Papers

In this paper, we are concerned with the existence and stability of temporal periodic solutions to a class of the Benjamin-Bona-Mahony (BBM) equation with damping in the torus $\mathbb{T}$, whose interior is applied with periodic force $f(x, t)$ with temporal period $\theta$. These types of solutions are established in $H^{\ell}, \ell\ge 0$, and it is noteworthy to see that the I-energy method is invoked when low regularity of $\ell\in[0, 1)$ is pursued.

We study the porous medium equation with a combustion-type reaction, \[ u_t=\Delta u^m+f(u),\qquad x\in\mathbb R^N,\ t>0, \] for radial, nonnegative, compactly supported initial data. A complete classification of the long-time behaviour of bounded solutions is established. In dimension $N=2$, every such solution converges locally uniformly to one of the constants $0$, $\theta$, or $1$; for ordered families of initial data there exists a unique threshold parameter separating vanishing from spreading, and the critical solution converges to the ignition temperature $\theta$. In dimensions $N\ge3$, under a natural total-disconnectedness condition on the set of central values of ground states, every bounded solution converges locally uniformly to either $0$, $\theta$, $1$, or a radial ground state $U\in\mathcal S$. Moreover, the ignition state $\theta$ is excluded as a transition limit in $N\ge3$ via a novel normalized annular perturbation argument. For transition solutions, we provide estimates for the propagation speed of the free boundary: in dimension two, \[ b(t)\asymp \frac{\sqrt t}{(\log t)^{\frac{m-1}{2m}}}, \] and in dimensions $N\ge3$, whenever the limit is a ground state, \[ b(t)\asymp t^{\frac{m}{N(m-1)+2}}. \] These results reveal a sharp dimensional dichotomy in the degenerate combustion dynamics.

At the critical Hardy Gaussian weight for the Schrödinger equation in one space dimension on $[0,1]$, the known nonzero scalar example in the weighted $L^2$ class carries a complex-valued potential. Cassano and Fanelli observed that the existence of a real-valued scalar endpoint example was open, and produced examples with real electric and magnetic potentials only after introducing a magnetic potential. We construct a nonzero smooth solution of $i\partial_t u+\partial_x^2 u=Vu$ on $\mathbb{R}\times[0,1]$ such that $e^{x^2/4}u(\cdot,0),e^{x^2/4}u(\cdot,1)\in L^2(\mathbb{R})$ and the potential is bounded, smooth, real-valued, and purely scalar and is supported in one fixed compact spatial interval for all times. This compact-support endpoint example is the main result. We also record the explicit rational-tail realization underlying the construction. The main results of this paper were obtained by the multi-agent system Eureka and have subsequently been verified by the authors.

In this paper, we establish an explicit boundary-corrected Li–Yau type gradient estimate for positive solutions of the Dirichlet heat equation on the Euclidean half-space. The main idea is to exploit the reflection structure of the Dirichlet heat kernel and introduce a normalized kernel-induced probability measure. Under this representation, logarithmic derivatives of the heat kernel become expectations of explicit kernel quantities. The reflected Gaussian component generates a hyperbolic correction term involving \[ \coth\left(\frac{x_n y_n}{2t}\right), \] which has no analogue in the whole Euclidean heat equation. Using Jensen's inequality and the sharp estimate \[ 0<z\,\csch z<1 , \qquad z>0, \] we prove that every positive solution satisfies \[ \Delta\log w(x,t) \geq -\frac n{2t} -\frac1{x_n^2}. \] The first term represents the classical Euclidean Li–Yau diffusion scaling, while the second term is an explicit inverse-square correction determined by the distance to the Dirichlet boundary. Our approach provides a direct kernel interpretation of the boundary effect and suggests possible extensions to more general domains.

We develop a rigorous variational framework for uniaxial nematic liquid crystals interacting with an external electric field in the one-constant Oseen–Frank approximation. Equilibrium configurations are governed by a nonlocal, nonlinear energy with a challenging min-max saddle-point structure. Our first main result reformulates this problem as a pure double-minimization problem. Using convex duality and the Hodge decomposition, we replace the scalar electrostatic potential with a vector potential, yielding a closed-form dual functional with a unique minimizer. This direct energy-minimization principle is advantageous for both theoretical analysis and numerical simulation. Our second result rigorously quantifies the decoupling of the electrostatic back-reaction in the limit of small dielectric anisotropy. We establish a uniform quadratic energy bound between the exact nonlocal energy and its standard local approximation, formally justifying the widespread physics heuristic of neglecting induced depolarization fields. Finally, under a strict coercivity condition, we combine $\Gamma$-convergence, uniform Sobolev regularity, and a perturbative coercivity transfer to prove that physical minimizers converge to limiting harmonic maps at an optimal, quantitative linear rate.

