Total variation distance between SDEs with stable noise and Brownian motion with applications to Poisson PDEs

We consider a d-dimensional stochastic differential equation (SDE) of the form dUₜ = b(Uₜ)d t + σ d Zₜ with initial condition U₀ = u∈ Rᵈ and additive noise. Denote by Xₜˣ the (unique) solution with initial value X₀ˣ = x, if the driving noise Zₜ is a d-dimensional rotationally symmetric α-stable (1<α<2) L\'evy process, and by Yₜʸ the (unique) solution with initial value Y₀ʸ=y, if the driving noise is a d-dimensional Brownian motion. We give precise estimates of the total variation distance \| law (Xₜˣ)- law (Yₜʸ)\|TV for all $t>0$, and we show that the ergodic measures μ_α and μ₂ of Xₜˣ and Yₜʸ, respectively, satisfy \|μ_α-μ₂\|TV ≤ 2[Cd3/2α-1(2-α)]. As an application, we consider the following d-dimensional Poisson PDE {gather*} -(-Δ)α/2 f(x) + b(x), ∇ f(x) = h(x)-μ_α(h), {gather*} where h ∈ Cb(Rᵈ,R) and 1<α≤ 2. We show that this PDE admits a solution f_α, which is continuous, but may have linear growth. For fixed α₀∈(1,2), we show that for any α₀≤ α <2 there is a constant C(α₀) such that {gather*} |f_α(x)-f_2(x)| ≤ C(α_0)(1+|x|)\|h\|_∞ d3/2(2-α)log1/2-α all x. {gather*}