Global Martingale Entropy Solutions

• Global martingale entropy solutions are a class of stochastic weak solutions for nonlinear SPDEs that integrate probabilistic martingale formulations with entropy methods.
• They leverage energy-dissipation inequalities, detailed balance conditions, and tightness-compactness techniques to ensure the existence of global nonnegative solutions in cross-diffusion systems.
• Methodologies such as stochastic Galerkin approximations and entropy a priori estimates enable control over degenerate, non-symmetric diffusion matrices in high-dimensional models.

A global martingale entropy solution is a notion of stochastic weak solution for nonlinear stochastic partial differential equations (SPDEs)—in particular, cross-diffusion systems—incorporating both the martingale (probabilistic) and entropy (thermodynamic or gradient-flow) structures. This concept seamlessly combines rigorous stochastic integration, energy-dissipation inequalities, and tightness-compactness methods to establish the existence of global (in time) nonnegative solutions for highly degenerate, non-symmetric, even non-positive semi-definite SPDEs, typically with multiplicative noise. The variational entropy structure provides control over regularity and positivity, while the martingale formulation encodes the stochastic nature and allows passage to limits in nonlinear, high-dimensional, or degenerate systems. This framework is foundational for modern SPDE theory in mathematical biology, physics, and fluid dynamics.

1. Model Classes and the SPDE Framework

Global martingale entropy solutions arise naturally in the context of stochastic cross-diffusion systems. The general class of systems considered consists of interacting vector fields, $u=(u_1,\dots,u_n)$, evolving according to

$$d u_i(t) - \operatorname{div} \left( \sum_{j=1}^n A_{ij}(u(t)) \nabla u_j(t) \right) dt = \sum_{j=1}^n \sigma_{ij}(u(t)) dW_j(t),$$

with no-flux boundary conditions

$$\sum_{j=1}^n A_{ij}(u) \nabla u_j \cdot \nu = 0 \quad \text{on } \partial\Omega,$$

and initial data $u_i(0)=u_i^0$. Here $A(u)$ represents the (possibly neither symmetric nor positive-definite) cross-diffusion matrix, and $\sigma_{ij}(u)$ are multiplicative noise coefficients acting on a collection of independent or cylindrical Wiener processes $W_j$. For population models, $u_i$ represent densities of $n$ species, and the diffusion matrix can encode complicated mobility and interaction structure (Dhariwal et al., 2018, Dhariwal et al., 2020, Biswas et al., 2022, Braukhoff et al., 2022).

The formulations are robust enough to encompass multi-species models, such as the Shigesada-Kawasaki-Teramoto (SKT) population systems, Maxwell–Stefan models, stochastic thin-film equations, and more degenerate or kinetic problems (Dhariwal et al., 2020, Punshon-Smith et al., 2016, Dareiotis et al., 2020).

2. Entropy Structure and Entropy Functionals

The entropy structure is central to the theory. An entropy density $h(u)$ is convex, and the associated entropy functional $H(u)=\int_\Omega h(u(x))dx$ is non-increasing (in expectation) along solutions. A prototypical form is

$h(u)=\sum_{i=1}^n \pi_i (u_i \log u_i - u_i + 1)$

for relative entropy, where the $\pi_i>0$ are detailed-balance weights.

For quadratic structures (e.g., "Rao entropy"),

$$h(u) = \frac{1}{2}\sum_{i,j=1}^n \pi_i a_{ij} u_i u_j,$$

mixing all species (Biswas et al., 2022).

The entropy variable $w_i = \partial_{u_i} h(u)$ is introduced, and the cross-diffusion system is often reformulated in terms of $w$:

$$\partial_t u - \operatorname{div}(B(w)\nabla w) = \text{noise terms},$$

where $B(w) = A(u(w))[h''(u(w))]^{-1}$ is symmetric positive semi-definite under detailed balance, overcoming the lack of symmetry or positivity in $A(u)$. Testing by the entropy variable yields dissipation estimates

$$\frac{d}{dt}H(u) + \int_\Omega \nabla u : h''(u)A(u)\nabla u\,dx = 0,$$

in the deterministic case, and corresponding Itô inequalities in the stochastic case (Dhariwal et al., 2018, Dhariwal et al., 2020, Dareiotis et al., 2020, Braukhoff et al., 2022).