We introduce and study a pseudo-differential operator $W^\alpha$, defined via the $p$-adic Mellin transformation, acting on real- or complex-valued functions on the multiplicative group of the field of $p$-adic numbers. The operator $W^\alpha$ is maximally dissipative. An estimate of its heat kernel is given.

We establish an exact representation formula for the deficit in Hölder's inequality. Rather than estimating the deficit by auxiliary quantities, we show that it can be represented as an accumulated variance along the natural exponential interpolation connecting the two endpoint densities. More precisely, the logarithmic Hölder deficit is expressed as the integral of the variance of the logarithmic density ratio weighted by the Green kernel associated with the one-dimensional interpolation parameter. This identity reveals that Hölder's inequality is a consequence of the convexity of a logarithmic partition function and provides an intrinsic interpretation of the deficit as an interpolation energy. The representation suggests a broader framework for studying functional inequalities through variance identities.

On a locally finite connected graph $G=(V,E)$, using the first eigenvalue method introduced by Kaplan \cite{MR160044} and the discrete Phragmén-Lindelöf principle developed by Hu-Wang \cite{cvhuyuanyang}, we first establish the asymptotic behaviour of the lifespan of positive solutions to a semilinear heat equation with the power-logarithmic nonlinearity $u^p|\log u|^q$, provided that the initial datum is bounded below by a positive constant. These results extend those of Hu-Wang \cite{cvhuyuanyang} to equations with a power-logarithmic source term. Moreover, by means of a more direct argument, we show that analogous lifespan estimates remain valid for nonnegative initial datum $u(x,0)$, provided that $u(x_i,0)$ is suitably large at some vertex $x_i\in V$.

A seminal work by Dacorogna and Moser introduced a way of constructing regular transport maps from a probability distribution on a bounded domain to another one. In this work, we extend this construction to the whole $\mathbb{R}^d$ for strictly asymptotically log-concave measures, a wide class of distributions that encompasses Lipschitz-perturbations of log-concave measures. We then leverage this construction to study geodesics in the space of probability measures on a product set with imposed marginal laws (couplings), for which we derive optimality conditions, answering an open question in a recent work by Conforti, Lacker and Pal. Taking inspiration from Brenier's variational model for incompressible fluids and its regularization, we further introduce an entropic regularization of the geodesic problem, which can be seen as the Schrödinger bridge problem on the space of couplings, for which we also derive optimality conditions. We eventually study convergence of minimizers as the regularization vanishes, and we prove convergence of the related Lagrange multipliers. Our approach involves proving uniform-in-time global regularity estimates on elliptic and parabolic equations on $\mathbb{R}^d$, by exploiting the structure of asymptotically log-concave measures using the probabilistic notion of reflection coupling.

The well-posedness theory of very weak solutions is a central topic in mathematical hydrodynamics, especially in the regularity theory for Navier-Stokes equations. It has been fully developed for incompressible fluid flows on bounded domains in R^3 of C^{2,1}-regularity. In this paper, based on the analytic theories in [D. Breit and A. Gaudin, ArXiv Preprint: 2511.19091 (2025)] and [V.G. Maz'ya and T.O. Shaposhnikova, Vol.337, Grundlehren der mathematischen Wissenschaften (2009)], we establish the well-posedness theory of very weak solutions to the Navier-Stokes equations on bounded Lipschitz domains whose boundary has local graphing functions with sufficiently small Sobolev multiplier norm, which contain the bounded Lipschitz domains with sufficiently small Lipschitz constants as a special case.

