arXiv subjects

Jiguang Yu

Publications and source records attributed to Jiguang Yu.

35 records · Page 2Linked to original sources

arXiv2026

Killed resolvents and measure-valued stopping gains for reflected optimal stopping with max-type rewards

We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the positive quadrant with nonsmooth max-type reward $G(x_1,x_2)=x_1\vee αx_2$. The paper develops a conditional measure-theoretic framework for the associated reflected obstacle problem. The main innovation is to show that the stopping gain $Γ=c+rG-\mathcal LG$ is a signed measure, not a function: the kink of $G$ generates an explicit negative surface measure on $Δ=\{x_1=αx_2\}$. We then prove that the correct potential representation uses the resolvent of the reflected diffusion killed on first entry into the stopping set, rather than the unrestricted reflected resolvent. Under explicit monotonicity, regularity, and measure-superharmonicity assumptions, we derive an epigraph representation, a continuation-side boundary-trace condition, and a candidate verification theorem. The framework clarifies hidden regularity and uniqueness assumptions in multidimensional nonsmooth optimal stopping.

math.PR↗

arXiv2026

Spectral perturbation theory for wall-admittance effects on compressible boundary-layer instability

Thin wall treatments modify high-speed boundary-layer instability through the pressure they admit or absorb at the wall. This paper develops a unified admittance formulation for such effects on trapped compressible Rayleigh modes. For a simple rigid-wall eigenpair, we prove the spectral sensitivity law \[ c(A)=c_0+KA+\mathcal O(|A|^2), \qquad δσ=α\Imag(KA)+\mathcal O(|A|^2), \] where $A$ is the wall admittance and $K$ is an explicit functional of the rigid-wall eigenfunction. The formula separates wall physics from outer-mode physics and yields a phase criterion for stabilisation. Matched asymptotics show that viscous and thermal wall layers, blind-pore coatings and shallow non-separating roughness all reduce to this same boundary condition, with additive leading admittances. Mach-4.5 computations validate the sensitivity coefficient and demonstrate porous damping, viscous-wall damping and sign-changing reactive roughness effects.

physics.flu-dyn↗

arXiv2026

Structural dichotomy and mass criticality in indirect chemotaxis cascades: fourth-order ellipticity versus Volterra memory

We investigate the structural and operator-theoretic foundations of indirect signal-generation mechanisms in Keller--Segel-type chemotaxis models. By analyzing a physically motivated multi-stage signaling cascade, we establish a precise mathematical dichotomy between instantaneous equilibration and transient kinetic memory. Specifically, we prove that the fully equilibrating parabolic--elliptic--elliptic (PES) cascade reduces to a static fourth-order elliptic interaction. In dimension four, an exact algebraic cancellation of the leading Newtonian singularity yields a purely logarithmic kernel, shifting the mass-critical dimension from $N=2$ to $N=4$. Through an $L^2$-gradient flow formulation, we identify the corresponding concentration-scaling candidate critical mass $M_* = 64π^2τ/χ$. In sharp contrast, we demonstrate that the mixed elliptic--parabolic (MEP) cascade retains a genuine Volterra memory effect that defies static reduction. Its interaction drift acts as a singular perturbation in time-exhibiting classical two-dimensional Keller--Segel principal order near the time diagonal, yet providing fourth-order smoothing in its frozen-time average, necessitating a mixed space-time threshold theory. These results isolate the physical origin of mass-critical dimensional shifts in multiscale biological systems and formulate the specific adapted Adams/logarithmic Hardy--Littlewood--Sobolev (log-HLS) inequalities and mixed-norm criteria required to close the threshold problems.

math.AP↗

arXiv2026

A Diagnostic Framework for Implementation Risk in Bilevel Decision Problems: The Ambiguity Premium and the Robustness--Efficiency Frontier

