K$, and let $\theta(K)$ be the corresponding expectation for normalized surface measure on $\partial K$. The Zaporozhets-Tarasov conjecture asserts $\Delta(K) \le \theta(K)$. We prove this conjecture in case $d=2$. In addition, we give a six-vertex convex polytope in $\mathbb R^3$ for which the reverse strict inequality holds, and obtain counterexamples in every dimension $d \ge 3$ by taking products with segments. Finally, we show that $\theta(K)> \frac{\operatorname{per}K}{6}$ for every planar convex body.

We develop a framework for proving universality results in sparse random graphs. As a first application, we show that there exists an absolute constant $C>1$ such that, with high probability, for every fixed constant $\Delta$, the binomial random graph $G(n,C\ln n/n)$ contains every $n$-vertex tree with maximum degree at most $\Delta$. This answers a question of Montgomery (Advances in Mathematics, 2019). We also determine, for every $p$ satisfying $C\ln n/n\leq p=n^{-1+o(1)}$, the minimum girth $\ell$ (up to an absolute multiplicative constant) for which with high probability $G(n,p)$ contains all cycle factors of girth at least $\Omega(\ell)$. In particular, with high probability $G(n,C\ln n/n)$ contains all cycle factors of girth at least $100\ln n/\ln\ln n$, which is optimal up to a constant factor. This extends an earlier result of Ferber, Kronenberg, and Luh (Transaction of the American Mathematical Society, 2019) and significantly improves a corollary of a deep result of Kahn, Lubetzky, and Wormald (Communications on Pure and Applied Mathematics, 2017). One of the key ingredients in the proofs is establishing the optimal depth of linking systems in sparse random graphs.

We develop a common framework for random width laws, spectral populations, and geometric reconstruction. For a $d$-dimensional orthotope, we prove an exact parity law for the maximal $\pi^{-1}$-grade of every spherical width cumulant, including noncancellation and sign in all dimensions and orders. The first $d$ scalar width moments recover the unordered side vector, and $d-1$ moments are generically insufficient. Each Laplace mode generates an auxiliary width law whose upper endpoint satisfies $M_{n,a} = \lambda_n(a)^{1/2}/\pi$. At high energy the modal coordinate partitions converge to a universal Dirichlet law, while an unsmoothed measure-valued cutoff expansion retains the first geometric memory at face scale. Its simplex moment determines, up to an explicit nonzero factor and a separate off-diagonal argument, a basis-independent projector-gradient Weyl tensor that reconstructs the orthotope. Genuine edge-scale jumps obstruct a third coefficient for the total raw cutoff; exact mixed-boundary Mobius inversion isolates every coordinate stratum and restores a recursive bulk-boundary expansion with a smaller remainder. Beyond orthotopes, we prove direction-labelled identifiability for a canonical linear-quadratic class and finite recovery from direction-sensitive ridge moments under a generator bound. In dimension three, a global great-circle incidence calculus gives the exact step, fold, endpoint-fold, and corner coefficients of reduced zonotopal width densities, including an explicit non-simple corner cancellation. The results distinguish universal aggregation, recoverable geometric memory, and the remaining scalar inverse problem.

We study multinomial and biased-progressive No-U-Turn Samplers for strongly log-concave targets satisfying $mI_d\preceq \nabla^2U(x)\preceq LI_d,$ and $\|\nabla^2U(x)-\nabla^2U(y)\|_{\mathrm F}\le \gamma L^{3/2}\|x-y\|$ with \(\kappa\coloneqq L/m\). We introduce profile separation, a sufficient sign condition on the stationary mean U-turn diagnostics, and combine it with diagnostic concentration, leapfrog fidelity, and whole-orbit energy control to show that on a high-probability certification event, every doubling realization reaches a common terminal depth through a genuine U-turn. If \(T_\star\) is the selected physical trajectory length and \(a_\star=\sqrt m\,T_\star\), a terminal-depth transfer argument yields restricted conductance and warm-start mixing without lazifying either kernel. Up to logarithmic warm-start and accuracy factors, the transition bounds are \[ \widetilde O\!\left( 1+a_\star^2\kappa^2(1+\gamma)^{4/3} \right) \quad\text{and}\quad \widetilde O\!\left( 1+a_\star^4\kappa^3(1+\gamma)^2 \right) \] for multinomial and biased-progressive selection, respectively. These transition bounds are unconditional. Gradient-work bounds are deterministic when the maximum-depth cap is comparable to the certified depth and otherwise take cap-aware expected and high-probability forms. The framework recovers the Gaussian dimension dependence under these work-accounting conditions, provides population-profile and exact-diagnostic verification for nonlinear product targets, a near-isotropic specialization of the practical-tree certificate, and quantifies when a fixed post-warmup metric removes linear anisotropy.

