Stochastic PDEs in 𝒮' for SDEs driven by Lévy noise

In this article we show that a finite-dimensional stochastic differential equation driven by a Lévy noise can be formulated as a stochastic partial differential equation (SPDE) driven by the same Lévy noise. We prove the existence result for such an SPDE by Itô’s formula for translation operators, and the uniqueness by an adapted form of “Monotonicity inequality”, proved earlier in the diffusion case. As a consequence, the solutions that we construct have the “translation invariance” property.

Dual mixed complex brightness integrals

In this paper, we define the dual mixed complex brightness integrals and establish related Brunn-Minkowski type inequality, Aleksandrov-Fenchel inequality, cyclic inequality and monotonicity inequality, respectively. As applications, we give the analogous version of the differences inequalities for the dual mixed complex brightness integrals

QUANTUM f-DIVERGENCES AND ERROR CORRECTION

Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.

THE MONOTONICITY INEQUALITY FOR LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

We prove the monotonicity inequality for differential operators A and L that occur as coefficients in linear stochastic partial differential equations associated with finite-dimensional Itô processes. We characterize the solutions of such equations. A probabilistic representation is obtained for solutions to a class of evolution equations associated with time dependent, possibly degenerate, second-order elliptic differential operators.