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We prove a Wegner estimate for discrete Schrödinger operators with a potential given by a Gaussian random process. The only assumption is that the covariance function decays exponentially; no monotonicity assumption is required. This improves earlier results where abstract conditions on the conditional distribution, compactly supported and non-negative, or compactly supported covariance functions with positive mean are considered.

In this paper, we deal with an optimal control, where the system is driven by a mean-field forward-backward doubly stochastic differential equation with jumps diffusion. We assume that the set of admissible control is convex, and we establish a necessary as well as a sufficient optimality condition for such system.

This paper proves the existence and uniqueness of a solution to reflected backward stochastic differential equations with a lower obstacle, which is assumed to be right upper-semicontinuous. The result is established where the coefficient is stochastic Lipschitz by using some tools from the general theory of processes such as Mertens decomposition of optional strong supermartingales and other tools from optimal stopping theory.

We study random Anderson Hamiltonians in Euclidean spaces with a long-range particle-media interaction potential 𝔲⁢(r)=r-A{\mathfrak{u}(r)=r^{-A}}. Improving earlier results, for any A>2⁢d{A>2d}, we establish spectral and strong dynamical localization with sub-exponential decay of eigenfunction correlators, both in the strong disorder regime and at low energies.

We analyze two weak random operators, initially motivated from processes in random environment. At first glance, these operators are ill-defined, but using bilinear forms, one can deal with them in a rigorous way. This point of view can be found, for instance, in [A. V. Skorohod, Random Linear Operators, Math. Appl. (Sov. Ser.), D. Reidel Publishing, Dordrecht, 1984], and it remarkably helps to carry out specific calculations. In this paper, we find explicitly the inverse of such weak operators by providing the closed forms of the so-called Green kernel. We show how this approach helps to analyze the spectra of the operators. In addition, we provide the existence of strong operators associated to our bilinear forms. Important tools that we use are the Sturm–Liouville theory and the stochastic calculus.

The limit G -Law for the solutions of the systems of linear algebraic equations with independent random coefficients is proven under the G -Lindeberg condition.