变分与奇异随机偏微分方程的对数导数
Logarithmic derivatives of variational and singular stochastic partial differential equations
中文总结 AI 辅助
该研究针对满足特定条件的随机偏微分方程，推导了其解律对数导数的内禀公式，详细处理了随机p-拉普拉斯方程和二维Navier-Stokes方程，还为奇异方程的重正化极限提供了分部积分恒等式。
中文摘要 AI 辅助
对于定义在Gelfand三元组上且满足Röckner、Shang和Zhang提出的完全局部单调条件的随机偏微分方程，我们计算了固定时刻解的律沿状态空间指定方向的对数导数。该公式具有内禀性，仅通过Hilbert-Schmidt Malliavin导数Φᵣ = 𝒟ᵣX(t)和协方差γₜ = ∫₀ᵗ ΦᵣΦᵣ* dr表达，既无需有限维中使用的一阶变分的逆，也无需Tikhonov族的一致Malliavin-Sobolev界。它通过抽象Hilbert空间上的分部积分恒等式、Wiener空间上覆盖场的Moore-Penrose构造以及Tikhonov极限的迹公式得到，并借助一阶变分对Malliavin导数的表示Φᵣ = Y(t,r)ℬ(r,X(r))应用于该方程；文中对随机p-拉普拉斯方程和二维Navier-Stokes方程进行了详细处理。除变分类外，标量约简为配对⟨u(t),φ⟩的律提供了分部积分恒等式，该配对传递到Bruned、Chandra、Chevyrev和Hairer的奇异方程的重正化极限，且在二阶Malliavin光滑性和负矩假设下由对数导数表示。
英文摘要
For a stochastic partial differential equation posed on a Gelfand triple and satisfying the fully local monotone conditions of Röckner, Shang and Zhang, we compute the logarithmic derivative of the law of the solution at a fixed time along a prescribed direction of the state space. The formula is intrinsic, being expressed through the Hilbert-Schmidt Malliavin derivative $Φ_r = \mathcal{D}_r X(t)$ and the covariance $γ_t = \int_0^t Φ_r Φ_r^{*} \,\mathrm{d}r$ alone, so that neither the inversion of the first variation used in finite dimensions nor uniform Malliavin-Sobolev bounds on the Tikhonov family are called upon. It is obtained from an integration-by-parts identity on an abstract Hilbert space, a Moore-Penrose construction of a covering field on Wiener space, and a trace formula for the Tikhonov limit, specialised to the equation through the representation $Φ_r = Y(t,r)\mathcal{B}(r,X(r))$ of the Malliavin derivative by the first variation; the stochastic $p$-Laplacian and the two-dimensional Navier-Stokes equation are treated in detail. Beyond the variational class, a scalar reduction gives an integration-by-parts identity for the law of a pairing $\langle u(t),φ\rangle$ which passes to the renormalised limit for the singular equations of Bruned, Chandra, Chevyrev and Hairer, and which is represented by a logarithmic derivative under second-order Malliavin smoothness and negative-moment hypotheses.
