jats:pA general model of a branching Markov process on jats:italicℝ</jats:italic> is considered. Sufficient and necessary conditions are given for the random variable jats:disp-formula

<jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" position="anchor" xlink:href="graphic/ista210796-math-0001.png">jats:alt-texturn:x-wiley:20903332:media:ista210796:ista210796-math-0001</jats:alt-text></jats:graphic> </jats:disp-formula>

to be finite. Here jats:italicΞ</jats:italic>jats:subjats:italick</jats:italic></jats:sub>(jats:italict</jats:italic>) is the position of the jats:italick</jats:italic>th particle, and jats:italicN</jats:italic>(jats:italict</jats:italic>) is the size of the population at time jats:italict</jats:italic>. For some classes of processes (smooth branching diffusions with Feller‐type boundary points), this results in a criterion stated in terms of the linear <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/ista210796-math-0002.png" xlink:title="urn:x-wiley:20903332:media:ista210796:ista210796-math-0002"/>. Here jats:italicσ</jats:italic>(jats:italicx</jats:italic>) and jats:italica</jats:italic>(jats:italicx</jats:italic>) are the diffusion coefficient and the drift of the one‐particle diffusion, respectively, and jats:italicλ</jats:italic>(jats:italicx</jats:italic>) and jats:italick</jats:italic>(jats:italicx</jats:italic>) the intensity of branching and the expected number of offspring at point jats:italicx</jats:italic>, respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integral jats:italicμ</jats:italic>(jats:italicx</jats:italic>)∫jats:italicπ</jats:italic>(jats:italicx</jats:italic>, jats:italicd</jats:italic>jats:italicy</jats:italic>)(jats:italicf</jats:italic>(jats:italicy</jats:italic>) − jats:italicf</jats:italic>(jats:italicx</jats:italic>)) and the product jats:italicλ</jats:italic>(jats:italicx</jats:italic>)(1 − jats:italick</jats:italic>(jats:italicx</jats:italic>))jats:italicf</jats:italic>(jats:italicx</jats:italic>), where jats:italicλ</jats:italic>(jats:italicx</jats:italic>) and jats:italick</jats:italic>(jats:italicx</jats:italic>) are as before, jats:italicμ</jats:italic>(jats:italicx</jats:italic>) is the intensity of jumping at point jats:italicx</jats:italic>, and jats:italicπ</jats:italic>(jats:italicx</jats:italic>, jats:italicd</jats:italic>jats:italicy</jats:italic>) is the distribution of the jump from jats:italicx</jats:italic> to jats:italicy</jats:italic>.</jats:p>