35.
Abstract: Many of the most consequential dynamics in human cognition occur \emph{before} events become explicit: before decisions are finalized, emotions are labeled, or meanings stabilize into narrative form. These pre-event states are characterized by ambiguity, contextual tension, and competing latent interpretations. Rogue Variable Theory (RVT) formalizes such states as Rogue Variables: structured, pre-event cognitive configurations that influence outcomes while remaining unresolved or incompatible with a system's current representational manifold. We present a quantum-consistent information-theoretic implementation of RVT based on a time-indexed Mirrored Personal Graph (MPG) embedded into a fixed graph Hilbert space, a normalized Quantum MPG State (QMS) constructed from node and edge metrics under context, Hamiltonian dynamics derived from graph couplings, and an error-weighted `rogue operator'' whose principal eigenvectors identify rogue factor directions and candidate Rogue Variable segments. We further introduce a Rosetta Stone Layer (RSL) that maps user-specific latent factor coordinates into a shared reference Hilbert space to enable cross-user comparison and aggregation without explicit node alignment. The framework is fully implementable on classical systems and does not assume physical quantum processes; collapse is interpreted as informational decoherence under interaction, often human clarification.
34.
Abstract: Organizations increasingly operate in environments characterized by volatility, uncertainty, complexity, and ambiguity (VUCA), where early indicators of change often emerge as weak, fragmented signals. Although artificial intelligence (AI) is widely used to support managerial decision-making, most AI-based systems remain optimized for prediction and resolution, leading to premature interpretive closure under conditions of high ambiguity. This creates a gap in management science regarding how human-AI systems can responsibly manage ambiguity before it crystallizes into error or crisis. This study addresses this gap by presenting a proof of concept (PoC) of the LAIZA human-AI augmented symbiotic intelligence system and its patented process: Systems and Methods for Quantum-Inspired Rogue Variable Modeling (QRVM), Human-in-the-Loop Decoherence, and Collective Cognitive Inference. The mechanism operationalizes ambiguity as a non-collapsed cognitive state, detects persistent interpretive breakdowns (rogue variables), and activates structured human-in-the-loop clarification when autonomous inference becomes unreliable. Empirically, the article draws on a three-month case study conducted in 2025 within the AI development, involving prolonged ambiguity surrounding employee intentions and intellectual property boundaries. The findings show that preserving interpretive plurality enabled early scenario-based preparation, including proactive patent protection, allowing decisive and disruption-free action once ambiguity collapsed. The study contributes to management theory by reframing ambiguity as a first-class construct and demonstrates the practical value of human-AI symbiosis for organizational resilience in VUCA environments.
33.
Abstract: This article develops the concept of Person-AI bidirectional fit, defined as the continuously evolving, context-sensitive alignment-primarily cognitive, but also emotional and behavioral-between a human decision-maker and an artificial intelligence system. Grounded in contingency theory and quality theory, the study examines the role of P-AI fit in managerial decision-making through a proof-of-concept case study involving a real hiring process for a Senior AI Lead. Three decision pathways are compared: (1) independent evaluations by a CEO, CTO, and CSO; (2) an evaluation produced by an augmented human-AI symbiotic intelligence system (H3LIX-LAIZA); and (3) an assessment generated by a general-purpose large language model. The results reveal substantial role-based divergence in human judgments, high alignment between H3LIX-LAIZA and the CEOs implicit decision model-including ethical disqualification of a high-risk candidate and a critical false-positive recommendation from the LLMr. The findings demonstrate that higher P-AI fit, exemplified by the CEO H3LIX-LAIZA relationship, functions as a mechanism linking augmented symbiotic intelligence to accurate, trustworthy, and context-sensitive decisions. The study provides an initial verification of the P-AI fit construct and a proof-of-concept for H3LIX-LAIZA as an augmented human-AI symbiotic intelligence system.
32.
