Volume 26, Issue 2

In this work, we establish some new forms of stochastic Gronwall–Bellman inequalities. We illustrate our work with examples of applications.

In this paper, we give a probabilistic representation of the heat equation associated with the quantum K-Gross Laplacian using infinite-dimensional stochastic calculus in two variables. Applying the heat semigroup to the particular case where the operator is the multiplication one, we establish a relation between the classical and the quantum heat semigroup. Finally, using a combination between convolution calculus and the generalized stochastic calculus, we give a generalization of the Feynman–Kac formula.

The Circular Law under Lindeberg’s condition for the independent blocks of random matrices having zero expectations and double stochastic matrix of covariances of their array is proven.

The lower bounds for the minimal singular eigenvalue of the matrix are obtained under the G -Lindeberg condition and the G -double stochastic condition for the variances of the matrix entries. The new method is based on the G -method of perpendiculars, the REFORM method, the martingale method, and the theory of canonical spectral equations.

The Inverse Tangent Law for the solutions of systems of linear algebraic equations with independent random coefficients is proven under Lindeberg’s condition.