3. Definition and Characterization of Global Martingale Entropy Solutions

Let $$(\Omega, \mathcal{F}, \{\mathcal{F}_t\}, \mathbb{P})$$ be a filtered probability space with cylindrical Wiener process $W$. A quintuple $$(\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t), W, u)$$ is a global martingale entropy solution if (Dhariwal et al., 2018, Braukhoff et al., 2022):

• $u$ is progressively measurable, valued in $$L^2(\Omega; C([0,T]; L^2(\Omega))) \cap L^2(\Omega; L^2(0,T; H^1(\Omega)))$$.
• For all test functions $\phi \in H^1(\Omega)$,

$$(u_i(t), \phi)_{L^2} = (u_i(0), \phi)_{L^2} - \int_0^t \langle \operatorname{div}(A(u(s))\nabla u_i(s)), \phi \rangle ds + \sum_j \int_0^t (\sigma_{ij}(u(s))dW_j(s), \phi)_{L^2}$$

holds almost surely for all $t$ and species $i$.

• The entropy inequality is satisfied:

$$\mathbb{E} \left[ \sup_{t \leq T} H(u(t)) \right] + \mathbb{E} \left[ \int_0^T D(u(s)) ds \right] \leq C,$$

where $D(u)$ is a coercive dissipation functional, possibly involving $|\nabla \sqrt{u_i}|^2$ and cross-gradients (Dhariwal et al., 2020, Biswas et al., 2022, Lin, 2022).

• Nonnegativity: Under suitable structural conditions on $\sigma_{ij}$, the solution is nonnegative in the sense $u_i(t,x)\geq0$ almost everywhere, a.s., proved via Stampacchia-type truncation methods (Dhariwal et al., 2018, Lin, 2022, Dareiotis et al., 2020).

This definition is tailored to ensure the solution exists globally in time and dissipates entropy in expectation (and almost surely under suitable conditions).

4. Existence Theory: Stochastic Galerkin and Tightness Approaches

The existence proofs for global martingale entropy solutions are based on several core steps:

• Stochastic Galerkin approximation: The SPDE is projected onto a finite-dimensional subspace (e.g., $H_N \subset L^2(\Omega)$), yielding a system of finite-dimensional SDEs. Existence and uniqueness follow from classical SDE theory under Lipschitz and growth hypotheses (Dhariwal et al., 2018, Lin, 2022).
• Entropy a priori estimates: Itô's formula is applied to the entropy functional or a regularized version, providing uniform bounds for Galerkin solutions, crucial for compactness (Dhariwal et al., 2020, Braukhoff et al., 2022, Dareiotis et al., 2020).
• Tightness and compactness: The uniform estimates yield tightness of the sequence of laws of Galerkin solutions in path spaces tailored to the required regularity (e.g., Sobolev spaces with weak topologies, Jakubowski's spaces). The Skorokhod–Jakubowski representation theorem allows passage to a subsequential limit (Dhariwal et al., 2018, Biswas et al., 2022, Dhariwal et al., 2019).
• Identification and martingale formulation: Limits are shown to solve the SPDE in the sense of martingale solutions, with entropy inequalities propagated to the limit (Dhariwal et al., 2020, Huber, 29 Jan 2025, Chen et al., 27 Dec 2025).

A distinguishing technical feature is that non-symmetric and non-positive-definite diffusion matrices are handled by leveraging the entropy/gradient-flow structure, not via classical L²-theory.

5. Structural Conditions, Entropy-Noise Interaction, and Nonnegativity

The main structural hypotheses ensuring the well-posedness of global martingale entropy solutions include (Dhariwal et al., 2018, Dhariwal et al., 2020, Braukhoff et al., 2022):

• Detailed balance or self-diffusion dominance: Existence of positive weights $\pi_i>0$ such that $\pi_i A_{ij}(u)=\pi_j A_{ji}(u)$ or sufficient self-diffusion ensures the positive definiteness of the weighted diffusion matrix $PA(u)$ or $B(w)$, giving access to entropy dissipation estimates.
• Entropy–noise compatibility: The Itô correction (arising from the stochastic calculus) must be controlled by the entropy functional. Typically, this requires

$$\operatorname{tr}(\sigma(u)^\top h''(u)\sigma(u)) \leq C (1 + H(u))$$

for all $u$.

• Nonnegativity preservation: Additional bounds

$$\sum_j \|\sigma_{ij}(u)\|_{L_2(Y;L^2)}^2 \leq C u_i(x)$$

guarantee that nonnegative initial data remains nonnegative. This is established via extensions of the Stampacchia truncation principle adapted to SPDEs (Dhariwal et al., 2018, Dhariwal et al., 2020).

