Stochastic Compensated Compactness Framework

• Stochastic compensated compactness framework is a set of analytical techniques that ensure existence, compactness, and limit passage for measure-valued or entropy solutions in stochastic PDEs.
• The framework replaces unavailable uniform L∞ bounds with uniform L^p estimates and employs martingale entropy inequalities alongside stochastic div–curl lemmas.
• It rigorously identifies solutions in stochastic conservation laws and homogenization problems by leveraging tools such as random Young measures and the Jakubowski–Skorokhod theorem.

The stochastic compensated compactness framework is a collection of analytical techniques enabling the existence, compactness, and meaningful passage to the limit for measure-valued or entropy solutions of stochastic partial differential equations, particularly stochastic conservation laws and stochastic isentropic Euler equations. The framework addresses key challenges where uniform $L^{\infty}$ bounds for approximate solutions are unavailable due to stochastic forcing, and replaces these with uniform higher-integrability ($L^p$) estimates and stochastic analogues of classical compactness tools. It unites techniques from martingale analysis, random Young measures, stochastic Sobolev spaces, and adapted versions of div–curl and interaction identities, permitting rigorous limit passage and solution identification across stochastic PDEs in fluid dynamics and homogenization of random heterogeneous media.

1. Analytical Foundation and Rationale

The central analytical challenge in stochastic PDEs is the lack of an invariant region or uniform $L^\infty$ bound for the class of approximate solutions generated by vanishing-viscosity regularization. Unlike deterministic settings, where compensated compactness and div–curl arguments ride on uniform control and strong compactness, stochastic systems often offer only $L^p$-based estimates due to the nature of randomness introduced via (possibly multiplicative) noise terms. As developed for the stochastic isentropic Euler equations (Chen et al., 27 Dec 2025), the stochastic compensated compactness framework establishes tightness and convergence in probability law by leveraging:

• Parabolic (vanishing-viscosity) regularizations with stochastic noise,
• Martingale entropy inequalities for all convex entropy pairs derived via kinetic representation,
• Uniform, $\varepsilon$-independent energy and higher-order relative energy bounds,
• Tightness in negative-order Sobolev spaces, circumventing lack of strong compactness,
• Jakubowski–Skorokhod representation theorem for non-metric spaces, enabling random Young measure convergence.

This suite of methods facilitates the passage to limit laws and solution identification in regimes where classical tools falter, enabling the construction of global martingale entropy solutions with finite relative energy.

2. Vanishing-Viscosity Approximations and Martingale Solutions

For a given stochastic system (e.g., isentropic Euler equations with general or polytropic pressure law), one studies vanishing-viscosity regularizations: $$\begin{cases} \partial_t \rho^\varepsilon + \partial_x m^\varepsilon = \varepsilon\,\partial_{xx} \rho^\varepsilon, \ \partial_t m^\varepsilon + \partial_x\left(\frac{(m^\varepsilon)^2}{\rho^\varepsilon} + P(\rho^\varepsilon)\right) = \varepsilon\,\partial_{xx} m^\varepsilon + \Phi^\varepsilon(\rho^\varepsilon, m^\varepsilon)\,\dot W(t), \end{cases}$$ with suitably mollified initial data (Chen et al., 27 Dec 2025). The lack of $L^\infty$ bounds is addressed by establishing uniform $L^p$–based energy and higher integrability estimates: $\E\left[\sup_{t\in [0,T]} \int_{\R} \frac12\,\rho^\varepsilon(u^\varepsilon)^2 + e^*(\rho^\varepsilon,\rho_\infty)\,dx \right] + \varepsilon\,\E\int_{0}^{T}\int_{\R} |\partial_x u^\varepsilon|^2\,dx\,dt \leq C.$

Key technical ingredients include:

• The use of martingale/Itô entropy inequalities valid for all convex entropy pairs,
• BDG-type (Burkholder–Davis–Gundy) estimates to control stochastic integrals,
• Identification of limit objects as martingale entropy solutions satisfying local mechanical energy inequality.

3. Tightness, Young Measures, and Stochastic Compactness

Uniform integrability and tightness in $L^p$-type spaces, together with BDG-type stochastic estimates, allow one to apply the Jakubowski–Skorokhod representation theorem for random Young measures in sub-Polish (non-metric) spaces. This facilitates the extraction, along subsequences, of almost sure pointwise and $L^p$ convergence for solution pairs: $$(\tilde{\rho}^\varepsilon, \tilde{m}^\varepsilon) \rightarrow (\tilde{\rho}, \tilde{m}), \quad \text{a.s.}$$ Random Young measures $\tilde{\mu}_{t,x}$ appear as measurable maps into the space of probability measures on $\R^2_+$.