Hierarchical decision problems are often modeled as bilevel programs in which a leader commits to a policy and a follower responds optimally. When the follower's optimal response is nonunique, or when only near-optimal follower behavior can be verified, the same leader decision may induce a range of upper-level outcomes. This paper develops a diagnostic framework for quantifying that exposure. For a leader decision $x$, we evaluate the optimistic and pessimistic upper-level values over the $ε$-optimal follower response set $S_ε(x)$ and use their difference, \[ Δ_ε(x):=ψ_ε^p(x)-ψ_ε^o(x), \] as an ambiguity premium. The premium itself is classical in the optimistic--pessimistic bilevel distinction; the contribution here is to make it operational as an implementation-risk diagnostic. We establish a diameter bound $Δ_ε(x)\le L_F(x)\,\mathrm{diam}(S_ε(x))$ and an $\mathcal{O}(\sqrtε)$ estimate under quadratic lower-level growth. We then organize existing bilevel--GNEP reformulations by their computational roles and propose a screening workflow that reports, for each candidate policy, nominal value, ambiguity exposure, and a first-order residual. Two stylized case studies -- a parallel-link Stackelberg pricing problem and a convex generation-planning model with diversification constraints -- show how the resulting robustness--efficiency frontier can identify policies that are nominally attractive but sensitive to near-optimal follower responses.

math.OC↗

arXiv2026

Controlling the Swarm: Sparse Actuation and Collision Avoidance under Stochastic Delay

Classical flocking models demonstrate how local interactions generate emergent order, but real-world multi-agent deployments are bound by severe constraints: limited actuator availability, heterogeneous communication latencies, and environmental noise. In this talk, we present a unified finite-N framework that tackles the interplay of these exact mechanisms. We study a delayed stochastic leader-follower particle system featuring topological communication, singular repulsion, and bounded sparse leader actuation. A central challenge in such systems is mathematical well-posedness, as discontinuous communication laws and singular repulsions clash with standard strong Ito frameworks. We resolve this by introducing an augmented Lyapunov functional that simultaneously enforces a strict collision barrier and closes a uniform Gronwall estimate. Building on this rigorous foundation, we formulate a free-terminal-time, chance-constrained optimal control problem. We show that temporally sparse, bang-off-bang leader actuation not only drastically reduces control effort compared to continuous baselines, but also reveals non-monotone sensitivities to leader density. Ultimately, we demonstrate that in delayed stochastic swarms, adding more direct actuation is not strictly optimal -- highlighting a highly non-trivial resource allocation paradox in cooperative control.

math.OC↗

arXiv2026

Optimization Workshop Notes for Mathematical Programming with Equilibrium Constraints (MPECs): Verification of MPEC Hypotheses

In this workshop, we present a compact but rigorous introduction to the basic language of nonlinear programming, variational inequalities, and complementarity systems. The goal is twofold. First, we explain the mathematical logic of hypotheses under which first-order optimality conditions for MPECs become valid. Second, we explain how to use that theory in research practice: how to classify a model, choose the appropriate verification route, prove the right hypotheses, and derive a correct first-order analysis.

math.OC↗

arXiv2026

Optimization Workshop Notes for Mathematical Programming with Equilibrium Constraints (MPECs): Second-Order Optimality Conditions

In this workshop, we present a compact but rigorous introduction to second-order optimality conditions for mathematical programs with equilibrium constraints (MPECs). We start from the classical nonlinear programming template, then explain why it fails in the equilibrium-constrained setting, and develop the three main viewpoints used in the literature: (i) multiplier-based conditions, (ii) implicit-programming conditions based on the solution map of the lower-level equilibrium system, and (iii) piecewise-programming conditions obtained by decomposing complementarity structure into smooth pieces. The emphasis is on conceptual structure, critical cones, strong regularity, and the exact role of curvature terms.

math.OC↗

arXiv2026

Optimization Workshop Notes for Mathematical Programming with Equilibrium Constraints Algorithms: Penalty Interior-Point, Implicit-Programming, and Piecewise SQP