Consider $n$ independent, non-negative, mean at most one random variables, $X_1,X_2,\ldots$. We show the following bound on the probability of their sum exceeding a threshold $t$: \[ \mathbb{P}\left[\sum_{i=1}^n X_i\ge t\right] \leq 1-\left(1-\frac{1}{t}\right)^n \text{ for all } t\ge 2n+1 \,. \] To prove this, we consider a relaxed optimization problem over a set of sequences of ordered, but non-independent random variables. This allows us to reformulate it recursively as dynamic programming problem. The bound becomes an equality for the binary i.i.d.~random variables satisfying $\mathbb{P}\left[X_i=0\right]= 1-\frac{1}{t}$ and $\mathbb{P}\left[X_i=t\right]=\frac{1}{t}$, which remains the maximizer in the relaxed problem.

Let $\alpha_1,\ldots,\alpha_r$ be independent Haar-random fractional Laurent series over $\mathbb{F}_q$, and let $L_r(n)$ be the least coefficient length of a polynomial denominator that simultaneously cancels the first $n$ negative coefficients. We prove that the minimal kernel is a line and that the residual vectors revealed immediately after the stopping times $T_n=n+L_r(n)-1$ are iid uniform on $\mathbb{F}_q^r$. Hence the jump indicators of $L_r(n)$ are iid Bernoulli variables with parameter $1-q^{-r}$; conditionally on a jump, the residual direction is uniform on $\mathbb P^{r-1}(\mathbb{F}_q)$. We also give an exact kernel-growth clock for positive jump sizes and a geometric tail bound uniform in the depth; for two series the jump is decided at the first or second kernel-growth epoch with probabilities $q^{-1}$ and $1-q^{-1}$. The marked renewal law yields exact binomial and fluctuation laws in the depth variable and the density of newly attained minimal denominator lengths $\frac{1-q^{-r}}{r}$ in the coefficient-length variable. For $r=1$ this is the classical iid partial-quotient degree law in the depth coordinate, for which we give an exact dictionary. The new probabilistic content is the simultaneous common-denominator law for $r\ge2$. We also establish exact profile-correspondence and record-duality formulas with joint linear complexity.

We study the problem of sampling from target distributions whose potentials are simultaneously non-smooth, subject to superlinear gradient growth, and non-convex. We introduce the Subgradient Tamed Unadjusted Langevin Algorithm (SG-TULA), a discretisation of the Langevin diffusion that operates directly on subgradients, without relying on computationally demanding smoothing procedures. To handle the superlinear regime, taming techniques are employed to produce a stable, explicit scheme. We derive non-asymptotic convergence bounds in Wasserstein-2 distance, with all constants tracked explicitly in terms of dimension and inverse temperature, improving upon the currently known rates for subgradient-based Langevin algorithms. We further provide excess risk estimates for the associated optimisation problem. We verify the assumptions, with explicit constants, for the regularized pretraining potential of a LLM in the GPT-2 lineage and the boosted coordinate-wise variant of SG-TULA pretrains the former competitively against finetuned AdamW and Muon, for which no comparable non-asymptotic guarantees are presently available.

The low-degree heuristic has become a widely used framework for predicting computational thresholds in average-case planted-versus-null problems. However, a recent sequence of counterexamples shows that low-degree indistinguishability does not, in general, rule out efficient noise-tolerant distinguishers; see Buhai et al. (2025) and Mao (2026). Motivated by these developments, Hsieh et al. (2026) initiated the study of rigorous consequences of the low-degree heuristic. In this work, we continue this program for planted-graph problems. Let $Q_n=G(n,c/n)$, and let $P_n$ be obtained by planting a uniformly random copy of a deterministic graph $\Gamma_n$ into an independent sample from $Q_n$. In the supercritical regime $c>1$, we show that if $P_n$ is degree-$D_n$ indistinguishable from $Q_n$ and $\operatorname{tw}(\Gamma_n)=o(D_n/\log n)$, then a noisy version of $P_n$ is asymptotically indistinguishable from $Q_n$. Here $\operatorname{tw}(\Gamma_n)$ denotes the treewidth of $\Gamma_n$, a measure of how efficiently the graph can be decomposed into tree-like pieces. In the critical and subcritical regimes $0<c\leq 1$, the same conclusion holds whenever $D_n=\omega(\log n)$, without any treewidth assumption. Our proof has two main ingredients. First, we uncover a correspondence between the subgraph-count and automorphism factors in the Fourier expansion and counts of isomorphism triples. Second, we cut the decomposition tree into subtrees, breaking each large Fourier support into low-degree pieces that meet at only a few interface vertices, and use noise to absorb the cost of reassembling them. At and below criticality, the low-degree assumption rules out short cycles, while noise destroys the remaining long cycles.