Reduction of measure-valued solutions via a stochastic version of Tartar's commutation relation (stochastic div–curl lemma), combined with analysis of kernel support, shows either collapse to vacuum ($\rho=0$) or reduction to Dirac masses, yielding pointwise identification of strong or weak solutions in the non-vacuum region (Chen et al., 27 Dec 2025).

4. Quantitative Compactness in Stochastic Conservation Laws

A quantitative form of stochastic compensated compactness for scalar stochastic conservation laws is achieved by combining spatial modulus of continuity estimates with stochastic Kružkov interpolation (Karlsen, 2023). In the setting

$$du + \mathrm{Div}_x f(x,u)\,dt = \sigma(x,u)\,dW(t),$$

the main theorem establishes explicit rates for the compactness modulus: $\E\int_0^T\int_{\R} |u^\varepsilon(t,x+z) - u^\varepsilon(t,x)|\,\chi(x)\,dx\,dt \lesssim |z|^{\mu_x}, \quad \forall |z|<1,$ with constants and exponents determined by the growth properties of the flux and entropies, via interaction identities adapted for stochastically forced systems. Stochastic Kružkov interpolation, together with kinetic and measure-valued formulations, implies precompactness of laws and pathwise existence of entropy solutions using Skorokhod’s theorem. This framework directly generalizes the regularity and compactness results of Golse–Perthame for deterministic equations to their stochastic counterparts, highlighting the impact of noise on regularity exponents.

5. Stochastic Compensated Compactness in Homogenization Theory

In stochastic homogenization of high-contrast media (Cherdantsev et al., 2017), compensated compactness is established via stochastic two-scale convergence on spaces $L^2(S\times\Omega)$, where $S$ is a spatial domain and $\Omega$ a probability space with an ergodic dynamical system. For $H^1$-bounded sequence $u^\varepsilon$, the stochastic compactness theorem asserts: $$u^\varepsilon \stackrel{2s}{\to} u_0(x) + u_1(x,\omega),\quad \nabla u^\varepsilon \stackrel{2s}{\to} \nabla_x u_0(x) + \nabla_\omega u_1(x,\omega),$$ where $u_0$ solves a macroscopic equation and $u_1$ a microstructure-corrector cell problem. The macro/micro system couples homogenized coefficients and stochastic analogues of periodic correctors, leading to spectral convergence in the Hausdorff sense. The stochastic Zhikov function $B(\lambda)$ appears as a nonlocal operator accounting for averaged microstructure effects on dispersion and spectra.

6. Extension, Generalizations, and Associated Analytical Tools

The stochastic compensated compactness framework extends to:

• Strictly hyperbolic gases with general pressure laws and polytropic models for all adiabatic exponents,
• Multi-dimensional spherically symmetric fluid models,
• Stochastic magnetohydrodynamics under symmetry restrictions,
• Systems of stochastic conservation laws under symmetrizable or genuinely nonlinear hypotheses (Chen et al., 27 Dec 2025, Karlsen, 2023).

Crucial analytical tools underpinning the framework include:

• Murat’s compactness lemma and its stochastic generalizations for $W^{-1,q}$ negative-order Sobolev spaces,
• Jakubowski–Skorokhod realisation theorem for random Young measures,
• Stochastic div–curl lemma enabling convergence of nonlinear terms under minimal regularity,
• BDG/Itô estimates controlling all martingale and noise terms in stochastic balance laws.

7. Comparison with Classical and Deterministic Theories

The stochastic compensated compactness framework formally generalizes deterministic two-scale methods (e.g., classical Tartar–Murat div–curl, Zhikov periodic homogenization) to the stochastic setting by substituting periodicity and uniform cell bounds with ergodicity and random inclusion distribution. Compactness is no longer derived from strong $L^\infty$ bounds but from stochastic $L^p$ estimates and tightness in probability. In all settings, the critical role is played by energetic and entropy structures (quantitative and higher-order), together with stochastic analysis tools providing control over random measure evolution. The analytic machinery yields explicit rates of regularity, precise solution identification, and convergence of spectra, robustly extending the reach of compensated compactness to random environments and stochastically forced PDEs (Chen et al., 27 Dec 2025, Cherdantsev et al., 2017, Karlsen, 2023).