In this workshop, we discuss several algorithms for mathematical programs with equilibrium constraints (MPECs). The unifying theme is that MPECs are optimization problems whose feasible set contains a lower-level equilibrium system, often written through complementarity or variational-inequality conditions. This destroys the smooth manifold or convex structure that standard nonlinear programming methods rely on. We focus on four algorithmic viewpoints: (i) the classical penalty interior-point algorithm (PIPA); (ii) a monotone-linear complementarity problem (LCP) variant of PIPA that explicitly controls complementarity decay; (iii) an implicit-programming descent method for variational inequality (VI)-constrained MPECs; (iv) piecewise SQP (PSQP), which applies SQP on locally selected smooth pieces. For each method we explain the model, the search direction subproblem, the globalization mechanism, and the meaning of the convergence result. Particular emphasis is placed on what the assumptions are really doing and on the distinction between an attractive algorithmic idea and a fully valid convergence theorem.

math.OC↗

arXiv2026

Physics-grounded Mechanism Design for Spectrum Sharing between Passive and Active Users

We propose a physics-grounded mechanism design for dynamic spectrum sharing that bridges the gap between radiometric retrieval constraints and economic incentives. We formulate the active and passive users coexistence problem as a Vickrey-Clarke-Groves (VCG) auctions mechanism, where the radiometer dynamically procures ``quiet'' time-frequency tiles from active users based on the marginal reduction in retrieval error variance. This approach ensures allocative efficiency and dominant-strategy incentive compatibility (DSIC). To overcome the computational intractability of exact VCG on large grids, we derive an approximation algorithm by using the monotone submodularity induced by the radiometer equation. AMSR-2-based simulations show that the approach avoids high-cost tiles by aggregating low-cost spectrum across time and frequency. In an interference-trap case study, the proposed framework reduces procurement costs by about 60% over a fixed-band baseline while satisfying accuracy targets.

math.AP↗

arXiv2026

From Microscopic Damage to Macroscopic Games: A Dimensionality Reduction of Stem Cell Homeostasis

Tissues must maintain macroscopic homeostasis despite the continuous microscopic accumulation of cellular damage. Theoretical models of this process often suffer from a disconnect between microscopic biophysics and macroscopic phenomenological games. Here, we bridge this gap by deriving an exact dimensionality reduction of a physiologically structured partial differential equation (PDE) into a low-dimensional dynamical system. Under the condition of uniform mortality, we mathematically demonstrate that tissue homeostasis operates as an induced Nash equilibrium, where the per-capita net growth rates of stem and differentiated phenotypes perfectly equalize. This reduction yields closed-form algebraic rules, the Ratio and Equalization Laws, that map continuous microscopic state dynamics to measurable macroscopic observables. To demonstrate the biological utility of this framework, we present a concrete, falsifiable case study of the murine intestinal crypt. By modeling crypt regeneration following irradiation-induced stem cell depletion, our framework successfully recovers the experimentally observed reliance on progenitor dedifferentiation. Furthermore, the model generates explicit, testable predictions, enabling the in vivo estimation of hard-to-measure lineage plasticity rates directly from aggregate static cell counts. This work provides a rigorous, predictive mathematical foundation for understanding how fast-renewing tissues filter microscopic noise to sustain macroscopic regenerative capacity.

math.AP↗

arXiv2026

The Geometry of Quasi-Cycles: How Stoichiometric Covariance Alters Pre-Bifurcation Signatures

Environmental enrichment can destabilize predator--prey coexistence through a Hopf bifurcation, yet real ecosystems are finite and intrinsically stochastic. We investigate how mechanistically derived demographic noise shapes near-Hopf dynamics in the Rosenzweig--MacArthur model by systematically comparing two diffusion closures that share identical deterministic drift but differ solely in predation-induced covariance structure. Starting from a continuous-time Markov chain description, we derive a full-covariance stochastic differential equation whose diffusion tensor inherits stoichiometric coupling, generating a negative prey--predator cross-covariance. This model is contrasted with a drift-matched diagonal-noise comparator. Using linear noise approximation, Lyapunov analysis, and matrix-valued power spectral density formulations, we propagate local covariance structure through the entire diagnostic chain, including stochastic sensitivity ellipses and a dimensionless noisy-precursor indicator. The results highlight that drift equivalence does not imply covariance equivalence and show how event-level noise geometry influences macroscopic behavior in nonlinear ecological systems. This work integrates bifurcation theory and stochastic analysis to advance multi-scale modeling of complex interacting systems.