The evolution of the front of branching Brownian motion is determined by the limit of the derivative martingale. In this work, we characterise which particles contribute to this limit. Precisely, we establish a sharp almost sure path localisation result which shows that the limit is determined by those particles whose trajectory stays within a thin tube at distance $s^{1/2}$ from the extremal particle.

For each of the four classical discrete laws — binomial, Poisson, negative binomial and hypergeometric — the mean absolute deviation about the mean collapses to a single point mass. We give a common telescoping proof of these identities and interpret the resulting closed forms by size biasing. We then derive complete asymptotic expansions for the Poisson ($\lambda\to\infty$), negative binomial ($r\to\infty$, $p$ fixed) and hypergeometric ($N\to\infty$, margins in fixed proportion) cases, extending the binomial expansion from the companion papers. The coefficients are given in closed Bernoulli-polynomial form and carry the lattice displacement of the mean exactly. At integer means the expansions reduce to sign-alternating odd series, and a single Binet-kernel argument shows that these series envelop the logarithm of the normalised mean absolute deviation: successive partial sums bracket it.

Collective decisions in agent networks emerge from repeated local interactions, as agents update their opinions (colors) based on their neighbors, potentially leading to consensus. In sparse communication graphs, repeated interactions can amplify both the initial majority preference and the random fluctuations caused by individual agents. Understanding when the initial majority determines the final preference and how quickly an agreement emerges is, therefore, a basic question in collective decision-making. We study binary majority dynamics as an idealized model of this process. At each round, the interaction graph is independently resampled from the Erdős–Rényi model $\mathbb{G}(N,p)$, with $p=b\log N/N$ and fixed $b>1$. Each agent adopts the majority opinion among its neighbors, retaining its current opinion if there is a tie. For a fixed initial configuration, let $\Delta_0$ denote the initial difference between the number of blue agents and the number of red agents. We identify three regimes for the consensus time, determined by $|\Delta_0|$. First, when $\Delta_0\sqrt p=O(1)$, the probability of blue unanimity is $$ \Phi\left(\Delta_0\sqrt{2p/\pi}\right) + o(1), $$ where $\Phi$ is the cumulative distribution function of a standard Gaussian. Moreover, consensus is reached after $(1+o(1))\log N/\log\log N$ rounds with high probability. Second, in the intermediate regime $1\ll\Delta_0 \sqrt{p}\lesssim \sqrt{N}$, we establish explicit high probability upper and lower bounds on the time to blue-unanimity. Lastly, when the initial blue advantage is above an explicit constant multiple of $N/\sqrt{\log N}$, blue unanimity is guaranteed within two rounds with high probability. Our findings provide a complete description of how the initial advantage influences winner selection and the speed of consensus in this type of majority dynamics with sparse interactions.

In the $3$-neighbor bootstrap percolation process, a vertex becomes (and remains) infected if at least three of its neighbors are infected. We say that an initial configuration of infected vertices percolates if eventually all vertices are infected. We exactly determine the size of the minimum percolating set for the $3$-neighbor bootstrap percolation process on all remaining open cases for rectangular grid graphs $P_m\square P_n$. This extends earlier work of Dukes, Noel, and Romer. Additionally, we consider the same question for the toroidal grids $C_m\square C_n$, proving upper and lower bounds which are at most one apart and determining the answer precisely in many divisibility cases.

We consider a Poisson process $\eta$ on a general Borel space with intensity measure $\lambda$ and a stopping set $Z$ depending on $\eta$. Generalizing earlier results we prove a spatial Markov property for $Z$. In fact we establish this property for the larger class of finitely insertion tolerant random sets, which include pointwise limits of stopping sets. Using the spatial Markov property of $\eta$, we derive several distributional identities for the restrictions of $\eta$ and $\lambda$ to $Z$ and the complement of $Z$. An important special case in Euclidean space is the convex hull of a finite Poisson process. In this case our results generalize classical (and also more recent) identities connecting the number of vertices and the volume of the convex hull.

