arXiv subjects

Louis Shuo Wang

Publications and source records attributed to Louis Shuo Wang.

At least 37 records · Page 2Linked to original sources

arXiv2026

A Measure-Valued Obstacle Problem for an Obliquely Reflected Diffusion with a Max-Type Payoff

We study an obliquely reflected optimal stopping problem in the nonnegative quadrant with nonsmooth max-type payoff $G(x)=x_1\veeαx_2$, and we develop a measure-valued potential-theoretic formulation of the associated obstacle problem. The kink of $G$ on the diagonal $x_1=αx_2$ produces a singular surface measure in the distributional generator, while the oblique reflection directions generate boundary local-time contributions on the coordinate faces. Together with the absolutely continuous stopping gain, these terms define a total signed stopping measure $\Gtot$. We derive the corresponding reflected Itô--Tanaka identity, prove a killed-resolvent representation of the value function in the continuation region, and show that the unrestricted reflected resolvent is generally incorrect because the process is not absorbed on the stopping set. The free boundary is formulated through a continuation-side trace condition for the killed potential. Under a vertical monotonicity hypothesis on $V-G$, the stopping set is shown to have an epigraph form. We finally prove a verification theorem: any admissible epigraph candidate satisfying contact, strict continuation, reflected Neumann compatibility, growth, the trace condition, and measure-superharmonicity coincides with the value function, and its first entry time is optimal.

math.AP↗

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

Time and Killed Resolvents in Reflected Optimal Stopping with a Max Payoff

We study infinite-horizon optimal stopping for normally reflected two-dimensional diffusions in the positive quadrant with max payoff $G(x_1,x_2)=x_1\veeαx_2$. The non-smooth payoff produces a singular stopping-gain measure on the kink set $Δ=\{x_1=αx_2\}$. We prove $\displaystyle Γ^Δ(dx) = -\frac{n^\top a(x)n}{2\sqrt{1+α^2}}\,σ_Δ(dx)$, with $n=(1,-α)$, so the diagonal component is non-positive and strictly negative under local ellipticity. This implies that every interior kink point lies in the continuation region. We further show that the correct value representation uses the resolvent killed at first entry into the stopping set, $\displaystyle V=G-R_r^{\mathcal C}Γ$, and give a closed-form reflected Brownian counter-example showing that the unrestricted reflected resolvent is generally wrong. A reflected Brownian benchmark and numerical experiments illustrate the local-time, resolvent-gap, and diagonal-avoidance mechanisms.

math.AP↗

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

Elliptic criticality versus Volterra memory in indirect chemotaxis cascades

Indirect signal production is often treated as a higher-order variant of classical Keller-Segel chemotaxis, but its critical structure depends strongly on how the signal cascade is closed. This paper separates two asymptotic regimes of a two-stage signalling mechanism. In the parabolic--elliptic--elliptic limit, the chemoattractant is generated by the self-adjoint fourth-order operator $K_τ=(I-τΔ)^{-1}(I-Δ)^{-1}$. We prove its spectral positivity, entropy-dissipation structure, fourth-order principal scaling, and logarithmic kernel singularity in four dimensions. Consequently, the correct critical space is $L^{N/4}$, and $N=4$ is the mass-critical dimension. A concentration calculation identifies the natural threshold candidate $M_* = 64π^2τ/χ$, while the sharp threshold theorem is formulated as an Adams/logarithmic-HLS open problem. In contrast, the mixed elliptic--parabolic cascade cannot be reduced to a static fourth-order kernel. Its eliminated signal is a Volterra memory operator whose near-diagonal multiplier has the same order as the classical Keller-Segel drift. Thus its critical theory must be based on mixed space--time estimates, not static elliptic scaling. Numerical experiments support our operator-level distinction.

