The Limit Behavior Of The Trajectories of Dissipative Quadratic Stochastic Operators on Finite Dimensional Simplex

The limit behavior of trajectories of dissipative quadratic stochastic operators on a finite-dimensional simplex is fully studied. It is shown that any dissipative quadratic stochastic operator has either unique or infinitely many fixed points. If dissipative quadratic stochastic operator has a unique point, it is proven that the operator is regular at this fixed point. If it has infinitely many fixed points, then it is shown that $\omega-$ limit set of the trajectory is contained in the set of fixed points.Comment: 14 pages, accepted in Difference Eq. App

Quantum stochastic equation for the low density limit

A new derivation of quantum stochastic differential equation for the evolution operator in the low density limit is presented. We use the distribution approach and derive a new algebra for quadratic master fields in the low density limit by using the energy representation. We formulate the stochastic golden rule in the low density limit case for a system coupling with Bose field via quadratic interaction. In particular the vacuum expectation value of the evolution operator is computed and its exponential decay is shown.Comment: Replaced with version published in J. Phys. A. References are adde

On the Lebesgue nonlinear transformations

In this paper, we introduce a quadratic stochastic operators on the set of all probability measures of a measurable space. We study the dynamics of the Lebesgue quadratic stochastic operator on the set of all Lebesgue measures of the set [0,1]. Namely, we prove the regularity of the Lebesgue quadratic stochastic operatorsComment: 11 page

A nonlinear Bismut-Elworthy formula for HJB equations with quadratic Hamiltonian in Banach spaces

We consider a Backward Stochastic Differential Equation (BSDE for short) in a Markovian framework for the pair of processes $(Y,Z)$, with generator with quadratic growth with respect to $Z$. The forward equation is an evolution equation in an abstract Banach space. We prove an analogue of the Bismut-Elworty formula when the diffusion operator has a pseudo-inverse not necessarily bounded and when the generator has quadratic growth with respect to $Z$. In particular, our model covers the case of the heat equation in space dimension greater than or equal to 2. We apply these results to solve semilinear Kolmogorov equations for the unknown $v$, with nonlinear term with quadratic growth with respect to $\nabla v$ and final condition only bounded and continuous, and to solve stochastic optimal control problems with quadratic growth

Well Posedness of Operator Valued Backward Stochastic Riccati Equations in Infinite Dimensional Spaces

We prove existence and uniqueness of the mild solution of an infinite dimensional, operator valued, backward stochastic Riccati equation. We exploit the regularizing properties of the semigroup generated by the unbounded operator involved in the equation. Then the results will be applied to characterize the value function and optimal feedback law for a infinite dimensional, linear quadratic control problem with stochastic coefficients