We advance the general theory of flows of tensorial $\mathrm{H}$-structures, focusing on non-isometric flows and on the case $\mathrm{H}=\mathrm{SU}(m)\subset\mathrm{SO}(2m)$. After developing the relevant $\mathrm{SU}(m)$ algebra, we compare two natural evolutions: the unrestricted negative gradient flow of the intrinsic-torsion energy and a Ricci-harmonic flow. We prove short-time existence and uniqueness for the Ricci-harmonic $\mathrm{H}$-flow, with arbitrary lower-order torsion-quadratic terms, for every closed subgroup $\mathrm{H}\subset\mathrm{SO}(n)$. For groups for which the projection to $\mathfrak{h}^\perp$ defines a $4$-form, including $\{1\}$, $\mathrm{SU}(2)$, $\mathrm{G}_2$, and $\mathrm{Spin}(7)$, we express the negative gradient flow in Ricci-harmonic form up to explicit lower-order torsion terms and prove short-time existence and uniqueness by a modified DeTurck argument. We treat the genuinely different $\mathrm{SU}(m)$ case by a separate principal-symbol computation, proving short-time existence and uniqueness for the unrestricted negative gradient flow of $\mathrm{SU}(m)$-structures. The same computation identifies the natural negative gradient flow of $\mathrm{U}(m)$-structures as a borderline case, which cannot be made strictly parabolic by first-order diffeomorphism gauges. For the modified Ricci-harmonic flow, we derive heat-type evolution equations for the intrinsic torsion, a doubling-time estimate and Shi-type derivative estimates for $(|\mathrm{Rm}|^2+|\nabla T|^2+|T|^4)^{1/2}$, and a finite-time continuation criterion. In dimension six, we translate the formalism into the standard torsion forms of an $\mathrm{SU}(3)$-structure and describe, to highest order, the corresponding family of second-order quasilinear $\mathrm{SU}(3)$-flows.

We study the Cauchy problem for the generalized porous medium equation on infinite weighted graphs. For a general nonlinearity, we establish Dirichlet comparison and weak maximum principles on finite subgraphs and, through an exhaustion argument, construct minimal and maximal pointwise solutions for arbitrary $\ell^\infty$ initial data, controlled by explicit, possibly time-dependent, barriers. For the porous nonlinearity $\phi(s)=s|s|^{m-1}$, assuming a $\nu$-Sobolev inequality with $\nu>2$, we derive quantitative energy estimates for $\ell^1$-mild solutions. These yield finite-time extinction in the fast diffusion range $0<m<2/\nu$ and $\ell^1$-$\ell^q$ smoothing in the range $m>2/\nu$. Interestingly, we recover the Euclidean critical exponent for several model graphs. Finally, we prove an exact generalized mass balance for nonnegative $\ell^1$-mild solutions on graphs that are stochastically complete at infinity, allowing for an arbitrary killing term. The same balance is established for suitable classical and bounded pointwise solutions. In the absence of killing, these reduce to conservation of mass.

We derive the $\mathcal{P}(\Phi)_2$ measure on the two-dimensional unit torus as the rigorous limit of many-body quantum Gibbs states. In particular, the quantum model corresponding to the $\Phi^{2p}_2$ measure is formulated in the grand-canonical ensemble with a general symmetric $p$-body interaction potential which is not required to have a factorized form. Unlike in the $\Phi^4_2$ case investigated by Fröhlich–Knowles–Schlein–Sohinger in \cite{FKSS25}, in the treatment of higher-order interactions, Wick renormalization generates a full hierarchy of lower-order interactions which we organize systematically using a graphical formalism. One key ingredient of our analysis is a logarithmic stability estimate for both the nonlocal Hartree functional and the many-body Hamiltonian that is uniform in the interaction range, allowing us to use the positivity of the leading $p$-body interaction to control all lower-order terms generated by the Wick counterterms.

We determine the sharp asymptotic behavior of the optimal constants in the discrete Hardy-Rellich inequalities on the lattice $\mathbb{Z}^d$. For every fixed integer $m\ge 1$, let ${\mathcal C}_{m,d}$ be the best constant in the $m$-th order Hardy-Rellich inequality. We prove that $$\lim_{d\to\infty}\frac{\mathcal C_{m,d}}{d^m}=2^m.$$ Our approach combines a Fourier reduction to weighted inequalities with the flat torus and general weighted Hardy-Rellich identities of first and second order. A key novelty is the use of probabilistic concentration estimates, specifically Hoeffding's inequality and entropy methods, to handle estimates involving the anisotropic weight $\omega^\gamma$ $(\gamma\geq 1)$ where $$\omega(x)=\sum_{j=1}^{d} \Big(\sin\frac{x_j}{2}\Big)^2$$ in a dimension-uniform manner. These tools yield asymptotically sharp weighted estimates on the torus, from which the lattice inequalities follow by iteration.

