Let \((E,{\mathcal F})\) be a measurable space and \({\mathfrak M}\) the collection of all probability measures on \((E,{\mathcal F})\). Denote by \(P(s,x,y,t,A)\), \(x,y\in E\), \(A\in{\mathcal F}\), the probability of obtaining an element from the set \(A\) at the moment \(t\) provided that the elements \(x\) and \(y\) interact at the moment \(s\). Let \(m_ 0\in{\mathfrak M}\) be an initial measure. Let the system of transition functions \(\{P(s,x,y,t,A)\): \(x,y\in E\), \(A\in{\mathcal F}\), \(s,t\in R^ +\}\) satisfy the following conditions:
1. \(P(s,x,y,t,A)=P(s,y,x,t,A)\) and it is defined for \(t-s\geq 1\), \(x,y\in E\), \(A\in{\mathcal F}\).
2. \(P(s,x,y,s+1,A)=P(0,x,y,1,A)\) for any \(s\geq 1\) and arbitrary \(x,y\in E\) and \(A\in {\mathcal F}\).
3. \(P(s,x,y,t,\cdot)\) is a probability measure on \((E,{\mathcal F})\) for any \(x,y\in E\).
4. \(P(s,x,y,t,A)\) as a function of two variables is measurable with respect to \((E\times E,{\mathcal F}\otimes{\mathcal F})\) for any \(A\in{\mathcal F}\).
5. Analogues of Kolmogorov-Chapman equation. There are two versions. For any \(s<\tau<t\) such that \(t-\tau\geq 1\) and \(\tau-s\geq 1\) either \[ P(s,x,y,t,A)=\int_ E\int_ E P(s,x,y,\tau,du)P(\tau,u,v,t,A)m_ \tau(dv),\tag{5a} \] where the measure \(m_ t\) on \((E,{\mathcal F})\) is defined as \[ m_ t(B)=\int_ E\int_ E P(0,x,y,t,B)m_ 0(dx)m_ 0(dy),\qquad B\in{\mathcal F}, \] or \[ P(s,x,y,t,A)=\int_ E\int_ E\int_ E\int_ E P(s,x,z,\tau,du)P(s,y,v,\tau,dw)P(\tau,u,w,t,A)m_ s(dz)m_ s(dv).\tag{5b} \] Then the process defined by the transition functions \(P(s,x,y,t,A)\) is called a quadric stochastic process of type \(A\) or type \(B\), respectively, in accordance with the fundamental condition (5a) or (5b).
Equations (5a) or (5b) can be interpreted as difference rules of the appearance of the “grandson”. These equations have implications also for chemical treatments. So the appearance of a particle in reactions passing in ordinary chemical kinetics is described by equations of type (5a), and the equations of type (5b) reflect the appearance of particles in processes of catalysis.
Theorem. Let \(P(s,x,y,t,A)\) be a transition function determining quadric stochastic process of type \(A\) or \(B\) and \(m_ 0\in{\mathfrak M}\) an initial distribution. Then the function \[ Q(s,x,t,A)=\int_ E P(s,x,y,t,A)m_ s(dy) \] is the transition function for some Markov process with the initial distribution \(m_ 0\). For quadric stochastic processes differential equations similar to Kolmogorov equations are deduced.