The stochastic formalism of inflation allows us to describe the scalar-field dynamics in a non-perturbative way. The correspondence between the diffusion and Schrödinger equations makes it possible to exhaustively construct analytical solutions in stochastic inflation. Those exact statistical quantities such as distribution and correlation functions have one-to-one correspondence to the exactly solvable solutions in non-relativistic quantum mechanics in terms of classical orthogonal polynomials. A class of such solutions is presented by means of isospectral Hamiltonians with an underlying symmetry called shape invariance.

中文翻译：

---

完全可解的随机观察者


通货膨胀的随机形式主义使我们能够以非扰动的方式描述标量场动态。扩散方程和薛定谔方程之间的对应关系使得在随机膨胀中穷举地构建解析解成为可能。这些精确的统计量，如分布和相关函数，与非相对论量子力学中经典正交多项式的精确可解解一一对应。一类这样的解是通过等谱哈密顿量表示的，其基本对称性称为形状不变性。