Abstract: We present a tripartite cognitive architecture that unifies somatic grounding, symbolic inference and metacognitive control within an extended SORK-N loop. Cognition is cast as the interaction of a Somatic layer (biophysical and affective inputs), a Symbolic layer (linguistic, logical, and representational processes), and a Metacognitive layer (global coherence estimation and policy adjustment), coordinated by a methodological framework that time-locks and analyzes multimodal physiological, linguistic and self-report data. Mathematical structure is provided by the Mirrored Profile Graph (MPG), an evidence-linked, hierarchical state space, and Rogue Variable (RV) analysis, which together localize structural sources of prediction-observation gaps and support falsifiable tests. This framework enables reproducible tests of intuition, pre-event cognition, and collective coherence, while remaining compatible with empirical scrutiny. We further discuss implications for symbiotic human-AI systems and argue that intentional co-evolution of biological and artificial cognition offers a practical route toward robust, reflective intelligence.
31.
Abstract: We consider a system of stochastic interacting particles with general diffusion coefficient and drift functions and we study the types of collisions that arise in them. In particular, interactions between particles are inversely proportional to their separation, and the coupling function of interaction is also considered in great generality. Our main result indicates that under very mild conditions, all collisions are simple almost surely, namely, only one pair of particles collides at any time, while more complicated collisions such as three-body or disjoint two-body collisions occur with zero probability. In order to obtain our results we make use of symmetric polynomials on the square of particle separations; the degree of these polynomials indicates the type of collision, and by a locality argument we show that polynomials indicating a non-simple collision almost surely do not cancel. We make use of our main result to study the Hausdorff dimension of times at which collisions occur, and we show that this dimension is given by the ratio between the interaction coupling and diffusion functions. Our results cover many of the most well-known particle systems, such as the Dyson model and Wishart processes and their extensions to non-constant diffusion coefficients and background drifts.
30.
Abstract: We prove an analogue of Ehrhard’s inequality for the two dimensional isotropic Cauchy measure. In contrast to the Gaussian case, the inequality is not valid for non-convex sets. We provide the proof for rectangles which are symmetric with respect to one coordinate axis.
29.
Abstract: We prove Archimedes’ principle for a macroscopic ball in ideal gas consisting of point particles with non-zero mass. The main result is an asymptotic theorem, as the number of point particles goes to infinity and their total mass remains constant. We also show that, asymptotically, the gas has an exponential density as a function of height. We find the asymptotic inverse temperature of the gas. We derive an accurate estimate of the volume of the phase space using the local central limit theorem.
28.
Abstract: We consider matrix-valued processes described as solutions to stochastic differential equations of very general form. We study the family of the empirical measure-valued processes constructed from the corresponding eigenvalues. We show that the family indexed by the size of the matrix is tight under very mild assumptions on the coefficients of the initial SDE. We characterize the limiting distributions of its subsequences as solutions to an integral equation. We use this result to study some universality classes of random matrix flows. These generalize the classical results related to Dyson Brownian motion and squared Bessel particle systems. We study some new phenomena as the existence of the generalized Marchenko–Pastur distributions supported on the real line. We also introduce universality classes related to generalized geometric matrix Brownian motions and Jacobi processes. Finally we study, under some conditions, the convergence of the empirical measure-valued process of eigenvalues associated to matrix flows to the law of a free diffusion.
27.
Abstract: Let p_t(x), f_t(x) and q_t^∗(x) be the densities at time t of a real Lévy process, its running supremum and the entrance law of the reflected excursions at the infimum. We provide relationships between the asymptotic behaviour of p_t(x), f_t(x) and q_t^∗(x), when t is small and x is large. Then for large x, these asymptotic behaviours are compared to this of the density of the Lévy measure. We show in particular that, under mild conditions, if p_t(x) is comparable to tν(x), as t→0 and x→∞, then so is f_t(x).
26.
