Real-Space renormalization for the drawups and drawdowns of stochastic trajectories: universality classes for diffusion processes and random walks via the scale function

For one-dimensional diffusion processes with arbitrary diffusion coefficient $D(x)$ and Ito force $F_I(x)$, we analyze the statistical properties of the renormalized trajectories at scale $Γ$, where one keeps only the alternating sequence of local minima and local maxima that are separated by drawups and drawdowns of amplitude bigger than $Γ$. The elementary RG-step between $Γ$ and $Γ+d Γ$ thus amounts to eliminate the drawups and drawdowns of amplitude $Γ$, a real-space-RG-procedure that is well-known for Brownian trajectories in the context of the Strong-Disorder-RG approach applied to the disorder landscape of random models. The main result is that the universality classes are determined by the scale function $s(x)$ satisfying $D(x) s''(x)+F_I(x) s'(x)=0$ that governs the exit probabilities from an interval : all the diffusions sharing the same scale function $s(x)$ have the same statistics of renormalized trajectories, even if their initial microscopic trajectories are completely different. The simplest universality classes for diffusions on the whole line $x \in ]-\infty,+\infty[$ and on the half-line $x \in ]0,+\infty[$ are illustrated with several examples of processes, with or without steady states: Cauchy, Student, Bessel, Fisher-Snedecor, Geometric Brownian. This framework concerning diffusion processes is then adapted to discrete-time and to continuous-time random walks on the one-dimensional lattice with arbitrary nearest-neighbor jump probabilities or rates.