Applying the resolution–scale relativity principle to develop a mechanics of non-differentiable dynamical paths, we find that, in one dimension, stationary motion corresponds to an Itô process driven by the solutions of a Riccati equation. We verify that the corresponding Fokker–Planck equation is solved for a probability density corresponding to the squared modulus of the solution of the Schrödinger equation for the same problem. Inspired by the treatment of the one-dimensional case, we identify a generalization to time dependent problems in any number of dimensions. The Itô process is then driven by a function which is identified as establishing the link between non-differentiable dynamics and standard quantum mechanics. This is the basis for the scale relativistic interpretation of standard quantum mechanics and, in the case of applications to chaotic systems, it leads us to identify quantum-like states as characterizing the entire system rather than the motion of its individual constituents.

中文翻译：

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Riccati 方程作为量子力学的尺度相对论门户

应用分辨率-尺度相对性原理来发展不可微动力学路径的力学，我们发现，在一维中，静止运动对应于由 Riccati 方程的解驱动的 Itô 过程。我们验证了对应的 Fokker-Planck 方程求解的概率密度与相同问题的薛定谔方程解的平方模相对应。受到一维案例处理的启发，我们确定了对任意数量维度上的时间相关问题的概括。然后，Itô 过程由一个函数驱动，该函数被确定为在不可微动力学和标准量子力学之间建立联系。这是标准量子力学的尺度相对论解释的基础，