Abstract: For an open subset Ω of R^d, symmetric with respect to a hyperplane and with positive part Ω_+, we consider the Neumann/Dirichlet Laplacians −Δ_{N/D,Ω} and −Δ_{N/D,Ω_+}. Given a Borel function Φ on [0,∞) we apply the spectral functional calculus and consider the pairs of operators Φ(−Δ_{N,Ω}) and Φ(−Δ_{N,Ω_+}), or Φ(−Δ_{D,Ω}) and Φ(−Δ_{D,Ω_+}). We prove relations between the integral kernels for the operators in these pairs, which in the particular cases of Ω_+=R^{d−1}×(0,∞) and Φ_t(u)=exp(−tu), u≥0, t>0, were known as reflection principles for the Neumann/Dirichlet heat kernels. These relations are then generalized to the context of symmetry with respect to a finite number of mutually orthogonal hyperplanes.
25.
Abstract: We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.
24.
Abstract: We provide sharp two-sided estimates on the Dirichlet heat kernel k1(t, x, y) for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel k1(t, x, y) in terms of its global two-sided sharp estimates. Such precise estimates were possible to obtain due to the enrichment of analytical methods with probabilistic tools.
23.
Abstract: We study the existence and uniqueness of solutions of SDEs describing squared Bessel particle systems in full generality. We define nonnegative and non-colliding squared Bessel particle systems and we study their properties. Particle systems dissatisfying non-colliding and unicity properties are pointed out. The structure of squared Bessel particle systems is described.
22.
Abstract: We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall’s identity for spectrally positive Lévy processes. We obtain the later result as a direct consequence of the former. The ballot theorem is extended to processes having possible negative jumps. Then we prove through straightforward arguments based on the law of bridges and Kendall’s identity, Theorem 2.4 in [20] which gives an expression for the law of the supremum of spectrally positive Lévy processes. An analogous formula is obtained for the supremum of spectrally negative Lévy processes.
21.
Abstract: A characterization of the existence of non-central Wishart distributions (with shape and non-centrality parameter) as well as the existence of solutions to Wishart stochastic differential equations (with initial data and drift parameter) in terms of their exact parameter domains is given. These two families are the natural extensions of the non-central chi-square distributions and the squared Bessel processes to the positive semidefinite matrices.
20.
Abstract: We prove a two-term Weyl-type asymptotic law, with error term O(1/n), for the eigenvalues of the operator ψ(-Δ) in an interval, with zero exterior condition, for complete Bernstein functions ψ such that \psi ψ'(\psi) converges to infinity as \psi→∞. This extends previous results obtained by the authors for the fractional Laplace operator (ψ(\psi)=\psi^α/2) and for the Klein–Gordon square root operator (ψ(\psi)=(1+\psi)^1/2-1). The formula for the eigenvalues in (-a,a) is of the form λ_n = \psi(\mu_n^2)+O(1/n), where \mu_n is the solution of \mu_n = n π/2a - 1/a \nu(\mu_n), and \nu(\mu) \in [0,π/2) is given as an integral involving ψ.
19.
Abstract: We provide sharp two-sided estimates of the Fourier–Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0,1) killed at 1 and killed or reflected at 0.
18.
Abstract: In the paper we consider the Bessel differential operator L^{(\mu)} = \dfrac{d^2}{dx^2}+\frac{2\mu+1}{2}\dfrac{d}{dx} in half-line [a,∞), a>0, and its Dirichlet heat kernel. For \mu=0, by combining analytical and probabilistic methods, we provide sharp two-sided estimates of the heat kernel for the whole range of the space parameters x,y>a and every t>0, which complements the recent results given in [1], where the case \mu\neq 0 was considered.
17.
Abstract: Let us consider a real valued Lévy process X, whose transition probabilities are absolutely continuous and have bounded densities. Then the law of the past supremum of X before any deterministic time t is absolutely continuous on (0,∞). We show that its density f_t(x) is continuous on (0,∞) if and only if the potential density h' of the upward ladder height process is continuous on (0,∞). Then we prove that f_t behaves at 0 as h′. We also describe the asymptotic behaviour of f_t, when t tends to infinity. Then some related results are obtained for the density of the meander and this of the entrance law of the Lévy process conditioned to stay positive.
16.
Abstract: The main objective of the work is to provide sharp two-sided estimates of the λ-Green function, λ≥0, of the hyperbolic Brownian motion of a half-space. We rely on the recent results obtained by K. Bogus and J. Małecki (2015), regarding precise estimates of the Bessel heat kernel for half-lines. We also substantially use the results of H. Matsumoto and M. Yor (2005) on distributions of exponential functionals of Brownian motion.
