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Abstract: This paper is concerned with the distribution properties of the binomial , where is a gamma random variable. We show in particular that is infinitely divisible for all and , and that for the second order polynomial is a generalized gamma convolution whose Thorin density and Wiener–gamma integral representation are computed explicitly. As a byproduct we deduce that fourth order multiple Wiener integrals are in general not infinitely divisible. [Copyright &y& Elsevier]

Abstract: Given a divergence operator δ on a probability space such that the law of is infinitely divisible with characteristic exponent we derive a family of Laplace transform identities for the derivative when u is a non-necessarily adapted process. These expressions are based on intrinsic geometric tools such as the Carleman–Fredholm determinant of a covariant derivative operator and the characteristic exponent (0.1), in a general framework that includes the Wiener space, the path space over a Lie group, and the Poisson space. We use these expressions for measure characterization and to prove the invariance of transformations having a quasi-nilpotent covariant derivative, for Gaussian and other infinitely divisible distributions. [Copyright &y& Elsevier]

Abstract: We investigate explosion in finite time of one-dimensional semilinear equations of the form with initial value , where is positive and , are constants. In the free case we provide conditions on φ under which any positive nontrivial solution is non-global. In the case and , , which includes in the special case the equation we use the Feynman–Kac formula for Bessel processes to give conditions on the equation parameters ensuring finite-time blowup and existence of nontrivial positive global solutions. [Copyright &y& Elsevier]

Abstract: We investigate finite-time blow-up and stability of semilinear partial differential equations of the form , , , where Γ is the generator of the standard gamma process and , , are constants. We show that any initial value satisfying , , for some positive constants , , , yields a non-global solution if . If , where , and , then the solution is global and satisfies , for some constant . This complements the results previously obtained in [M. Birkner et al., Proc. Amer. Math. Soc. 130 (2002) 2431; M. Guedda, M. Kirane, Bull. Belg. Math. Soc. Simon Stevin 6 (1999) 491; S. Sugitani, Osaka J. Math. 12 (1975) 45] for symmetric α-stable generators. Systems of semilinear PDEs with gamma generators are also considered. [Copyright &y& Elsevier]

The non-commutative Malliavin calculus on the Heisenberg–Weyl algebra is extended to the affine algebra. A differential calculus and a non-commutative integration by parts are established. As an application we obtain sufficient conditions for the smoothness of Wigner-type laws of non-commutative random variables with gamma or continuous binomial marginals. [Copyright &y& Elsevier]

We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any orders, by combining the use of neural networks with a Monte Carlo branching algorithm. In comparison with other deep learning PDE solvers, it also allows us to check the consistency of the learned neural network function. Numerical experiments presented show that this algorithm can outperform deep learning approaches based on backward stochastic differential equations or the Galerkin method, and provide solution estimates that are not obtained by those methods in fully nonlinear examples. [ABSTRACT FROM AUTHOR]