\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
We study multiple fields inflation in the diffusion-dominated regime using stochastic ÃÂ´ N formalism. The fields are under pure Brownian motion in a de Sitter background with boundaries in higher-dimensional field space. This setup can be realized toward the final stages of the ultra-slow-roll setup where the classical drifts fall off exponentially and the perturbations are driven by quantum kicks. We consider both symmetric and asymmetric boundaries with absorbing and reflective boundary conditions and calculate the average number of
e-folds, the first crossing probabilities, and the power spectrum. We study the primordial black holes (PBHs) formation in this setup and calculate the mass fraction and the contribution of PBHs in dark matter energy density for various higher-dimensional field spaces.
\end{document}