We study the critical branching random walk on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {Z}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> started from a distant point x and conditioned to hit some compact set K in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {Z}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>. We are interested in the occupation time in K and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Vert x\Vert ^{4-d}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">‖</mo> <mi>x</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <mn>4</mn> <mo>-</mo> <mi>d</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> in dimensions <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$d\le 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, of order <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\log \Vert x\Vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mo stretchy="false">‖</mo> <mi>x</mi> <mo stretchy="false">‖</mo> </mrow> </math></EquationSource> </InlineEquation> in dimension <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$d=4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, and of order 1 in dimensions <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$d\ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle [<CitationRef CitationID="CR37">37</CitationRef>].

Yaglom theorem for critical branching random walk on $\mathbb {Z}^{d}$

期刊论文
发表日期：

2026年7月27日

查看全文

Xinxin Chen,
Shen Lin

摘要

We study the critical branching random walk on $\mathbb {Z}^d$ $Z^{d}$ started from a distant point x and conditioned to hit some compact set K in $\mathbb {Z}^d$ $Z^{d}$ . We are interested in the occupation time in K and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order $\Vert x\Vert ^{4-d}$ ${‖ x ‖}^{4 - d}$ in dimensions $d\le 3$ $d \leq 3$ , of order $\log \Vert x\Vert $ $log ‖ x ‖$ in dimension $d=4$ $d = 4$ , and of order 1 in dimensions $d\ge 5$ $d \geq 5$ . The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle [37].

Yaglom theorem for critical branching random walk on \(\mathbb {Z}^{d}\)

摘要