Approximation numérique et comportement en temps long de quelques équations stochastiques singulières

In this thesis, we explore the numerical approximation and long-term behavior of certain stochastic equations driven by dissipative and/or distributional drift terms.Firstly, stochastic differential equations (SDEs) with fractional Brownian motion (fBm) and distributional drift are studied. The convergence of a tamed-Euler scheme is quantified. Similar techniques are applied to the stochastic heat equation (SHE) with space-time white noise and a distributional reaction term, and the same results are obtained for a tamed-Euler scheme with finite differences.Secondly, the focus shifts to SDEs with fBm and dissipative drift, establishing, in the long-term regime, the almost-sure regularity of solutions and their ergodic means with respect to time and the Hurst parameter. These results are applied in a statistical context to estimate the parameters of the equations through an approximation of their invariant measures.Finally, we combine dissipative and distributional reaction terms in SHE, presenting preliminary results in the long-term regime, the well-posedness of the equation is proven and the moments of the solution are uniformly bounded over time.