Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2024_442_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$B^{a,b}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>B</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> be a weighted-fractional Brownian motion with Hurst indexes a and b such that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2024_442_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$a>-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>></mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2024_442_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="138" /> </InlineMediaObject> <EquationSource Format="TEX">$0<b<1\wedge (1+a)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>b</mi> <mo><</mo> <mn>1</mn> <mo>∧</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we consider the linear self-attracting diffusion <Equation ID="Equ78"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2024_442_Article_Equ78.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="341" /> </MediaObject> <EquationSource Format="TEX">$ dX^{a,b}_t=dB^{a,b}_t-\theta \left( \int _0^t(X^{a,b}_t-X^{a,b}_s)ds-\nu \right) dt $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>d</mi> <msubsup> <mi>X</mi> <mi>t</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <mi>d</mi> <msubsup> <mi>B</mi> <mi>t</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msubsup> <mo>-</mo> <mi>θ</mi> <mfenced close=")" open="("> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>t</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>X</mi> <mi>t</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>X</mi> <mi>s</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>s</mi> <mo>-</mo> <mi>ν</mi> </mfenced> <mi>d</mi> <mi>t</mi> </mrow> </math></EquationSource> </Equation>with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2024_442_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$X^{a,b}_0=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>X</mi> <mn>0</mn> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2024_442_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2024_442_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu \in {{\mathbb {R}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> are two real parameters. The model is an analogue of the linear self-attracting diffusion (Cranston and Le Jan in Math Ann 303:87–93, 1995). We study large time behavior of the model. For this seemingly trivial generalization, we find that the large time behavior is much more complex than that of fractional Brownian motion, which can not be observed in the model driven by fractional Brownian motion.

The Linear Self-Attracting Diffusion Driven by Weighted-Fractional Brownian Motion I: Large Time Behaviors

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发表日期：

2025年7月31日

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Litan Yan,
Rui Guo,
Wenyi Pei

摘要

Let $B^{a,b}$ $B^{a, b}$ be a weighted-fractional Brownian motion with Hurst indexes a and b such that $a>-1$ $a > - 1$ and $0<b<1\wedge (1+a)$ $0 < b < 1 \land (1 + a)$ . In this paper, we consider the linear self-attracting diffusion $ dX^{a,b}_t=dB^{a,b}_t-\theta \left( \int _0^t(X^{a,b}_t-X^{a,b}_s)ds-\nu \right) dt $ $d X_{t}^{a, b} = d B_{t}^{a, b} - θ (\int_{0}^{t} (X_{t}^{a, b} - X_{s}^{a, b}) d s - ν) d t$ with $X^{a,b}_0=0$ $X_{0}^{a, b} = 0$ , where $\theta >0$ $θ > 0$ , $\nu \in {{\mathbb {R}}}$ $ν \in R$ are two real parameters. The model is an analogue of the linear self-attracting diffusion (Cranston and Le Jan in Math Ann 303:87–93, 1995). We study large time behavior of the model. For this seemingly trivial generalization, we find that the large time behavior is much more complex than that of fractional Brownian motion, which can not be observed in the model driven by fractional Brownian motion.