Search Results (235)

The standard models of stock-price dynamics and option valuation rest on stochastic processes postulated at the outset; here, we lay down an entropic-inference framework that derives these processes rather than assuming them, by making explicit the information each one encodes. A symmetry comes first: markets reward returns rather than price levels, which selects the logarithm of price as the dynamical variable. The price then evolves through two channels, a continuous one carrying the constraints of continuity and directionality, and a jump channel carrying the arrival rate and the first two moments of the jump size. Because these constraints act on disjoint parts of the microstate, the channels factorize as a theorem, and the dynamics is the Merton jump-diffusion, with Geometric Brownian Motion as its no-jump limit; the log-price density obeys a Kolmogorov–Feller equation, of which the Fokker–Planck equation is the no-jump limit. The same principle, now imposing no-arbitrage through the mean log-return, selects the Esscher transform from among the many martingale measures an incomplete market admits, here derived rather than borrowed; the premium then satisfies Merton’s partial integro-differential equation, and the risk-neutral mixture of lognormals generates the implied-volatility smile, the Black–Scholes results returning when jumps vanish. What changes from one model to the next is never the inference but the information supplied to it. Full article

This study examines how misinformation can persist when broadcast exposure and social feedback reinforce one another under stochastic platform conditions. Text classifiers and single-layer cascade models omit latent exposure, reply-driven amplification, random attention shocks, and uncertainty in intervention response. A stochastic duplex SEIR model is developed on heterogeneous networks, with an information exposure layer for broadcast and recommendation channels and a social feedback layer for replies, discussion, and amplification. The analysis combines degree-weighted mean-field equations, next-generation threshold calculations, Lyapunov stability arguments, Fokker–Planck linear noise approximation, Milstein simulation, and Wasserstein distributionally robust control. Theoretical results provide positivity, stochastic threshold conditions, extinction and persistence regimes, and sufficient conditions for stationary behavior and robust control stability. Numerical simulations show extinction–persistence transitions, cross-layer resonance, noise-induced threshold shifts, stationary bands, control cost–safety trade-offs, and sensitivity to unidentifiable stochastic parameters. A CoAID tweet–reply case study maps public interaction traces to observable duplex indicators, including tweet–reply densities, propagation elasticities, coupling proxies, and classifier features. Duplex observable features improve over a single-layer public data baseline, while model-assisted stochastic features add modest gains in the available public projection. Structural fitting of the stochastic duplex process would require time-stamped user-level multiplex trajectories, recommendation exposures, and intervention logs. The case study also clarifies the data granularity needed for future platform-level calibration and operational readiness. The framework supports data-informed platform governance by linking propagation thresholds, algorithmic down-ranking, reply thread moderation, intervention cost, and robustness bounds within a common threshold control language for practical settings. Full article

We investigate the dynamics of a driven quantum field coupled to a finite-temperature reservoir. The corresponding master equation is solved using superoperator techniques, yielding an analytical expression for the density operator. To obtain a compact and physically transparent description of the dynamics, we adopt a phase-space representation based on the Husimi Q-function. For an initially coherent state, we derive a closed-form Gaussian expression for the Husimi Q-function whose stationary limit corresponds to a displaced thermal state. This approach also enables an analytical study of quantum-interference dynamics for an initial superposition of coherent states. Furthermore, we derive the corresponding Fokker–Planck equation for the Husimi Q-function and obtain closed-form expressions for relevant statistical quantities, including the mean photon number, the photon-number standard deviation, and the Mandel parameter. We also investigate the Wehrl and linear entropies, which quantify the loss of phase-space information and purity induced by the thermal environment. The framework provides a complete analytical characterization of the phase-space dynamics, photon statistics, and entropic properties of driven–dissipative quantum fields while avoiding the explicit manipulation of the density operator. Full article

