Integrable Ito equations and properties of the associated Fokker-Planck equations (Q6580257)

The authors start from the scalar Ito equations written as: \( dx=f(x,t)dt+\sigma (x,t)dw\), where \(w=w(t)\) is the driving Wiener process, \(f \) the drift coefficient and \(\sigma \) the noise coefficient, especially considering the case where \(f\) and \(\sigma \) depend only on \(x\). Looking for Lie-point symmetries of this equation, the authors consider vector fields of the form: \(X=\vartheta (t)\partial _{t}+\varphi (x,t;w)\partial _{x}\), especially focusing on the case where \(\vartheta =0\) (time-preserving symmetries). They classify the symmetries according to the value of the derivative \(\varphi _{w}\) and they write the associated determining equations. They then consider a general Fokker-Planck equation for \(u=u(x,t)\) written in the form: \(\frac{ \partial u}{\partial t}+\frac{\partial }{\partial x}[\alpha (x,t)u]-\frac{1}{ 2}\frac{\partial ^{2}}{\partial x^{2}}[\beta ^{2}(x,t)u]=0\), and they recall some known results concerning the symmetries. Considering a unit noise (\( \sigma =1\)) for the scalar Ito equations, the authors recall the classification of time-preserving symmetries they obtained in their paper [Open Commun. Nonlinear Math. Phys. 2, 53--101 (2022; Zbl 1547.34088)]. They also describe the autonomous case (\(f\) independent of \(t\)). Still in the case \(\sigma =1\), they rewrite the Fokker-Planck equations in the form: \(u_{t}+fu_{x}+f_{x}u-\frac{1}{2} u_{xx}=0\), and define \(\gamma =-\frac{1}{2}(f^{2}+f_{x})\). They consider three cases: (A) \(f(x,t)=h(t)\), (B) \(f(x,t)=h(t)+k(t)x\), (C) \( f(x,t)=h(t)+k(t)e^{\beta x}\), and they draw computations in each case to determine the symmetries. The paper ends with an example.