math.AP↗

arXiv2026

Finite-sheeted Cauchy operator at rational corners

We study Cauchy singular integral operators on planar wedges whose opening angle is a rational multiple of $π$. For $θ=pπ/q$, the covering $w=ζ^q$ yields an exact finite-sheeted factorization of the wedge Cauchy transform into $2q$ interval Cauchy transforms with explicit algebraic recombination coefficients. The factorization is formulated on weighted conormal Hölder spaces. We prove that the lifting operator preserves conormal order, lowers the Hölder exponent from $β$ to $β/q$, and has sharp $\ell^1$ sheet norm $q$. Combining this operator factorization with a Mellin model for interval Cauchy transforms, we derive a mode-by-mode propagation rule for polyhomogeneous endpoint expansions. Nonresonant powers preserve their logarithmic order, while integer exponents raise it by one. The results also give a local singular decomposition for Cauchy operators on piecewise analytic curves with rational corner angles.

math.FA↗

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

Introduction to Mathematical Programming with Equilibrium Constraints (MPECs) and Bilevel Optimization

Our aim is to explain mathematical programs with equilibrium constraints (MPECs), motivate them through applications, present the main equivalent formulations of equilibrium constraints, and summarize the basic existence theory for optimal solutions. The central message is that an MPEC is an optimization problem whose feasible set is partly defined by another optimization, variational inequality, complementarity system, or equilibrium model.

math.OC↗

arXiv2026

Introduction to Exact Penalization for Mathematical Programming with Equilibrium Constraints

We present a focused introduction to exact penalty methods for nonlinear programs and mathematical programs with equilibrium constraints (MPECs), emphasizing their connection to modern error bound theory. The goal is twofold. First, we explain how classical optimality conditions can be interpreted through exact penalization, and why such results typically rely on constraint regularity conditions that can be understood as error bounds on perturbations of feasible sets. We then highlight how recent developments based on subanalytic geometry and Lojasiewicz-type inequalities extend this framework beyond classical regularity assumptions, enabling exact penalization under broader analytic conditions. Second, we demonstrate how this theory can be applied in practice to MPECs by reformulating them via KKT systems and constructing exact penalty functions based on residual mappings. Particular attention is given to fractional-order penalties arising from Lojasiewicz error bounds, as well as to improved formulations for special problem classes where sharper exponents can be obtained. These developments provide both theoretical insight and practical guidance for analyzing and solving challenging constrained optimization problems.

math.OC↗

arXiv2026

First-Order Optimality Conditions for Mathematical Programming with Equilibrium Constraints

We present a systematic introduction to first-order optimality conditions for mathematical programs with equilibrium constraints (MPECs), emphasizing the limitations of classical nonlinear programming techniques. The goal is twofold. First, we explain why a direct application of standard optimality conditions -- based on reformulating MPECs via KKT systems or differentiable exact penalty functions -- is often inadequate, as such approaches typically require strong and restrictive assumptions, including nondegeneracy and smoothness conditions. Second, we develop a first-principles framework for analyzing MPECs by focusing on the geometric structure of the feasible region. In particular, we study stationarity concepts and provide a detailed characterization of the tangent cone at feasible points, which leads to appropriate constraint qualifications tailored to MPECs. These results form the foundation for rigorous first-order analysis and clarify the relationship between the original MPEC formulation and its KKT-based representation, offering practical guidance for handling these inherently challenging optimization problems.

math.OC↗

arXiv2026

Lecture Note for Bounded Controls in Continuous-Time and Control of Several Variables

In this note, we develop the first-order theory of optimal control problems with box constraints on the control. We emphasize the precise modification of Pontryagin's maximum principle when the admissible control set is compact, the projection/clamping formula for scalar quadratic Hamiltonians, the distinction between intrinsic projection inside the optimality system and post hoc truncation of an unconstrained solution, and the corresponding forward-backward sweep implementation. The presentation is pitched at senior PhD students who are already comfortable with variational arguments, adjoint systems, and basic nonlinear analysis. These notes are mainly based on the book ``optimal control applied to biological models'' of Suzanne Lenhart and John T. Workman.

math.OC↗

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↗