Bound for an Approximation of Invariant Density of Diffusions via Density Formula in Malliavin Calculus

The Kolmogorov and total variation distance between the laws of random variables have upper bounds represented by the $L^1$-norm of densities when random variables have densities. In this paper, we derive an upper bound, in terms of densities such as the Kolmogorov and total variation distance, for several probabilistic distances (e.g., Kolmogorov distance, total variation distance, Wasserstein distance, Forter–Mourier distance, etc.) between the laws of F and G in the case where a random variable F follows the invariant measure that admits a density and a differentiable random variable G, in the sense of Malliavin calculus, and also allows a density function.