The initial motivation for this work was the linguistic case of the spread of Germanic syntactic features into Romance dialects of North-Eastern Italy, which occurred after the immigration of German people to Tyrol during the High Middle Ages. To obtain a representation of the data over the territory suitable for a mathematical formulation, an interactive map is produced as a first step, using tools of what is called Geographic Data Science. A smooth two-dimensional surface G is introduced, expressing locally which fraction of territory uses a given German language feature: it is obtained by a piecewise cubic curvature minimizing interpolant of the discrete function that says if at any surveyed locality that feature is used or not. This surface G is thought of as the value at the present time of a function describing a diffusion-convection phenomenon in two dimensions (here said tidal mode), which is subjected in a very natural way to the same equation used in physics, introducing a contextual diffusivity concept: it is shown that with two different assumptions about diffusivity, solutions of this equation, evaluated at the present time, fit well with the data interpolated by G, thus providing two convincing different pictures of diffusion-convection in the case under study, albeit simplifications and approximations. Very importantly, it is shown that the linguistic diffusion model known to linguists as Schmidt waves can be counted among the solutions of the diffusion equation

翻译：本工作的最初动机源自语言学的案例：中世纪中期德意志人迁徙至蒂罗尔后，德语语法特征扩散至意大利东北部罗曼语方言的现象。为获得适用于数学表述的地域性数据表征，我们首先运用“地理数据科学”工具构建交互式地图。通过引入一个光滑的二维曲面G，该曲面以分片三次曲率最小化插值函数表达局部地区使用的德语特征所占比例——该插值基于对每个调查地点是否使用该特征的离散函数进行拟合。我们将曲面G视作描述二维扩散-对流现象（此处称为潮汐模式）的函数在现时刻的值，并自然地遵循物理学中的相同方程，引入情境化扩散系数概念。研究表明，在两种不同扩散系数假设下，该方程的解在现时刻的估计值与G所插值的数据吻合良好，从而为所研究案例提供了两种具有说服力的扩散-对流图景（尽管存在简化和近似）。尤为重要的是，本文证明语言学界熟知的“施密特波”语言扩散模型可被视为该扩散方程的解之一。