We study the high-temperature mean-field limit of grand-canonical bosonic Gibbs states on the torus with renormalized nonlocal three-body interactions. In dimensions two and three, we construct the limiting nonlinear classical Gibbs measure and prove convergence of the relative free energy and of the reduced density matrices of every fixed order; in three dimensions, a smallness condition on the interaction is imposed. The proof combines the density-channel representation of the interaction with a coherent-state variational method based on the upper-symbol representation of the free Gibbs state. The same framework also contains, as a special case, the homogeneous positive-type model studied by Lewin, Nam, and Rougerie (2021).

We study a parabolic-elliptic forager-exploiter model describing the cascade interaction between two species through a shared environmental resource with a constant renewal rate. We first establish the global existence and uniform boundedness of classical solutions in arbitrary dimensions {\color{black}for large initial population data and large taxis sensitivity coefficients. We then investigate the long-time behavior of solutions.} When the resource renewal rate is small, all solutions will converge exponentially to the unique constant steady state. Furthermore, for large resource renewal rates, the parabolic-elliptic system does not undergo Hopf bifurcation, which is in sharp contrast to the corresponding fully parabolic case, where temporal oscillations arise via Hopf bifurcation.

The Constantin-Iyer formula represents a smooth solution of the incompressible Navier-Stokes equations on $\mathbb{T}^3$ as $u=\mathbb{P}\,\mathbb{E}[(\nabla A_t)^\top(u_0\circ A_t)]$, the projected expectation of a stochastic Weber field. We separate this expectation into the product of the means and a centred covariance, and show that the covariance – together with the second moment of the inverse deformation gradient and the two-point covariance – satisfies a closed deterministic advection-diffusion equation: because every Lagrangian label is driven by the same Brownian path, the quadratic covariation collapses into a Laplacian and no moment hierarchy appears. From these equations we obtain an energy/production balance, pointwise and Holder covariance bounds, and an exact degeneracy of the two-point operator in the separation variable. A second group of results concerns the mean displacement, which solves a forced heat equation along the flow; its $L^2$ norm is controlled unconditionally by the initial energy, while an exact family of Navier-Stokes shear flows shows that no analogous Holder bound can hold uniformly down to $t=0$. We further isolate several obstructions, each backed by an explicit counterexample – the norm of an expected deformation gradient does not control the expected norm, even for genuine stochastic flows of smooth divergence-free drifts; a spatial Holder bound on the mean does not imply parabolic Campanato decay – and we prove a gauge-invariant quotient criterion equivalent to the Serrin norm of the velocity, together with a Liouville-type rigidity theorem for ancient solutions whose mean coordinate is affine of rank at least one. No claim is made regarding global regularity; the missing steps are stated explicitly as open problems.

Understanding the transport of semiclassical measures along generalized bicharacteristics is a key ingredient in the proof of observability for the wave equation. Here, the coefficients are only assumed to be of class $C^1$, which is a limit case for the existence of bicharacteristics. We consider boundary conditions of the form $\partial_\nu u + i \partial_t u = 0$, as a first step towards more general Lopatinskii-type conditions. We derive the transport equation satisfied by the measure arising from the concentration of high-frequency waves that obstruct observability. Observability of solutions associated with positive time-frequencies is then obtained by contradiction under the geometrical control condition.

Let $\Omega \Subset \mathbb{R}^{n}$ be a strictly convex bounded domain. Suppose $\mathbf{A} : \mathbb{R}^{Nn} \longrightarrow \mathbb{R}^{Nn}$, $\mathbf{B}: \mathbb{R}^{Nn} \longrightarrow \mathbb{R}^{N}$, $\mathbf{C}: \mathbb{R}^{N} \longrightarrow \mathbb{R}^{N}$ are linear maps, where $\mathbf{A}$ is symmetric and non-negative definite. Given $f \in L^2(\Omega, \mathbb{R}^N)$, we consider the problem of existence of solutions $u: \Omega \longrightarrow \mathbb{R}^N$ to the PDE system \[ \left\{ \begin{array}{rl} \displaystyle\sum_{\beta = 1}^{N}\sum_{i, j = 1}^{n} \mathbf{A}_{\alpha i \beta j}\mathrm{D}_{ij}^{2}u_{\beta} + \sum_{\beta = 1}^{N} \sum_{i=1}^{n} \mathbf{B}_{\alpha \beta i}\mathrm{D}_{i}u_{\beta} + \sum_{\beta = 1}^{N} \mathbf{C}_{\alpha \beta}u_{\beta} = f_\alpha, &\text{ in $\Omega$}, \\ u = 0,\ \,& \text{ on $\partial \Omega$}. \end{array} \right. \] This is a linear degenerate elliptic system, and it has not been considered before without the assumption of strict rank-one convexity. In general, it may not possess not even distributional solutions. By introducing some natural structural assumptions, we prove the existence of an appropriately defined unique generalised solution $u\in L^2(\Omega, \mathbb{R}^N)$, satisfying additional partial regularity properties. This paper extends earlier work of the first appearing author [N. Katzourakis, On linear degenerate elliptic PDE systems with constant coefficients, Adv.\ in Calc.Var.\ 9:3, 283-291 (2016)] to include lower-order terms.

