Multivariate simultaneous approximation

Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

“…Observe now that the choice (25) is optimal for the bound (26), and the result follows from Theorem 5. Finally, the last inequality in (24) follows from (26) recalling the inequality…”

Characterization of Stationary Distributions of Reflected Diffusions

“…To see that µ is a sigma-finite measure, fix any constant, say C = 1, and a compact subset K ⊂ K 1 . Let r, s > 0 be such that (34) holds, and let f = f r,s ∈ H be the function from Assumption 1 (2). Then h f ∈ T 0 , and substituting f in (32), we obtain…”

“…The first main result of this paper, Theorem 1, shows that under some analytical conditions (see Assumption 1), a probability measure π onḠ is a stationary distribution for a reflected diffusion defined by a well-posed submartingale problem if and only if π satisfies π(∂G) = 0 and (2) Ḡ Lf (x) dπ(x) ≤ 0 for all f belonging to H, a certain class of test functions defined in (3). A subtlety in this result lies in the correct choice of test functions in (2). See Remarks 2.4 and 5.2 for further discussion of this issue.…”

“…It follows from Echeverria's work that, given drift and dispersion coefficients b(·) and σ(·) that are associated with a well-posed martingale problem, a probability measure π is a stationary distribution for the corresponding diffusion if and only if (2) holds with inequality replaced by equality and for test functions f ∈ C 2 c (E), the space of twice continuously differentiable functions with compact support on E. Extensions of Echeverria's criterion were obtained by Bhatt and Karandikar [5], who relaxed the local compactness condition on E, and by Stockbridge [41] and Bhatt and Borkar [4], who extended it to controlled processes. An extension of Echeverria's criterion to reflected diffusions in smooth domains defined by well-posed submartingale problems was first obtained by Weiss in his PhD thesis [47].…”

Small perturbations of polynomial meshes

“…[21]) and euclidean balls (with m k = k, cf. [25]); see [2,23] for some recent results on the multivariate Jackson inequality.…”