Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators

The Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable U ( x , ω ) is M δ ( y , ω ) = A ( U ( x , ω ) ) + δ  ( y , ω ) , where A is a finitely many orders smoothing linear hypoelliptic operator and δ > 0 is the noise magnitude. The covariance operator C U of U is smoothing of order 2 r , self-adjoint, injective and elliptic pseudodifferential operator. If  was taking values in L 2 then in Gaussian case solving the conditional mean (and maximum a posteriori ) estimate is linked to solving the minimisation problem T δ ( m δ ) = arg min u ∈ H r { ∥ Au − m δ ∥ L 2 2 + δ 2 ∥ C U − 1