math.DS↗

arXiv2026

Beyond Diagonal Noise: A Better Predator-Prey Modeling Framework with Cross-Covariance

The introduction of stochasticity into continuous ecological models frequently relies on phenomenological, diagonal diffusion terms that lack a rigorous microscopic basis. We demonstrate that this standard practice fundamentally misrepresents the geometry of demographic fluctuations. By deriving a stochastic Rosenzweig--MacArthur model directly from an integer-valued, Bernoulli-coupled continuous-time Markov chain, we isolate the exact diffusion covariance structure dictated by event stoichiometry. We mathematically prove that coupled predation--conversion events inherently generate a structurally negative predator--prey cross-covariance, exposing the severe mathematical and biological limitations of standard diagonal-noise approximations. Furthermore, we resolve a persistent ambiguity in stochastic population modeling by explicitly formalizing the bifurcation between open-domain formulations (for survival-conditioned interior dynamics) and absorbed formulations (for extinction-permitting dynamics). To rigorously support this distinction, we develop a tailored two-stage Lyapunov well-posedness architecture that separates non-explosion criteria from boundary-barrier positivity invariance. By bridging microscopic event stoichiometry with macroscopic boundary-degenerate diffusions, this work replaces ad hoc noise constructs with a definitive, mathematically exact template for covariance-consistent and boundary-aware ecological modeling.

q-bio.PE↗

arXiv2026

Full-Covariance Chemical Langevin Predator--Prey Diffusion with Absorbing Boundaries

Many stochastic Rosenzweig--MacArthur predator--prey models inject ad hoc independent (diagonal) noise and therefore cannot encode the event-level coupling created by predation and biomass conversion. We derive an absorbed, fully mechanistic diffusion approximation and its extinction structure from a continuous-time Markov chain on $\mathbb N_0^2$ with four reaction channels: prey birth, prey competition death, predator death, and a coupled predation--conversion event. Absorbing coordinate axes are imposed to represent the irreversibility of demographic extinction. Under Kurtz density-dependent scaling, the law-of-large-numbers limit recovers the classical RM ODE, while central-limit scaling yields a chemical-Langevin diffusion with explicit drift and full state-dependent covariance. A distinctive signature is the strictly negative cross-covariance $Σ_{12}(N,P)=-mNP/(1+N)$ induced solely by the predation--conversion increment $(-1,1)$. We define the absorbed Itô SDE by freezing trajectories at the first boundary hit and prove strong well-posedness, non-explosion, and moment bounds up to absorption. Extinction has positive probability from every interior state, and predator extinction is almost sure when $m\le c$.

math.PR↗

arXiv2026

Mode-Wise Spectral Criteria for Coupled Mass Transport in Hybrid PDE--ODE Tumor Microenvironments

We study coupled mass transport in a tumor--microenvironment setting with two motile densities $(S,R)$ and non-motile state switching $(P,A)$. The populations diffuse and undergo chemotactic drift; $(P,A)$ follow pointwise ODE switching. A decoupled inhibitory field $D$ satisfies a damped Neumann heat equation, giving maximum-principle bounds and exponential decay. Together with the pointwise invariant $P+A$, these identities yield global existence, positivity, and long-time reduction to limiting $(S,R)$ kinetics with a unique globally attracting coexistence state. Neumann eigenmode reduction gives closed dispersion relations. The base $(S,R)$ reaction--diffusion block remains stable for all nonconstant modes for any $d_S,d_R>0$, excluding classical Turing destabilization. Chemotaxis is posed via a diffusive cue $c$, since $\nabla A$ is undefined for non-diffusive $A$. In one-way damped coupling, the linearized mode matrix is block triangular and leaves the $(S,R)$ spectrum unchanged. Two-way coupling adds a feedback rank-one mobility correction, induces effective cross-diffusion, and admits mode growth. We give explicit trace/determinant criteria for unstable Laplacian modes and the resulting instability thresholds.

math.AP↗

arXiv2026

Dedifferentiation stabilizes stem cell lineages: From CTMC to diffusion theory and thresholds