The roll equation of a port–starboard symmetric ship is a clean physical realization of an order-two reflection (${\mathbb{Z}}_2$)-equivariant nonlinear oscillator: odd restoring and damping make the upright state a symmetric equilibrium whose two angles of vanishing stability form a heteroclinic-connected pair. A steady heeling action—beam wind, off-center load or list—enters as a single symmetry-breaking parameter c. Although the qualitative effect of such a bias is known, we show that this one parameter organizes the whole capsize problem in closed form. Equivariant singularity theory identifies c as the imperfection that unfolds the symmetric pitchfork of equilibria into a cusp, turning the heteroclinic pair into a homoclinic loop. The biased Melnikov boundary then yields two directional capsize thresholds and a damping-independent asymmetry index $\Delta f_{cr}=2c{I}_c/A\left(\mathsf{\Omega }\right)$, exactly linear in the bias: for a lightly damped hull a heel below $0.1°$ already halves the port/starboard split. For parametric roll, Floquet analysis gives the bias-corrected stability boundary, recovering $\Delta GM/GM>4\zeta$ in the symmetric limit and translating the principal tongue; for the dead-ship condition, the stationary Fokker–Planck solution—exact for the adopted one-degree-of-freedom (1-DOF) model—shows that, in the light-damping energy-diffusion regime, lightly damped capsize occurs over the lowered barrier with probability approaching one. A small-bias expansion reveals a sensitivity hierarchy—beam-sea capsize and dead-ship survival are first-order in the heel, and parametric detuning is only second-order—and a two-parameter cusp links the bias to pure loss of stability. All results are validated against safe-basin erosion, Floquet multipliers and Monte-Carlo simulation for a real vessel, offering the International Maritime Organization (IMO) second-generation intact-stability criteria (SGISC) a transparent, analytically based correction for the port–starboard asymmetry their symmetric assumption omits. Full article

In this paper, we apply the double Fourier transform method to the two-dimensional Vlasov equations for a charged particle subjected to white noise, exponentially correlated Gaussian forces, trap forces and thermal and active noises in a magnetic field. By deriving the corresponding Fokker–Planck equation, analytical solutions for the joint probability density are obtained in different time domains. The mean squared displacement and velocity of a charged particle driven by white noise exhibits a super-diffusive behavior, scaling as $~t^2$ in the short-time regime, while it grows linearly with time ($~t$) in the long-time regime, in agreement with numerical simulations of the mean squared displacement. When thermal noise is included together with harmonic trap and viscous forces, the characteristic time scale increases as $~t^{2h+1}$ in the corresponding time domains, whereas the mean squared velocity scales as $~t^{2h+3}$. The moments of the joint probability density under thermal noise scale as ${~t}^{2h+5}$. Furthermore, when the persistent Hurst exponent $h\to 1/2$, the entropy of the joint probability density associated with thermal noise coincides with that obtained for active noise in both the short-time ($t\ll \tau$) and long-time ($t\gg \tau$) limits. Full article

The fractional Whitham–Broer–Kaup model governs nonlinear wave propagation in memory-dependent media, including porous structures, viscoelastic fluids, and irregular seabeds, yet the full dynamical spectrum from quasi-periodicity to deterministic chaos, the role of stochastic forcing, and reliable identification from noisy data remains insufficiently explored, particularly how the fractional order $\beta$ influences these regimes. This study addresses these gaps through a comprehensive, multi-method dynamical analysis of a representative nonlinear oscillator embodying key FWBK features. Three-dimensional attractor visualizations, return maps, and surrogate data tests demonstrate a transition from quasi-periodic toroidal attractors to fully developed chaos via torus breakdown, confirming that observed complexity originates from deterministic nonlinearity. Poincaré sections reveal multistability and KAM-type structures, where coexisting attractors depend on initial conditions, while increasing noise progressively disrupts coherent dynamics. The OGY control method effectively stabilizes unstable periodic orbits across chaotic regimes with minimal perturbation, and Lyapunov analysis indicates that stochastic forcing attenuates chaos while enhancing dissipation. The Fokker–Planck framework shows that noise reshapes probability landscapes, driving transitions from unimodal to bimodal distributions. Comparative analysis of SINDy, JMAP and VBA highlights trade-offs in interpretability, computational efficiency, and uncertainty quantification, while an integrated Bayesian–PCE–Sobol approach quantifies parametric uncertainty and reveals time-dependent sensitivity variations. Additionally, the overlapping of soliton solutions extracted via the enhanced modified Sardar sub-equation method reveals structural relationships among soliton families and their stability under interaction. Soliton branches that maintain high overlap under noise correspond to stable regimes, while those losing coherence indicate the onset of chaos. Furthermore, while the reduced dynamics in $\eta$-space are independent of $\beta$, the fractional order controls spatial compression and temporal scaling in physical coordinates, directly influencing observable wave localization. These results imply that fractional effects can modify chaos transitions, support controllability through OGY, and influence noise–instability interactions depending on $\beta$. This framework provides a robust, transferable methodology for analyzing and controlling nonlinear oscillatory systems under deterministic and stochastic conditions, with direct applications to FWBK-based models in coastal engineering, fiber optics, and quantum interference systems. Full article

