The infinitesimal generator of the Brox diffusion

Antoine Mouzard (1, 2)

1 MINGUS - Multi-scale numerical geometric schemes
2 IRMAR - Institut de Recherche Mathématique de Rennes

We construct the infinitesimal generator of the Brox diffusion on a line with a periodic Brownian environment. This gives a new construction of the process and allows to solve the singular martingale problem. We prove that the associated semigroup is strong Feller with Gaussian lower and upper bounds. This also yields a construction of the Brox diffusion on a segment with periodic or Dirichlet boundary conditions. In this bounded space, we prove that there exists a unique measure and the existence of a spectral gap giving exponential ergodicity of the diffusion.

Mots clés: Singular diffusions; Distributional drift; Singular stochastic operators; Paracontrolled calculus; Brox diffusion

Domaines: Probabilités [math.PR]; Théorie spectrale [math.SP]

Antoine Mouzard. The infinitesimal generator of the Brox diffusion. 2022. ⟨hal-03908296⟩