15.
Abstract: In this paper we study the Bessel process R_t^(\mu) with index μ ≠ 0 starting from x > 0 and killed when it reaches a positive level a, where x > a > 0. We provide sharp estimates of the transition probability density for the whole range of space parameters x,y > a and every t > 0.
14.
Abstract: We study systems of stochastic differential equations describing positions x_1,..., x_p of p ordered particles, with inter-particles repulsions of the form H_ij(x_i,x_j)/(x_i−x_j). We show the existence of strong and pathwise unique non-colliding solutions of the system with a colliding initial point x_1(0)≤...≤x_p(0) in the whole generality, under natural assumptions on the coefficients of the equations.
13.
Abstract: The eigenvalues and eigenfunctions of the one-dimensional quasi-relativistic Hamiltonian (-ℏ^2c^2d^2/dx^2 + m^2c^4)^(1/2) + V_well(x) (the Klein–Gordon square-root operator with electrostatic potential) with the infinite square well potential V_well(x) are studied. Eigenvalues represent energies of a "massive particle in the box" quasi-relativistic model. Approximations to eigenvalues λ_n are given, uniformly in n, ℏ, m, c and a, with error less than C_1ℏca^(-1)exp(-C_2ℏ-1mca)n^(-1). Here 2a is the width of the potential well. As a consequence, the spectrum is simple and the nth eigenvalue is equal to (nπ/2 - π/8)ℏc/a + O(1/n) as n → ∞. Non-relativistic, zero mass and semi-classical asymptotic expansions are included as special cases. In the final part, some L^2 and L^∞ properties of eigenfunctions are studied.
12.
Abstract: In this paper we study the supremum functional M_t=sup(X_s: 0≤s≤t), where X_t, t≥0, is a one-dimensional Lévy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of M_t. In the symmetric case we find an integral representation of the Laplace transform of the distribution of M_t if the Lévy–Khintchin exponent of the process increases on (0,∞).
11.
Abstract: Let T_1^(\mu) be the first hitting time of the point 1 by the Bessel process with index μ ∈ ℝ starting from x > 1. Using an integral formula for the density q_x^(\mu)(t) of T_1^(\mu), obtained in Byczkowski and Ryznar (Stud Math 173(1):19–38, 2006), we prove sharp estimates of the density of T_1^(\mu), which exhibit the dependence both on time and space variables. Our result provides optimal uniform estimates for the density of the hitting time of the unit ball by the Brownian motion in ℝ^n, which improve existing bounds. Another application is to provide sharp estimates for the Poisson kernel for half-spaces for hyperbolic Brownian motion in real hyperbolic spaces.
10.
Abstract: Let X_t be a subordinate Brownian motion, and suppose that the Lévy measure of the underlying subordinator has a completely monotone density. Under very mild conditions, we find integral formulae for the tail distribution P(\tau_x>t) of first passage times \tau_x through a barrier at x>0, and its derivatives in t. As a corollary, we examine the asymptotic behaviour of P(\tau_x>t) and its t-derivatives, either as t→∞ or x→0.
9.
Abstract: We prove a multidimensional version of the Yamada-Watanabe theorem, i.e., a theorem giving conditions on coefficients of a stochastic differential equation for existence and pathwise uniqueness of strong solutions. It implies an existence and uniqueness theorem for the eigenvalue and eigenvector processes of matrix-valued stochastic processes, called a “spectral” matrix Yamada-Watanabe theorem. The multidimensional Yamada-Watanabe theorem is also applied to particle systems of squared Bessel processes, corresponding to matrix analogues of squared Bessel processes, Wishart and Jacobi matrix processes. The β-versions of these particle systems are also considered.
8.
