A number of authors have proposed stochastic versions of the Schrödinger equation, either as effective evolution equations for open quantum systems or as alternative theories with an intrinsic collapse mechanism. We discuss here two directions for generalization of these equations. First, we study a general class of norm preserving stochastic evolution equations, and show that even after making...

Classical age-structured mass-action models such as the McKendrick-von Foerster equation have been extensively studied but are unable to describe stochastic fluctuations or population-size-dependent birth and death rates. Stochastic theories that treat semi-Markov age-dependent processes using, e.g., the Bellman-Harris equation do not resolve a population's age structure and are unable to quant...

The chemical master equation (CME) is the fundamental evolution equation of the stochastic description of biochemical reaction kinetics. In most applications it is impossible to solve the CME directly due to its high dimensionality. Instead indirect approaches based on realizations of the underlying Markov jump process are used such as the stochastic simulation algorithm (SSA). In the SSA, howe...

In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E,

BACKGROUND Migration between local populations plays an important role in evolution - influencing local adaptation, speciation, extinction, and the maintenance of genetic variation. Like other evolutionary mechanisms, migration is a stochastic process, involving both random and deterministic elements. Many models of evolution have incorporated migration, but these have all been based on simplif...

Evolution equations like the heat or diffusion equation or the Schrödinger equation can be associated with stochastic processes. We introduce a notion of nilpotency for quantum groups / braided groups that extends that for (simply-connected) Lie groups, and study the relations between evolution equations and (quantum) stochastic processes on these structures. We show on some examples how polyno...

2.1 Quantum stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Weyl operators and the Bose fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Quantum stochastic integrals . . . . . . . . . . . . . . . . . . . . . . . . . ...

We consider stochastic gradient descent for continuous-time models. Traditional approaches for the statistical estimation of continuous-time models, such as batch optimization, can be impractical for large datasets where observations occur over a long period of time. Stochastic gradient descent provides a computationally efficient method for such statistical learning problems. The stochastic gr...

*Correspondence: [email protected] School of Mathematics and Statistics, Chong Qing University, Chong Qing, P.R. China Abstract The existence and uniqueness of a mild solution to stochastic equations with jumps are established, a stochastic Fubini theorem and a type of Burkholder-Davis-Gundy inequality are proved, and the two formulas are used to study the regularity property of the mild sol...

Introduction Conclusions References Tables Figures Back Close Full Screen / Esc This discussion paper is/has been under review for the journal Atmospheric Chemistry and Physics (ACP). Please refer to the corresponding final paper in ACP if available. Abstract Introduction Conclusions References Tables Figures Back Close Full Screen / Esc Abstract In cloud modeling studies, the time evolution of...