Asymptotic behavior of the mean value for functionals of a random field of particles that is determined by a branching random walk

Consider a homogeneous Markov process with continuous time on the phase space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Subscript plus Baseline equals StartSet 0 comma 1 comma 2 comma ellipsis EndSet"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msub> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb Z_+ = \{0, 1, 2, \dots \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which is interpreted as the movement of a particle. The particle can only move to neighboring points of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Subscript plus"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb Z_+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , i.e., with each change in position, its coordinate changes by one. The process is equipped with a branching mechanism. Branching sources can be located at each point in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Subscript plus"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb Z_+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . At the moment of branching, new particles appear at the branching point and then evolve independently of each other (and of the other particles) in accordance with the same laws as the initial particle. At each time <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we have a random field on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Subscript plus"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb Z_+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of particles present in the system at that moment. Functionals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript left-parenthesis m Subscript j Baseline comma m Subscript k Baseline right-parenthesis Endscripts normal upper Phi left-parenthesis m Subscript j Baseline comma m Subscript k Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:munder> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum _{(m_j, m_k)}\Phi (m_j, m_k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of this field are considered, where the sum is taken over all ordered pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis m Subscript j Baseline comma m Subscript k Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(m_j, m_k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of different particles in the field. The asymptotic behavior of the mean value of this functional as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t right-arrow plus normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">t \to +\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is studied.