We establish dispersive decay estimates for two-dimensional Dirac equations with bounded and unbounded domain walls, together with estimates for the analogous one-dimensional problem. These are paradigmatic models of bulk-edge correspondence in topological insulators, yet their dispersive dynamics have not been previously studied. We show that topologically equivalent models may exhibit qualitatively different dynamics. Our new approach allows us to analyze models which are beyond the usual framework of localized perturbations of constant-coefficient Dirac operators.

The class AC of integrands satisfying the atomic condition was introduced by De Philippis, De Rosa, and Ghiraldin in 2018. These integrands give rise to Almgren elliptic geometric functionals as proven by the second author and De Rosa in 2020. Explicit examples in intermediate dimensions include small class 2 perturbations of the area integrand and the recent anisotropies constructed by De Rosa, Firester, and Tsiamis, among them the $\ell^q$ norms in Plücker coordinates for $q$ close to $2$. We reinterpret the atomic condition in terms of convex geometry. We also propose variants of AC, related to the SAC and USAC defined by De Rosa and Tione, which we call the exposed condition EC, the strong exposed condition SEC, and the quadratic exposed condition QEC. For integrands of class 2, QEC is equivalent to validity of a Caccioppoli-type inequality and is stable under class 2 perturbations; every such QEC integrand admits Lipschitzian normals. Combining these results with a result of De Rosa, Firester, and Tsiamis, we prove that a Lipschitz graph whose anisotropic mean curvature for a QEC integrand of class 2 is $p$-integrable, with $k < p < \infty$, is of class $\mathscr{C}^{1,\alpha}$ on an open set of full measure. Finally, for all $k \ge 2$ and $n - k \ge 3$ we construct strictly polyconvex integrands - associated to inner-product norms on $\bigwedge_k \mathbf{R}^n$ - which fail the atomic condition, showing that strict polyconvexity is necessary but not sufficient for AC.

Using the framework of critical point theory for multivalued functionals developed in \cite{Fri}, we establish some linking theorems for such functionals, which generalizes \cite[Theorem 2.11]{Wi}, \cite[Theorem 2.12]{Wi} and \cite[Theorem 2.12]{Fri}. The proofs of our abstract results rely on a general minimax principle for multivalued mappings. As an application, we obtain a nontrivial solution of a semilinear Dirichlet boundary value problem for partial differential inclusions.

We study the steady self-propelled motion of a rigid body immersed in an incompressible viscous fluid occupying an exterior domain in $\mathbb{R}^3$. In a body-fixed reference frame, the problem is described by a coupled fluid-rigid body system posed in a fixed exterior domain, where the Navier–Stokes equations are coupled with the unknown translational and angular velocities of the rigid body. The self-propulsion is generated by prescribing a boundary velocity on the body surface. Our main result asserts the existence of weak solutions under the sole assumption that the prescribed boundary flux is sufficiently small. The crucial step is the construction of a divergence-free extension of the boundary data that does not rely on either the zero-flux condition or the smallness of the boundary data. As a result, we generalize the result of Galdi [Theorem 5.1 in On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Rational Mech. Anal., 148 (1999), 53-88], where both assumptions were required.