This paper establishes the global-in-time existence and uniqueness of mild solutions to the relativistic Vlasov–Maxwell–Fokker–Planck (VMFP) system near a global Maxwellian equilibrium. We adopt a low-regularity functional framework, namely the mixed-norm space $L_k^1{L}_T^{\infty }{L}_p^2$ introduced for kinetic equations, which requires only integrability in the Fourier frequency variable and avoids high-order spatial differentiability. By employing a macro–micro decomposition, we derive macroscopic estimates for the hydrodynamic density and electric field, complemented by coercive estimates for the microscopic dissipation. Under a smallness assumption on the initial perturbation measured in this low-regularity norm, we derive a uniform a priori bound for the associated energy functional. This work provides the global existence result for the relativistic VMFP system in such low-regularity spaces, significantly relaxing the regularity requirements of previous classical Sobolev approaches. Full article

The sign of the maximal almost sure Lyapunov exponent determines the stability of stochastic systems, while its numerical computation for three-dimensional linear Stratonovich stochastic differential equations remains challenging due to the failure of classical two-dimensional strategies. The spherical angular motion of 3D systems produces a Fokker–Planck equation with intractable mixed partial derivatives, preventing conventional analytical solutions. This paper develops a unified computational framework for three-dimensional linear Stratonovich stochastic systems using analytical derivation for degenerate cases and physics-informed neural network (PINN) approximation for general non-degenerate scenarios. For degenerate systems, we reduce the coefficient matrix to a lower triangular form via orthogonal transformation and establish tight upper bounds based on the logarithmic growth property of the Wiener process, yielding closed-form expressions for the maximal almost sure Lyapunov exponent under all parameter sign configurations. For non-degenerate systems, we reformulate the Fokker–Planck equation in spherical coordinates and construct a customized PINN with trigonometric encoding to enforce periodic boundary conditions. The network is trained by joint loss functions of equation residuals, boundary constraints and normalization consistency, and the converged stationary density is substituted into the Furstenberg–Khasminskii formula to calculate the exponent via Gauss–Legendre quadrature. Monte Carlo simulations confirm the accuracy and robustness of the proposed method, which reliably identifies the sign of the maximal almost sure Lyapunov exponent even in near-critical regimes. Numerical experiments on a 3D stochastic Hopf bifurcation model show that noise negatively shifts the bifurcation point, with the offset linearly proportional to the squared noise intensity. This work extends Lyapunov stability analysis from two-dimensional to three-dimensional linear Stratonovich stochastic systems, offering an effective tool for stability evaluation of general three-dimensional stochastic dynamical models. Full article

In this work, we formulate a systematic expectation-value framework for dynamical systems whose probability densities evolve according to linear partial differential equations, such as the Fokker-Planck and Liouville equations. The approach is based on expectation-calculus identities associated with the Fluctuation-Dissipation Theorem and the Conjugate Variables Theorem, allowing the derivation of evolution equations directly for arbitrary observables and fluctuations without explicitly solving the full probability-density equation. The resulting relations provide a classical Ehrenfest-type formulation for observable dynamics and fluctuations under linear probability-density evolution. While the resulting equations are not closed in general, since they typically involve higher-order moments, correlations, or derivatives, the formalism offers a unified operational framework for studying observable dynamics under suitable approximations or closure assumptions. We illustrate the procedure with examples involving Fokker–Planck and Liouville dynamics and discuss the scope, limitations, and possible applications of the framework in nonequilibrium statistical mechanics. In particular, we emphasize that the method is intended as a systematic observable-based formulation for systems governed by linear evolution equations, rather than as a universal closure scheme for arbitrary nonequilibrium dynamics. Full article

Hysteresis seems to play a critical role in the generation and modulation of electrical signal events in neurons, muscles, and other excitable tissues. In voltage-gated ion channels, hysteretic conductance manifests under cycling changes in transmembrane voltage when conductance is delayed in response to voltage changes. Such dynamic behavior emerges naturally when the frequency of the oscillatory voltage becomes comparable to the characteristic relaxation time associated with transitions between channel conductance states and is reminiscent of hysteresis observed in transistors, memristors or solar cells. To investigate this delayed response, various discrete-state Markov models have been proposed. In these frameworks, an ion channel is represented as a finite set of states—typically corresponding to closed and open conformations—with transitions governed by voltage-dependent rates. As an alternative, the progress of activation and transition between opening and closing states of a channel is described in terms of a diffusive, collective “reaction coordinate” which fulfills a Langevin equation and the Smoluchowski–Fokker–Planck equation associated with it. Here we review this approach in modeling dynamic memory of ion channels. Full article

