We study the motion of the hypersurface $$(\gamma _t)_{t\ge 0}$$ ( γ t ) t ≥ 0 evolving according to the mean curvature perturbed by $$\dot{w}^Q$$ w ˙ Q , the formal time derivative of the Q -Wiener process $${w}^Q$$ w Q , in a two-dimensional bounded domain. Namely, we consider the equation describing the evolution of $$\gamma _t$$ γ t as a stochastic partial differential equation (SPDE) with a multiplicative noise in the Stratonovich sense, whose inward velocity V is determined by $$V=\kappa \,+\,G \circ \dot{w}^Q$$ V = κ + G ∘ w ˙ Q , where $$\kappa $$ κ is the mean curvature and G is a function determined from $$\gamma _t$$ γ t . Already known results in which the noise depends on only the time variable are not applicable to our equation. To construct a local solution of the equation describing $$\gamma _t$$ γ t , we derive a certain second-order quasilinear SPDE with respect to the signed distance function determined from $$\gamma _0$$ γ 0 . Then we construct the local solution making use of probabilistic tools and the classical Banach fixed point theorem on suitable Sobolev spaces.

中文翻译：

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由有色噪声随机扰动的平均曲率流

我们研究了超曲面 $$(\gamma _t)_{t\ge 0}$$ ( γ t ) t ≥ 0 根据 $$\dot{w}^Q$$ w 扰动的平均曲率演化的运动˙ Q ，Q -Wiener 过程$${w}^Q$$ w Q 的形式时间导数，在二维有界域中。即，我们将描述 $$\gamma _t$$ γ t 演化的方程视为一个随机偏微分方程 (SPDE)，在 Stratonovich 意义上具有乘法噪声，其向内速度 V 由 $$V=\kappa 决定\,+\,G \circ \dot{w}^Q$$ V = κ + G ∘ w ˙ Q ，其中 $$\kappa $$ κ 是平均曲率，G 是由 $$\gamma 确定的函数_t$$ γ t 。噪声仅取决于时间变量的已知结果不适用于我们的方程。要构造描述 $$\gamma _t$$ γ t 的方程的局部解，我们相对于由 $$\gamma_0$$ γ 0 确定的有符号距离函数推导出某个二阶拟线性 SPDE。然后我们在合适的 Sobolev 空间上利用概率工具和经典的 Banach 不动点定理构建局部解。