The Landau--Lifshitz--Bloch equation perturbed by a space-dependent noise was proposed in Garanin 1991 as a model for evolution of spins in ferromagnatic materials at the full range of temperatures, including the temperatures higher than the Curie temperature. In the case of a ferromagnet filling a bounded domain $D\subset \mathbb R^d$, $d=1,2,3$, we show the existence of strong (in the sense of PDEs) martingale solutions. Furthermore, in cases $d=1,2$ we prove uniqueness of pathwise solutions and the existence of invariant measures.

中文翻译：

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随机 Landau-Lifshitz-Bloch 方程的唯一解和不变测度的存在性

受空间相关噪声扰动的 Landau--Lifshitz--Bloch 方程于 1991 年在 Garanin 中提出，作为铁磁性材料在整个温度范围内（包括高于居里温度的温度）中自旋演化的模型。在铁磁体填充有界域 $D\subset\mathbb R^d$, $d=1,2,3$ 的情况下，我们展示了强（在偏微分方程的意义上）鞅解的存在。此外，在 $d=1,2$ 的情况下，我们证明了路径解的唯一性和不变测度的存在。