We consider the local times of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(1 + \beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-stable d-dimensional super-Brownian motion with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$0< \beta < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>β</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. It is known from Sugitani (J Math Soc Jpn 41(3):437–462, 1989) that for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the local time is differentiable for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$0< \beta < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>β</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, Mytnik and Perkins (Ann Probab 31(3): 1413–1440, 2003) proved that the local time, denoted by L(t, x), is jointly continuous for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$d = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, while it is locally unbounded in x for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$d \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> where it exists. This paper strengthens the results of Mytnik and Perkins for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$d=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> by showing that the local time L(t, x) is continuously differentiable in the spatial parameter x. Moreover, we give a representation of the spatial derivative, denoted by <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\frac{\partial }{\partial x}L(t, x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mi>∂</mi> <mrow> <mi>∂</mi> <mi>x</mi> </mrow> </mfrac> <mi>L</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and further prove that the derivative is locally <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>-Hölder continuous in x for any index <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\gamma \in (0, \frac{\beta }{1+\beta })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mfrac> <mi>β</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

On the Differentiability of Local Times of ($1+\beta $)-Stable Super-Brownian Motion

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发表日期：

2026年2月23日

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Ziyi Chen,
Jieliang Hong

摘要

We consider the local times of $(1 + \beta )$ $(1 + β)$ -stable d-dimensional super-Brownian motion with $0< \beta < 1$ $0 < β < 1$ . It is known from Sugitani (J Math Soc Jpn 41(3):437–462, 1989) that for $\beta = 1$ $β = 1$ , the local time is differentiable for $d=1$ $d = 1$ . For $0< \beta < 1$ $0 < β < 1$ , Mytnik and Perkins (Ann Probab 31(3): 1413–1440, 2003) proved that the local time, denoted by L(t, x), is jointly continuous for $d = 1$ $d = 1$ , while it is locally unbounded in x for $d \ge 2$ $d \geq 2$ where it exists. This paper strengthens the results of Mytnik and Perkins for $d=1$ $d = 1$ by showing that the local time L(t, x) is continuously differentiable in the spatial parameter x. Moreover, we give a representation of the spatial derivative, denoted by $\frac{\partial }{\partial x}L(t, x)$ $\frac{\partial}{\partial x} L (t, x)$ , and further prove that the derivative is locally $\gamma $ $γ$ -Hölder continuous in x for any index $\gamma \in (0, \frac{\beta }{1+\beta })$ $γ \in (0, \frac{β}{1 + β})$ .

On the Differentiability of Local Times of (\(1+\beta \))-Stable Super-Brownian Motion

摘要