论文标题
纠正“ Kolmogorov方程的最佳规律性结果和某些关键SPDE的唯一性弱”
Correction to "An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs"
论文作者
论文摘要
我们显示了关键SPDE \ begin {eqnarray} \ label {qq1} dx_t = ax_t dt dt dt +（-a）^{1/2} f（x（x（x（t））dt + dw_t，\; \; \; \; x_0 = x \ in H，\ end {eqnarray}其中$ a $：$ a $：\ text {dom}（a）\ subset H \ to h $是一个负面的自偶会在可分开的hilbert space $ h $上，$ a^{ - 1} $ w $ w $ a $ w $是$ a $ a $ h $ h $ h $ h $ h $ h $ h $ h $。这里$ f：h \ to h $可以在本地与最多线性增长相关（某些功能$ f $也可以考虑线性的增长更多）。这导致了广义随机汉堡方程以及具有有趣应用的三维随机Cahn-Hilliard类型方程的新独特性结果。我们不知道唯一的唯一假设是$ f $ $ f $加上生长条件的唯一假设，如[Priola，Ann。概率。 49（2021）]。为了获得虚弱的唯一性，我们使用无限的尺寸定位原则，以及当$ f = z \ in H $是常数和$λ> 0 $时，kolmogorov方程$λu -l u = f $是与SPDE相关的最佳规律性结果。该最佳结果类似于[Da Prato，J。Evol。等式。 3（2003）]。
We show uniqueness in law for the critical SPDE \begin{eqnarray} \label{qq1} dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, \end{eqnarray} where $A$ $ : \text{dom}(A) \subset H \to H$ is a negative definite self-adjoint operator on a separable Hilbert space $H$ having $A^{-1}$ of trace class and $W$ is a cylindrical Wiener process on $H$. Here $F: H \to H $ can be locally Hölder continuous with at most linear growth (some functions $F$ which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers equations and for three-dimensional stochastic Cahn-Hilliard type equations which have interesting applications. We do not know if uniqueness holds under the sole assumption of continuity of $F$ plus growth condition as stated in [Priola, Ann. of Prob. 49 (2021)]. To get weak uniqueness we use an infinite dimensional localization principle and an optimal regularity result for the Kolmogorov equation $ λu - L u = f$ associated to the SPDE when $F = z \in H$ is constant and $λ>0$. This optimal result is similar to a theorem of [Da Prato, J. Evol. Eq. 3 (2003)].
