# NOAA's National Service discovery kits. Volume 1, Corals, by United States. National Oceanic and Atmospheric

By United States. National Oceanic and Atmospheric Administration

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If P(x, E) :::; 1, then the kernel is s aid to be substochastic. Stochastic kernel P(x, B) determines transition (jump) probabilities from the state x into the set of states B for the stochastic system under consideration. 1. In a discrete phase space E = {I, 2,·· . , N} stochastic matrix P = [Pij; i,j E E] all elements of which are non-negative: i E E. 2. Let us have a probability distribution P(B) on the real line R = E. Then the stochastic kernel P(x, B) := P(B - x) defines a random walk on R: n /'l,n = L k=O ak, n ~ 0 where ak, k ~ 1 are independent identically distributed random variables with the distribution function P(B) = P{ak E B}.

IIcp = cp. 28) under condition IICP1 = O. 27). 29). 30) is obvious. 2. 3 is valid for the closed operator Q with the common dense domain Bo. o is a closed densely defined operator [12]. Phase merging algorithms 53 There are various situations when the operator Qo is not invertible. We will utilize the situation when the operator Ql is reducible-invertible. 4. 33) ~ -1 has the inverse operator Q2 . 4. 31) are valid. The following analysis is a somewhat different one. 31) gives us Ill{) = O. 31) with respect to the vector l{)1 has the following form: IlQIIll{) = After contraction on the subspace o.

Under some natural requirements on regularity the trajectories of Markov jump processes (MJP) behave as follows: an MJP occupies each state during a random time and has exponential distribution with some parameter depending on the state. Then a jump into a new state occurs according to the transition probabilities of a Markov chain with discrete time. Thus the MJP can be constructively defined by a stochastic kernel which determines the transition probabilities of the Markov chain and by non-negative function on states, which gives the intensities of exponential distributed sojourn times of the MJP in the states between two consecutive jumps.