We give a local geometric criterion ensuring that the edge-connectivity of a hypergraph equals its minimum incidence degree: every locally finite connected $r$-uniform linear hypergraph with $r\ge3$ and nonnegative Lin–Lu–Yau curvature has this property. Among the various extensions of Ollivier–Ricci curvature to hypergraphs, we work with the equal-edges random walk on hypergraphs \cite{CoupetteEtAl2023}. Moreover, we show that both uniformity and linearity are essential: if either assumption is removed, there exist positively curved hypergraphs for which the gap between minimum incidence degree and edge-connectivity is arbitrarily large. For arbitrary locally finite simple hypergraphs, we also determine the dependence on idleness completely: every idleness function is piecewise affine with at most three affine pieces and is affine on the universal interval $[1/2,1]$. These results extend the corresponding theory for graphs \cite{BourneEtAl2018}. They also provide two useful tools below: the $2$-section reduction underlying the edge-connectivity argument and a limit-free formula used in the sharpness constructions.

In this paper, we study the strong averaging principle for multiscale time-inhomogeneous stochastic systems driven by multiplicative $\alpha$-stable processes with $\alpha\in(1,2)$. Based on Khasminskii's discretization approach, we first establish that the fast component processes with a frozen slow variable admits a periodic measure. We then prove the strong convergence of the slow subsystem to an averaged system that depends on the time scale $\varepsilon$. For any fixed $\varepsilon$, if the reciprocals of the two periods $\tau_1$ and $\varepsilon \tau_2$ are rationally linearly independent, an important consequence is that the averaged system has random quasi-periodicity. Furthermore, by applying the ergodic theorem, we prove the strong convergence of the slow subsystem to another averaged system, a time-inhomogeneous SDEs independent of the time scale $\varepsilon$. Our result is also novel even in the time-homogeneous case for a fully coupled multiscale system with multiplicative $\alpha$-stable noises. Finally, we apply the result to a climate-weather system.

We study the behavior of the ordinary differential equations, flow maps, and continuity equations associated to autonomous random velocity fields that admit a natural multiscale finite range decomposition. The velocity fields we consider are only Hölder regular in space—$C^{\alpha-}(\mathbb{T}^d)$ for some $\alpha \in (0,1)$—thus the associated ODE and continuity equation are not a priori well-posed. However, above the critical threshold of $\alpha = 1/2$, due to multiscale stochastic cancellations, we prove well-posedness is almost surely restored away from the zero level set of the velocity field. This threshold marks a genuine transition, as demonstrated by examples lying below the threshold that exhibit robust ill-posedness. We additionally provide effective regularity estimates below the critical threshold and prove analogous results in the related "refreshing" regime.

We provide limit theory for the trace of the squared sample correlation matrix $\mathbf R$, constructed from $n$ observations of a $p$-dimensional random vector with iid components. If the entries have finite fourth moment and $p$ and $n$ grow proportionally, it is known that $\operatorname{tr}({\mathbf R}^2)$ satisfies a central limit theorem (CLT) and the centering and scaling sequences are universal in the sense that they do not depend on the entry distribution. Under symmetry and regular variation assumption with index $\alpha$ and any growth rate of the dimension, we prove that the universal CLT remains valid for $\alpha >3$. For $\alpha<3$, we identify a critical dimension growth at which the fluctuations of $\operatorname{tr}({\mathbf R}^2)$ become non-Gaussian. Moreover, if the dimension $p$ grows faster and $\alpha\le 3$ we establish a non-universal CLT with norming sequences depending on the value of $\alpha$. Our findings are illustrated in a simulation study.

We consider sums of freely independent self-adjoint random variables that are not necessarily identically distributed. Let $\mu_j$ denote the distribution of the $j$th summand. We assume that they have mean zero and finite absolute moments of order $2+\delta$, where $0<\delta\le 1$. Let $\Delta$ denote the Kolmogorov distance, let $\mu^{(n)}$ be the distribution of the normalized partial sum, let $\omega$ be the standard semicircle law, and let $B_n^2$ be the variance of the partial sum. The purpose of this paper is to prove the Berry–Esseen estimate in the free central limit theorem. Namely, there exists an absolute constant $C>0$ such that, for every $0<\delta\le 1$, \[ \Delta(\mu^{(n)},\omega) \le \frac{C}{B_n^{2+\delta}}\sum_{j=1}^n \int_{\R}|x|^{2+\delta}\,\mu_j(dx), \] Our result not only improves several known estimates for general non-identically distributed random variables, but also establishes exactly the same Berry–Esseen estimate as in classical probability theory. The proof combines truncation, a quantitative estimate for the $R$-transform, a stability analysis of a perturbed semicircle equation, and a Bai-type smoothing inequality.

