Taming discrete rough paths via strong Lyapunov functions

Based on the newly introduced concept of {\it strong Lyapunov functions} for rough differential equations \cite{ducjost25}, we study a tamed numerical scheme to approximate the solutions of the continuous system.

We derive explicit estimates of solution norms of the tamed system which look similar to those of the continuous system.

As a result, we prove the convergence of the tamed scheme in the $L^1$ sense.

For systems with the negative gradient condition, we prove the existence of a numerical pullback attractor for the generated random dynamical system from the tamed numerical scheme which is integrable and upper semi-continuous w.r.t. the scheme step size.