These conditions are not only technical artifacts—they delineate the regime in which entropy-based compactness can overcome degeneracy and ensure physically meaningful, nonnegative, and stable solutions.

6. Notable Examples and Model Classes

Global martingale entropy solutions have been constructed for a range of systems:

• Cross-diffusion population systems with arbitrary $n$ and multiple interaction and noise terms; the full existence theory is established under detailed balance or self-diffusion dominance, including bounds for higher moments and regularity (Dhariwal et al., 2018, Dhariwal et al., 2020, Biswas et al., 2022, Lin, 2022, Braukhoff et al., 2022).
• Degenerate-parabolic stochastic thin-film equations, with cubic and higher-order mobility, for which global nonnegative martingale solutions were constructed using entropy-energy methods adapted to degenerate nonlinearities (Dareiotis et al., 2020).
• Multi-component Maxwell–Stefan models and biofilm models: The boundedness-by-entropy method allows derivation of almost sure positivity and $L^\infty$-bounds for systems with volume-filling effects and highly nonlinear cross-diffusion (e.g., Maxwell–Stefan and $n$-species biofilm models) (Dhariwal et al., 2019).
• Stochastic segregation cross-diffusion systems: Use of the Rao entropy yields exponential equilibration results and supports the existence of global martingale entropy solutions even when the diffusion matrix is non-PSD and non-symmetric (Biswas et al., 2022).
• Stochastic isentropic Euler equations: Recent work extends the entropy solution framework to hyperbolic SPDEs with general pressure laws, using stochastic compensated compactness and relative energy functionals to establish the existence of global martingale entropy solutions with entropy inequalities (Chen et al., 27 Dec 2025).
• Kinetic equations: Global renormalized martingale entropy solutions for the Boltzmann equation with stochastic kinetic transport under sufficient integrability and coloring hypotheses for the noise coefficients (Punshon-Smith et al., 2016).

7. Methodological Significance and Future Directions

The emergence of global martingale entropy solutions as an organizing principle highlights:

• The precise interplay between stochastic analysis, entropy methods, and PDE regularity in establishing global-in-time well-posedness for classes of SPDEs not treatable by classical monotonicity or maximum principle techniques. This approach systematically incorporates both the martingale and entropy structures, leading to uniform in $N$ a priori estimates, tightness of solution laws, and control over strong nonlinearity or degeneracy.
• A plausible implication is that these techniques, in particular the entropy structure and entropy–noise compatibility, may extend to new regimes of SPDEs in fluid dynamics, biological aggregation, and kinetic theory, potentially encompassing further nonlocal, memory, or non-Markovian stochastic effects.
• Remaining open questions include uniqueness (often unresolved for these highly degenerate and nonlinear systems), fine regularity properties, and extension to higher dimensions and additional noise structures (Dareiotis et al., 2020, Chen et al., 27 Dec 2025).

Global martingale entropy solutions thus provide a mathematically robust and physically meaningful framework to analyze the long-time evolution of stochastic nonlinear systems in a variety of scientific fields.

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References:

• (Dhariwal et al., 2018) Global martingale solutions for a stochastic population cross-diffusion system
• (Dhariwal et al., 2020) Global martingale solutions for a stochastic Shigesada-Kawasaki-Teramoto population model
• (Biswas et al., 2022) Global martingale solutions to a segregation cross-diffusion system with stochastic forcing
• (Lin, 2022) Global martingale solutions to a stochastic cross-diffusion population system between superlinear and subquadratic transition rate case
• (Dhariwal et al., 2019) Global martingale solutions for quasilinear SPDEs via the boundedness-by-entropy method
• (Dareiotis et al., 2020) Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise
• (Braukhoff et al., 2022) Global martingale solutions for stochastic Shigesada-Kawasaki-Teramoto population models
• (Huber, 29 Jan 2025) Fluctuation Correction and Global Solutions for the Stochastic Shigesada-Kawasaki-Teramoto System via Entropy-Based Regularization
• (Chen et al., 27 Dec 2025) Global Martingale Entropy Solutions to the Stochastic Isentropic Euler Equations
• (Punshon-Smith et al., 2016) On the Boltzmann Equation with Stochastic Kinetic Transport: Global Existence of Renormalized Martingale Solutions
• (Shkolnikov et al., 2024) From rank-based models with common noise to pathwise entropy solutions of SPDEs