The quantum motion of a photon in an arbitrary medium was considered within the framework of the gauge symmetry group $SU\left(2\right)\otimes U\left(1\right)$ using the Yang–Mills (Y-M) equations for Abelian fields. A system of second-order partial differential equations (PDEs) for the vector wave function of a photon is derived using the first-order Y-M equations as identities. The full wave function of a photon was defined as the arithmetic mean of the components of the wave function. In a particular case, an equation is obtained for its full wave function, taking into account the structure of space-time in a plane perpendicular to the direction of propagation of the photon. The quantum state of a photon in a nanowaveguide was investigated, and it is shown that under certain conditions, it is reduced to the problem of two coupled 1D quantum harmonic oscillators (QHO) with variable frequencies. An explicit expression is obtained for the wave function of a photon, which is characterized by two vibrational quantum numbers. A quantum theory of a photon for a dissipative medium has been developed taking into account the processes of absorption and emission of photons. The mathematical expectation (ME) of the photon wave function is constructed as the product of two 2D integral representations in which the integrand is the solution of a system of two coupled second-order PDEs. The ME of the probability amplitude of the transition of a single-photon state into one of the two-photon entangled Bell states is constructed. Finally, it was proven that, in addition to frequency, spin, momentum and polarization, the photon also has a spatial structure responsible for the cross sections of processes in which this massless fundamental particle participates. Full article

In this article, a Fokker–Planck equation framework for the copula density associated with a two-dimensional stochastic differential equations system is developed. The different information pieces associated with the statistical interdependence properties and the marginal ones are separated explicitly, and the corresponding boundary conditions for the copula distribution are analyzed. Given the set of functions that defines the copula density dynamics and the marginal probability density functions, a Fokker-Planck equation for the multivariate density probability function of the stochastic volatility model is obtained. Full article

Underwater wireless optical communication (UWOC) for autonomous underwater vehicles is severely compromised by the coupling of oceanic optical turbulence and platform motion. Traditional static statistical models fail to capture the temporal evolution of these stochastic processes, hindering effective real-time beam tracking. This paper proposes a unified dynamic framework and a hybrid intelligent control strategy to address beam misalignment in turbulent environments. First, a physically motivated stochastic differential equation (SDE) model is derived from the Radiative Transfer Equation via diffusion approximation. Validated by an inverse Fokker–Planck approach, this model accurately reconstructs drift fields for diverse channel conditions, serving as a dynamic generator for time-varying fading. Second, to maintain robust link alignment, a hybrid Fuzzy-Reinforcement Learning control strategy is developed. This approach integrates the interpretability of fuzzy logic with the adaptive optimization of Q-learning, incorporating a supervisor mechanism to handle deep fading events. Numerical simulations and hardware-in-the-loop (HIL) experiments demonstrate the system’s efficacy. The proposed controller achieves a median alignment error of 3.64 mm and reduces transient errors by over 80% compared to classical PID controllers during signal recovery. These results confirm that the proposed framework significantly enhances link stability and tracking robustness for AUVs in complex random media. Full article

This paper presents unified counterexamples for which standard elliptic regularity results break down for linear elliptic equations with highly singular coefficients in dimensions $d\ge 3$. First, we establish well-posedness for the case where the drift vector field has merely $L^2$-integrability but can be expressed as the gradient of a bounded potential function. Subsequently, we investigate the critical endpoint cases of known regularity results where coefficients or data satisfy borderline integrability conditions. By using a single, explicit function, $\rho \left(x\right)=ln(2+{\displaystyle \frac{1}{\parallel x\parallel }})$, we present counterexamples to the regularity of solutions for divergence form equations and stationary Fokker–Planck equations. Full article

Cells adapt their phenotypes in noisy microenvironments while maintaining robust decision-making. We develop a coarse-grained theoretical framework in which cellular phenotypic adaptation is described as Bayesian decision-making coupled to replication and diffusion. This leads to an effective Fokker-Planck equation with an emergent fitness landscape governing phenotypic dynamics. We identify distinct phenotypic regimes homeostatic fixation, bistable decision-making, critical switching, and runaway explosion and propose a biological interpretation in which homeostatic and bistable landscapes correspond to healthy differentiated cell states, whereas explosive landscapes capture stem-like or cancer-like behavior. In the Gaussian setting, the correlation between intrinsic and extrinsic states directly encodes mutual information and acts as a bifurcation parameter: high correlation produces shallow or explosive landscapes associated with phenotypic plasticity, while reduced correlation stabilizes differentiated fates by deepening potential wells. We further show that proliferation reshapes these landscapes in a nontrivial manner. Proliferation conditionally stabilizes local homeostasis without altering global confinement, or cooperates with biased environmental sensing to eliminate homeostasis/bistability and drive cancer-like phenotypic explosion even at high phenotypic fidelity. Finally, we show that negative intrinsic–extrinsic correlations suppress explosive dynamics but also reduce bistable plasticity, suggesting a robustness–plasticity trade-off. Together, our results suggest that development, tissue homeostasis, and carcinogenesis can be understood as information-driven deformations of a Bayesian phenotypic fitness landscape. Full article