We study stem-terminally differentiated (TD) lineages in small niches where demographic noise from discrete division and death events is non-negligible. Starting from a mechanistic five-channel, density-dependent CTMC (symmetric self-renewal, symmetric differentiation, asymmetric division, dedifferentiation, TD death), we derive its mean-field limit and a functional CLT, obtaining a chemical Langevin diffusion whose explicit state-dependent covariance exactly matches the CTMC's aggregated channel-wise infinitesimal covariances. Within this diffusion approximation we remove the dedifferentiation flux and obtain a sharp dichotomy: in subcritical regimes the stem coordinate becomes extinct asymptotically almost surely, whereas in supercritical regimes polynomial moments diverge exponentially. This identifies, at the diffusion level, a structural failure mode of strictly hierarchical lineages under demographic fluctuations and clarifies how a cyclic return flux can rescue homeostasis. For interpretation we also derive an exact totals ODE backbone from a damage-structured transport model and obtain two steady-state constraints (ratio and equalization laws) linking compartment ratios to turnover and balancing dedifferentiation against fate bias. Numerical experiments corroborate the $Ω^{-1/2}$ fluctuation scaling, illustrate the pathology, and contrast theorem-regime global convergence with threshold (Allee-type) behaviour outside the theorem hypotheses.

physics.bio-ph↗

arXiv2025

A hybrid PDE-ABM model for angiogenesis and tumour microenvironment with application to resistance in cancer treatment

The main obstacle to effective cancer treatment is the development of drug resistance, which can be divided into two categories: spontaneous and acquired drug resistance. Non-small cell lung cancer (NSCLC) is the main cause of cancer-related deaths worldwide. A subset of lung cancer, adenocarcinomas, is characterised by mutations in the epidermal growth factor receptor (EGFR) gene. Treatment of EGFR-mutated lung adenocarcinomas has become less effective over time due to drug resistance development, which is associated with a second mutation in the EGFR gene. An important factor in the development of cancer is angiogenesis, which is the formation of blood vessels from the existing vasculature. These newly formed blood vessels provide oxygen and nutrients to tumour cells to maintain tumour growth and proliferation. We applied a hybrid discrete-continuous (HDC) model to capture the dynamic vasculature in the tumour microenvironment (TME). In the case of pre-existing resistance, the formation of angiogenic networks creates a microenvironment that supports tumour survival and enhances drug resistance. In the case of spontaneous mutation-induced resistance, earlier and more frequent mutations confer a greater survival advantage to the tumour population. There is also a mutually reinforcing relationship between a high proliferation rate and high resistance characteristics. These findings explain two conflicting experimental results about the second mutation in NSCLC.

math.NA↗

arXiv2025

Well-Posedness for the Rosenzweig-MacArthur Model with Internal Stochasticity

In this work, we propose a stochastic version of the Rosenzweig-MacArthur model solely driven by internal demographic noise, extending classical Lotka-Volterra-type systems focused on external noise. We give a criterion for the existence and uniqueness of autonomous stochastic differential equations (SDEs) on an open submanifold of $\mathbb{R}^{n}$, and the framework allows for a wider choice of Lyapunov functions. In the meantime, the invariance of open submanifolds, which is a biologically feasible result and has been implicitly incorporated into many biological and ecological models, facilitates the application of analytic tools typically suited to $\mathbb{R}^{d}$ and indicates the persistence of predator and prey populations, thus providing a criterion for determining whether a population will become extinct. We apply the well-posedness criterion to our stochastic Rosenzweig-MacArthur model and show the existence and uniqueness of solutions. Furthermore, the asymptotic estimates of solutions are obtained, indicating the at most exponential growth of the population with internal stochasticity. Some numerical experiments are performed, which illustrate the discrepancy between the deterministic and stochastic models. Overall, this work demonstrates the broad applicability of our results to ecological models with constrained dynamics, offering a foundation for analyzing extinction, persistence, and well-posedness in systems where internal randomness dominates. This paper not only promotes the development of stochastic modeling and stochastic differential equations in theoretical ecology but also proposes a rigorous mathematical methodology for studying the predator-prey system with internal stochasticity.

math.PR↗