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Wall turbulence consists of various sizes of vortical structures that induce flow circulation around a wide range of closed Eulerian loops. Here we investigate the multiscale properties of circulation around such loops in statistically homogeneous planes parallel to the wall. Using a high-resolution direct numerical simulation database of turbulent channels at Reynolds numbers of 
$Re_\tau =180$, 550, 1000 and 5200, circulation statistics are obtained in planes at different wall-normal heights. Intermittency of circulation in the planes of the outer flow (
$y^+ \gtrsim 0.1Re_\tau$) takes the form of universal bifractality as in homogeneous and isotropic turbulence. The bifractal character simplifies to space-filling character close to the wall, with scaling exponents that are linear in the moment order, and lower than those given by the Kolmogorov paradigm. The probability density functions of circulation are long-tailed in the outer bifractal region, with evidence showing their invariance with respect to the loop aspect ratio, while those in the inner region are closely Gaussian. The unifractality near the wall implies that the circulation there is not intermittent in character.
Let 
$(X_t)_{t \geqslant 0}$ be the solution of the stochastic differential equation where 
\[\mathrm{d} X_t = b(X_t) \,\mathrm{d} t+A\,\mathrm{d} Z_t, \quad X_{0}=x,\]
$b\,:\, \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a Lipschitz-continuous function, 
$A \in \mathbb{R}^{d \times d}$ is a positive-definite matrix, 
$(Z_t)_{t\geqslant 0}$ is a d-dimensional rotationally symmetric 
$\alpha$-stable Lévy process with 
$\alpha \in (1,2)$ and 
$x\in\mathbb{R}^{d}$. We use two Euler–Maruyama schemes with decreasing step sizes 
$\Gamma = (\gamma_n)_{n\in \mathbb{N}}$ to approximate the invariant measure of 
$(X_t)_{t \geqslant 0}$: one uses independent and identically distributed 
$\alpha$-stable random variables as innovations, and the other employs independent and identically distributed Pareto random variables. We study the convergence rates of these two approximation schemes in the Wasserstein-1 distance. For the first scheme, under the assumption that the function b is Lipschitz and satisfies a certain dissipation condition, we demonstrate a convergence rate of 
$\gamma^{\frac{1}{\alpha}}_n$. This convergence rate can be improved to 
$\gamma^{1+\frac {1}{\alpha}-\frac{1}{\kappa}}_n$ for any 
$\kappa \in [1,\alpha)$, provided b has the additional regularity of bounded second-order directional derivatives. For the second scheme, where the function b is assumed to be twice continuously differentiable, we establish a convergence rate of 
$\gamma^{\frac{2-\alpha}{\alpha}}_n$; moreover, we show that this rate is optimal for the one-dimensional stable Ornstein–Uhlenbeck process. Our theorems indicate that the recent significant result of [34] concerning the unadjusted Langevin algorithm with additive innovations can be extended to stochastic differential equations driven by an 
$\alpha$-stable Lévy process and that the corresponding convergence rate exhibits similar behaviour. Compared with the result in [6], our assumptions have relaxed the second-order differentiability condition, requiring only a Lipschitz condition for the first scheme, which broadens the applicability of our approach.
