The integral representation of the exponential product of stochastic semigroups

An integral representation is obtained for the exponential product of stochastic semigroups $$X_s^t \otimes Z_s^t = X_s^t + \mathop \smallint \limits_{s< u< t} X_u^t dV_u X_s^u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} X_{u_2 }^t dV_{u_2 } X_{u_1 }^{u_2 } dV_{u_1 } X_s^{u_1 } + \cdots ,$$ whereV _t is the generating process of the semigroupZ _s ^t and the integrals are understood in the sense of mean-square limits of the Riemann-Stieltjes sums. This representation is different from the traditional representation $$X_s^t \otimes Z_s^t = E + \mathop \smallint \limits_{s< u< t} dW_u + \mathop {\smallint \smallint }\limits_{s< u_1< u_2< t} dW_{u_2 } dW_{u_1 } + \cdots ,$$ in which the integration extends over the processW _t=Y_t+V_t that is the generating process of the exponential productX _s ^t ?Z _s ^t andY _t is the generator of the semigroupX _s ^t .     

Asymptotic behavior of a class of stochastic semigroups in the Bernoulli scheme

A family of subalgebras describing the space of complex-valued 2×2 matrices is selected. In this space, a stochastic semigroupY _n =X _n X _–1 ...X ₁,n= , is considered, where {X_i, i= } are independent equally distributed random matrices taking two values. For a stochastic semigroupY _n , whose phase space belongs to one of the subalgebras, the index of exponential growth is determined explicitly.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1580–1584, November, 1993.     

Slow recurrent regimes for a class of one-dimensional stochastic growth models

We classify the possible behaviors of a class of one-dimensional stochastic recurrent growth models. In our main result, we obtain nearly optimal bounds for the tail of hitting times of some compact sets. If the process is an aperiodic irreducible Markov chain, we determine whether it is null recurrent or positive recurrent and in the latter case, we obtain a subgeometric convergence of its transition kernel to its invariant measure. We apply our results in particular to state-dependent Galton–Watson processes and we give precise estimates of the tail of the extinction time.     

Interpolation solution in generalized stochastic exponential population growth model

In this paper, first we consider model of exponential population growth, then we assume that the growth rate at time t is not completely definite and it depends on some random environment effects. For this case the stochastic exponential population growth model is introduced. Also we assume that the growth rate at time t depends on many different random environment effect, for this case the generalized stochastic exponential population growth model is introduced. The expectations and variances of solutions are obtained. For a case study, we consider the population growth of Iran and obtain the output of models for this data and predict the population individuals in each year.     

Kernel estimates for a class of Kolmogorov semigroups

We prove short time pointwise upper bounds for the heat kernels of certain Kolmogorov operators. We use Lyapunov function techniques, where the Lyapunov functions depend also on the time variable. Received: 29 November 2007     

On exponential stability of contraction semigroups

A semigroup [T(t)] on a Hilbert space is exponentially stable if there exist real constants M≥1 and α>0 such that ∥T(t)∥≤Me ^−αt for every t≥0. If [T(t)] is a strongly continuous contraction semigroup, then it is proved that we can set M=1 in the definition of exponential stability if and only if the generator A of [T(t)] is boundedly strict dissipative (just a strict dissipative A is not enough).     

On a class of t-Archimedean semigroups

Communicated by H.-J. Hoehnke