# The two systems of differential equations

• Kai Lai Chung
Chapter
Part of the Die Grundlehren der Mathematischen Wissenschaften book series (volume 104)

## Abstract

Let us recall the dual equations: if qi<∞,
$$p'_{ij} \left( {s + t} \right) = \sum\limits_k {p'_{ik} } \left( s \right)p_{ij} \left( t \right);$$
(3.5 bis)
and if q i <∞,
$$p'_{ij} \left( {t + s} \right) = \sum\limits_k {p_{ik} } \left( t \right)p'_{kj} \left( s \right);$$
(3.11 bis)
both valid for s > 0, t ≧ 0. The limiting cases for s = 0 may be written as
$$p'_{ij} \left( t \right) = - q_i p_{ij} \left( t \right) + \sum\limits_{k \ne i} {q_{ik} } p_{kj} \left( t \right);$$
(1ij)
$$p'_{ij} \left( t \right) = - p_{ij} (t)q_j + \sum\limits_{k \ne i} {p_{ik} } (t)q_{kj} ;$$
(2ij)

## Keywords

Minimal Solution Continuous Parameter Sample Function Stationary Transition Probability Dual Equation
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