Strong semiclassical convergence from the Hartree equation to the Vlasov equation is established in three dimensions near Penrose-stable homogeneous steady states at positive density. For sufficiently regular integrable interaction kernels, an $O(\hbar^2)$ convergence rate in weighted Sobolev spaces is proved on finite time intervals. Combining this estimate with earlier uniform-in-$\hbar$ phase-mixing and scattering bounds for the Hartree equation yields global existence and scattering for the corresponding Vlasov solution through the semiclassical limit. If the quantum and classical initial perturbations are aligned, the semiclassical convergence is uniform for all times, and the Wigner transforms of the quantum scattering profiles converge to the classical scattering profile at the explicit rate $O((\ln\hbar^{-1})^{-1/3})$.

The aim of this paper is to obtain a version of the classical Szegö limit theorem, where instead of the operator of second derivative on a circle we consider the Heisenberg-Hörmander Laplacian in $L^2(\Bbb R^3)$. Besides, we derive a sharp inequality for convex functions that we call Szegö-type inequality.

The positivity of solutions to subdiffusion equations has been widely studied, mainly in the context of homogeneous problems for single equations. In this article, we fill the missing strict positivity for inhomogeneous subdiffusion equations with nonnegative and nontrivial source terms by connecting Green's functions for fractional and classical diffusion equations via special functions. As a direct application, we further investigate the strict positivity property of coupled subdiffusion systems with nonnegative and partially nontrivial initial values or sources. Under suitable cooperativeness and connectivity conditions, the strict positivity turns out to propagate not only in time but also across different components of the system, reflecting the intrinsic interactions induced by the coupling structure. These results provide a unified framework for understanding positivity properties of both scalar and coupled subdiffusion equations, offering new insights beyond the classical maximum principle approach.

We introduce and analyze a fluctuating homogeneous Landau equation with regularised thermal noise. The model is motivated by the nonlocal gradient flow structure of the deterministic Landau equation, the fluctuation–dissipation principle, and the covariance of the martingale fluctuations of a Kac-like conservative Landau particle system. The noise is written in Landau-divergence form, is antisymmetric in the pair of velocities, and is interpreted in the Stratonovich sense after introducing a velocity correlation. To handle the vacuum singularity of the square-root mobility and the nonlocal Stratonovich-to-Itô correction, we replace the mobility by a regular coefficient. For moderately soft potentials, we prove the existence of probabilistic weak solutions to the regularised fluctuating homogeneous Landau equation. The proof is based on a three-level approximation scheme combining Galerkin approximations, coefficient regularisations, artificial diffusion, and compactness in both $L^2$ and $L^1$ frameworks. The solutions satisfy mass conservation, an energy inequality, and the entropy dissipation estimate. Finally, for a special class of admissible noise bases satisfying a tangential divergence-free condition, we obtain a refined entropy inequality in which the expected entropy is non-increasing relative to the initial entropy.

We consider a model of thermal explosion in reactive turbulent jets introduced in [GHH], which falls into the general class of Gelfand-type problems. Our main result closes the gap in [GMN], where the question of finding an explicit sharp bound for the $L^{\infty}$ norm of the extremal solution in the asymptotic limit of strong flows was left partially unresolved. Namely, we obtain the matching upper and lower bounds under no additional assumptions on the nonlinear reaction rate.

This paper analyzes the regularity of solutions to a fluid-structure interaction (FSI) problem involving a Stokesian fluid and a linear elastic solid in a two-dimensional polygonal domain, seperated by an interface which intersects the boundary of the domain. The main challenges arise from the limited solution regularity caused by geometric singularities at the domain corners. To address this, we introduce tailored weighted Sobolev spaces, denoted by $\mathbf{H}^{k,l}_{\boldsymbol{\beta}}$, and a novel solution operator framework on these spaces. This framework provides the key insight for decoupling the coupled FSI system into tractable fluid and elastic solid subproblems in the regularity analysis. As our main result, we prove the existence and uniqueness of a solution in the weighted Sobolev space $\mathbf{H}^{2,2}_{\boldsymbol{\beta}}$.

We introduce a two-factor stochastic volatility model in which the mean-reversion speed of a Cox–Ingersoll–Ross variance process is an observable functional of a market attention field rather than a latent factor. The field admits an exact Feynman–Kac representation and generates closed-form volatility forecasts. The model nests the Heston volatility model, provides measurement-stable dynamics under observable data aggregation, and preserves its structure under an equivalent martingale measure. A mean-field analysis establishes convergence of the underlying particle system, while simulations demonstrate volatility and option-pricing differences following attention shocks.