Uniform law of large numbers
We review some recent results in quantitative stochastic homogenization for divergence-form, quasilinear elliptic equations. In particular, we are interested in obtaining -type bounds on the gradient of solutions and thus giving a demonstration of the principle that solutions of equations with random coefficients have much better regularity (with overwhelming probability) than a general equation with non-constant coefficients.
Suppose that is a finite, connected graph and is a lazy random walk on . The lamplighter chain associated with is the random walk on the wreath product , the graph whose vertices consist of pairs where is a labeling of the vertices of by elements of and is a vertex in . There is an edge between and in if and only if is adjacent to in and for all . In each step, moves from a configuration by updating to using the transition rule of and then sampling both...
We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.
We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.
We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the -measure.
For α∈(0, 1) an α-trimming, P∗, of a probability P is a new probability obtained by re-weighting the probability of any Borel set, B, according to a positive weight function, f≤1/(1−α), in the way P∗(B)=∫Bf(x)P(dx). If P, Q are probability measures on euclidean space, we consider the problem of obtaining the best L2-Wasserstein approximation between: (a) a fixed probability and trimmed versions of the other; (b) trimmed versions of both probabilities. These best trimmed approximations naturally...
In the limit theory for strictly stationary processes , the decomposition proved to be very useful; here is a bimeasurable and measure preserving transformation an is a martingale difference sequence. We shall study the uniqueness of the decomposition when the filtration of is fixed. The case when the filtration varies is solved in [13]. The necessary and sufficient condition of the existence of the decomposition were given in [12] (for earlier and weaker versions of the results see [7])....
We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpi´nski carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.
Consider an infinite dimensional diffusion process process on , where is the circle, defined by the action of its generator on local functions as . Assume that the coefficients, and are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that is only a function of and that . Suppose is an invariant product measure. Then, if is the Lebesgue measure or if , it is the unique invariant measure. Furthermore, if is translation invariant, then...