In this paper, we study a nonlinear state-space model represented as a perturbation of a linear model and investigate an asymptotic expansion of the nonlinear filter with respect to the perturbation parameter. The conditional expectation of the hidden state given the observations admits a closed-form representation via the Kallianpur-Striebel formula. We provide a rigorous justification of the expansion of this representation in probability under very general assumptions. Our framework applies not only to nearly linear models but also to more general situations, including nonlinear systems with small system noise. In these cases, the coefficients of the resulting expansion can be computed through systems of ordinary differential equations. Although the explicit computational procedure will be presented in a subsequent paper, the present work establishes the theoretical foundation of our asymptotic expansion approach to nonlinear filtering.

Consider the Gaussian Volterra process introduced by Mishura and Shklyar \cite{MS22a,MS22b}, $$ X(t) = \int_0^t r^\alpha \left( \int_r^t u^\beta (u-r)^\gamma\,du \right)dW_r, \qquad t\ge 0, $$ where $$ \alpha>-\frac12, \quad \gamma\in\left(-1,-\frac12\right), \quad H:=\alpha+\beta+\gamma+\frac32>0. $$ We obtain two-sided estimates for the small ball probabilities of $X$. As applications, we prove Chung's laws of the iterated logarithm (Chung's LILs) at every fixed point $t>0$, at the origin, and at infinity. The fixed-time result follows from the small ball estimates and the Lamperti transformation, whereas the results at the origin and infinity follow from Talagrand's lower-class criteria \cite{talagrand1996lower}. These results show that $\gamma+\frac32$ determines the local roughness and the small ball exponent, $\alpha+\beta$ determines the scale of local fluctuations at fixed positive times, and $H$ governs the self-similar scaling at the origin and infinity.

Using the theory of stochastic integration in duals of nuclear spaces with respect to cylindrical martingale-valued measures, a vector-valued Itô formula is proved for generalized Itô processes defined with respect to these integrals. The abstract result is then applied to prove an Itô formula for Lévy-Itô processes defined with respect to Lévy processes taking values in the dual of a reflexive nuclear space.

Two-phase phase-type ($PH_2$) service distributions are widely used in call center, healthcare and manufacturing models: they capture coefficients of variation both above and below unity while remaining parsimonious enough for reliable statistical fitting. We derive explicit closed-form queue-length distributions $p_n = A_1 r_1^n + A_2 r_2^n$ and sojourn-time densities $f_{W_s}(t) = c_1 e^{\alpha_1 t} + c_2 e^{\alpha_2 t}$ for the complete $M/PH_2/1$ family (Erlang-2, hypoexponential-2, hyperexponential-2 and Coxian-2), with every coefficient given as an explicit function of the system parameters and consolidated in ready-to-use reference tables. Because the results are functions rather than numerical values, they support analytical operations that numerical output cannot deliver directly: exact sensitivity derivatives, closed-form threshold optimization, capacity sizing by bisection on an exact cumulative distribution, and $O(1)$ tail-probability evaluation at arbitrary queue length. We further prove that $PH_2$ marks the boundary of algebraic tractability: the three-phase queue $M/E_3/1$ has a universally negative discriminant, forcing complex roots at every traffic intensity, so its distribution admits no representation as a sum of real geometric terms, and for $k \geq 5$ phases the Abel-Ruffini theorem precludes radical solutions. Validation against the matrix-analytic library BuTools confirms the derivations and characterizes where such computation degrades: queue-length evaluation holds machine precision throughout, while sojourn-time evaluation via the matrix exponential loses up to ten significant digits when high traffic intensity and high service variability act jointly. An application calibrated to published surgical time data for 46,322 cases yields an exact affine law for the sensitivity of overflow risk to case mix.

We study utility maximization for high-frequency trading in fractional Brownian motion models. We first consider exponential utility in the discretized arithmetic model. Using spectral methods for stationary Gaussian sequences, we derive the asymptotic growth rate of the optimal certainty equivalent and the scaling limit of the optimal positions. We then consider power and logarithmic utility in a positive-price model with positive wealth and investment fractions in $[0,1]$. In this case, the logarithm of the optimal certainty equivalent grows at rate $n^{1-H}$ for $H\neq1/2$, with a coefficient determined by the one-sided prediction error of fractional Gaussian noise. We also give a simple binary strategy which attains this leading rate, without claiming exact finite-horizon optimality.