[Paper Review] Stochastic averaging lemmas for kinetic equations

Lions, Pierre-Louis, Benoı̂t Perthame+1 more|Apr 2, 2012

Gas Dynamics and Kinetic Theory13 references4 citations

TL;DR

This paper establishes stochastic averaging lemmas for kinetic equations with Stratonovich noise, showing that the time and space regularity of velocity averages in $L^2$ are altered by the stochastic time scale: time decay is slower (e.g., $\lambda^{-3/2}$ vs. $\lambda^{-1/2}$ in deterministic case), and regularity gains differ due to parabolic vs. hyperbolic scaling in the Fourier multiplier. The results are derived via stochastic characteristics and Fourier analysis, with applications to scalar conservation laws with stochastic fluxes.

ABSTRACT

We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale. Compared to the deterministic case and as far as we work in $L^2$, the nature of regularity on averages is not changed in this stochastic kinetic equation and stays in the range of fractional Sobolev spaces at the price of an additional expectation. However all the exponents are changed; either time decay rates are slower (when the right hand side belongs to $L^2$), or regularity is better when the right hand side contains derivatives. These changes originate from a different space/time scaling in the deterministic and stochastic cases. Our motivation comes from scalar conservation laws with stochastic fluxes where the structure under consideration arises naturally through the kinetic formulation of scalar conservation laws.

Motivation & Objective

• To develop averaging lemmas for stochastic kinetic equations driven by Stratonovich noise, particularly for scalar conservation laws with stochastic fluxes.
• To characterize how stochastic time scaling (Brownian motion) alters the regularity and decay properties of velocity averages compared to the deterministic case.
• To establish $L^2$ estimates for velocity averages $\rho_\psi$ in fractional Sobolev spaces, quantifying the change in exponents due to stochasticity.
• To clarify the role of Stratonovich calculus and stochastic characteristics in preserving the structure of kinetic equations under noise.

Proposed method

• Use of stochastic characteristics to represent solutions via $f(x+B(t)\circ\xi,\xi,t)$, valid under Stratonovich convention.
• Fourier analysis in space and time to estimate $L^2$ norms of velocity averages $\rho_\psi = \int \psi(\xi)f(x,\xi,t)d\xi$.
• Application of Calderón-Zygmund theory to control singular integrals arising from stochastic time evolution.
• Introduction of time-damping $e^{-\lambda t}$ to isolate time decay behavior and compare deterministic vs. stochastic scaling.
• Comparison of Fourier multipliers: hyperbolic $1/(2\lambda - i(\xi_1-\xi_2)\cdot k)$ in deterministic case vs. parabolic $1/(2\lambda + |(\xi_1-\xi_2)\cdot k|^2)$ in stochastic case.
• Use of $L^2$ energy estimates and duality to derive bounds on $\mathbb{E}\|\rho_\psi\|_{L^2(\mathbb{R}^+; \dot{H}^s)}^2$ for various regularity gains.

Experimental results

Research questions

• RQ1 How does the addition of Stratonovich noise to the kinetic equation alter the regularity of velocity averages in $L^2$?
• RQ2 What are the precise time decay rates for velocity averages in the stochastic case compared to the deterministic case?
• RQ3 How do the exponents in fractional Sobolev regularity estimates change due to stochastic time scaling?
• RQ4 Can the averaging lemma framework be extended to stochastic kinetic equations with $L^2$ right-hand sides and stochastic fluxes?
• RQ5 What is the role of the Stratonovich convention in preserving the structure of kinetic equations under noise?

Key findings

• For $f^0 = 0$, the stochastic averaging lemma yields $\mathbb{E}\|e^{-\lambda t}\rho_\psi\|_{L^2(\mathbb{R}^+; \dot{H}^{1/2})}^2 \leq C(\text{supp}\,\psi) \lambda^{-3/2} \|e^{-\lambda t}\psi g\|_{L^2}^2$, showing slower time decay than the deterministic case.
• When $g = \text{div}\,h$, the lemma gives $\mathbb{E}\|\rho_\psi\|_{L^2(\mathbb{R}^+; \dot{H}^{1/3})}^2 \leq C(\psi) \left(\|h\|_{\dot{H}^{-2/3}_{x} \cap L^2}^2 + \|\psi f\|_{L^2}^2\right)$, indicating improved regularity compared to deterministic $\dot{H}^{1/2}$.
• The stochastic Fourier multiplier $1/(2\lambda + |(\xi_1 - \xi_2)\cdot k|^2)$ leads to parabolic scaling, causing slower decay and different regularity exponents than the deterministic hyperbolic multiplier $1/(2\lambda - i(\xi_1 - \xi_2)\cdot k)$.
• In the time regularity case, the stochastic lemma yields a gain of $1/4$ derivative in $H^{1/4}(\mathbb{R}^+)$, while the deterministic case achieves $1/2$ derivative gain, reflecting the stronger cancellation in the deterministic setting.
• The results are robust under $L^2$ assumptions and hold globally in time, with bounds depending on the support of $\psi$ and initial data or source terms.
• The method confirms that stochasticity fundamentally alters the time-scale structure, changing the nature of regularity from hyperbolic to parabolic in the Fourier domain.