Abstract: The purpose of the paper is to provide a general method for computing the hitting distributions of some regular subsets D for Ornstein–Uhlenbeck type operators of the form \frac{1}{2}Δ+F⋅∇, with F bounded and orthogonal to the boundary of D. As an important application we obtain integral representations of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion and for the classical Ornstein–Uhlenbeck process. The method developed in this paper is based on stochastic calculus and on the skew product representation of multidimensional Brownian motion and yields more complete results than those based on the Feynman–Kac technique.
7.
Abstract: The spectral properties of the pseudo-differential operator (-d^2/dx^2)^(1/2) + x^2 are analyzed by a combination of functional integration methods and direct analysis. We obtain a representation of its eigenvalues and eigenfunctions, prove precise asymptotic formulae, and establish various analytic properties. We also derive trace asymptotics and heat kernel estimates.
6.
Abstract: Let X^(μ)={X^(μ)_t;t≥0}, μ >0, be then-dimensional hyperbolic Brownian motion with drift, that is a diffusion on the real hyperbolic spaceHnhaving the Laplace–Beltrami operator with drift as its generator. We prove the reflection principle for X^(μ), which enables us to study the process X^(μ) killed when exiting the hyperbolichalf-space, that is the set D={x∈H^n:x_1>0}. We provide formulae, uniform estimates and describe asymptotic behavior of the Green function and the Poisson kernel of D for the process X^(μ). Finally, we derive formula for the λ-Poisson kernel of the set D.
5.
Abstract: We study the spectral properties of the transition semigroup of the killed one-dimensional Cauchy process on the half-line (0, ∞) and the interval (−1, 1). This process is related to the square root of one-dimensional Laplacian A=−(-d^2/dx^2)^(1/2) with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula for generalized eigenfunctions ψλ of 𝒜 is derived, and then used to construct a spectral representation of 𝒜. Explicit formulas for the transition density of the killed Cauchy process on the half-line (or the heat kernel of 𝒜 in (0, ∞)), and for the distribution of the first exit time from the half-line follow. The formula for ψλ is also used to construct approximations to eigenfunctions of 𝒜 in the interval. For the eigenvalues λn of 𝒜 in the interval the asymptotic formula λn = n π/2 − π/8 + O(1/n) is derived, and all eigenvalues λn are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λn are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to the ninth decimal point.
4.
Abstract: We apply the Feynman–Kac formula to compute the λ-Poisson kernels and λ-Green functions for half-spaces or balls in hyperbolic spaces. We present known results in a unified way and also provide new formulas for the λ-Poisson kernels and λ-Green functions of half-spaces in H^n and for balls in real and complex hyperbolic spaces.
3.
Abstract: The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place for half-spaces of codimension one for a diffusion in ℝ^{n+1}, composed of one-dimensional Bessel process and independent n-dimensional Brownian motion. The most important argument is carried out for the two-dimensional situation. We show that this amounts to computation of distributions of various integral functionals with respect to a two-dimensional process with independent Bessel components. As a result, we provide a formula for the Poisson kernel of a half-space or of a strip for the operator (I − Δ)^α/2, 0 < α < 2. In the case of a half-space, this result was recently found, by different methods, in Byczkowski et al. (Trans Am Math Soc 361:4871–4900, 2009). As an application of our method we also compute various formulas for first hitting places for the isotropic stable Lévy process.
2.
Abstract: The purpose of the paper is to find explicit formulas for basic objects pertaining to the potential theory of the operator (I − Δ)^α/2, which is based on Bessel potentials J_α = (I − Δ)^(-α/2), 0 < α < 2. We compute the harmonic measure of the half-space and obtain a concise form for the corresponding Green function of the operator (I − Δ)^α/2. As an application we provide sharp estimates for the Green function of the half-space for the relativistic process.
1.
Abstract: Let (X_t)_(t⩾0) be the n-dimensional hyperbolic Brownian motion, that is the diffusion on the real hyperbolic space D^n having the Laplace–Beltrami operator as its generator. The aim of the paper is to derive the formulas for the Gegenbauer transform of the Poisson kernel and the Green function of the ball for the process (X_t)_(t⩾0). Under additional hypotheses we prove integral representations for the Poisson kernel. This yields explicit formulas in D^4 and D^6 spaces for the Poisson kernel and the Green function as well.