Random linear operators

TL;DR: In this article, the authors define a generalization of the Stochastic Integral Integral with respect to an X-valued Martingale, and define a linear transformation of the solution of this transformation.

Abstract: 1. Random Operators in Hilbert Space.- 1. Basic Definitions.- 1.1 Strong Random Operator.- 1.2 Weak Random Operator.- 1.3 Product of Random Operators.- 2. Characteristic Functions of Random Operators.- 2.1 Definition.- 2.2 Characteristic Functions of Strong and Bounded Operators.- 2.3 Gaussian Random Operators.- 3. Convergence of Random Operators.- 3.1 Weak Convergence of Random Operators.- 3.2 Strong Convergence of Random Operators.- 3.3 Convergence of Distributions corresponding to Random Operators.- 2. Functions of Random Operators.- 4. Spectral Representation for Symmetric Random Operators.- 4.1 Symmetric Random Operators and Selfadjoint Extensions.- 4.2 Spectral Representation of a Selfadjoint Random Operator.- 4.3 Spectral Representation of a Strong Symmetric Operator.- 5. Equations with Symmetric Random Operators.- 5.1 Evolution Equations.- 5.2 Schrodinger-type Equations.- 5.3 Spectral Moment Functions.- 5.4 Equation of Fredholm Type.- 6. Equations with Semi-Bounded Random Operators.- 6.1 Nonnegative Closed Random Operators.- 6.2 Resolvent of a Nonnegative Operator.- 6.3 Resolvent of a Nonnegative Random Operator.- 6.4 Equations of Fredholm Type.- 6.5 Equations of Evolution Type.- 3. Operator-Valued Martingales.- 7. Operator-Valued Martingale Sequences.- 7.1 Weak Operator-valued Martingale.- 7.2 Strong Operator-valued Martingales.- 7.3 Operator-valued Martingale.- 8. Convergence of Infinite Products of Independent Random Operators.- 8.1 Infinite Products as Martingales.- 8.2 Convergence of Infinite Products given the Existence of Two Moments.- 8.3 Convergence of Infinite Products in Absolute Norm.- 9. Continuous Operator-Valued Martingales.- 9.1 Some Properties of Continuous Real-valued Local Martingales.- 9.2 Continuous Martingales with values in X.- 9.3 Operator-valued Continuous Martingales.- 9.4 Strong Operator-valued Wiener Processes.- 4. Stochastic Integrals and Equations.- 10. Stochastic Integrals with Respect to an X-Valued Martingale.- 10.1 Definition.- 10.2 Integrals for Processes with Regular Characteristics.- 10.3 Stochastic Integral with respect to a Wiener Process.- 11. Stochastic Integral with Respect to an Operator-Valued Martingale.- 11.1 Integrals of X-valued Functions.- 11.2 Integrals of Operator-valued Functions.- 12. Stochastic Operator Equations.- 12.1 Operator-valued Functions of Random Operators.- 12.2 Stochastic Equations Involving I(Z, Y)t.- 12.3 Stochastic Equations Involving I*(Z, Y)t.- 12.4 Some Generalizations.- 5. Linear Stochastic Operator Equations.- 13. Generalization of the Stochastic Operator Integral.- 13.1 General Form of the Linear Equation.- 13.2 A Generalization of the Stochastic Integral.- 14. Linear Differential Operator Equations.- 14.1 Definition of a Linear Equation.- 14.2 Existence of Uniqueness of Solution.- 14.3 Linear Transformations of Solutions.- 14.4 Equations for Moments of the Solution of a Stochastic Equation.- 15. Continuous Stochastic Semigroups.- 15.1 Solutions of Simple Linear Equations -Stochastic Semigroups.- 15.2 Time Reversal in Stochastic Differential Equations.- 15.3 Definition of Stochastic Semigroups.- 15.4 Semigroups which are Martingales.