| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>B</mi><mi>H</mi></msup><mo>=</mo><mrow><mo>{</mo><msubsup><mi>B</mi><mi>t</mi><mi>H</mi></msubsup><mo>,</mo><mi>t</mi><mo>≥</mo><mn>0</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> be a fractional Brownian motion with Hurst index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>≤</mo><mi>H</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In this paper, we consider the linear self-attracting diffusion: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><msubsup><mi>X</mi><mrow><mi>t</mi></mrow><mi>H</mi></msubsup><mo>=</mo><mi>d</mi><msubsup><mi>B</mi><mrow><mi>t</mi></mrow><mi>H</mi></msubsup><mo>+</mo><mi>σ</mi><msubsup><mi>X</mi><mrow><mi>t</mi></mrow><mi>H</mi></msubsup><mi>d</mi><mi>t</mi><mo>−</mo><mi>θ</mi><mfenced separators="" open="(" close=")"><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mi>t</mi></msubsup><mfenced separators="" open="(" close=")"><msubsup><mi>X</mi><mrow><mi>s</mi></mrow><mi>H</mi></msubsup><mo>−</mo><msubsup><mi>X</mi><mrow><mi>u</mi></mrow><mi>H</mi></msubsup></mfenced><mi>d</mi><mi>s</mi></mfenced><mi>d</mi><mi>t</mi></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>+</mo><mspace width="3.33333pt"></mspace><mi>ν</mi><mi>d</mi><mi>t</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>X</mi><mrow><mn>0</mn></mrow><mi>H</mi></msubsup><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>,</mo><mi>ν</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> are three parameters. The process is an analogue of the self-attracting diffusion (Cranston and Le Jan, <i>Math. Ann.</i><b>303</b> (1995), 87–93). Our main aim is to study the large time behaviors. We show that the solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open="(" close=")"><mi>t</mi><mo>−</mo><mfrac><mi>σ</mi><mi>θ</mi></mfrac></mfenced><mi>H</mi></msup><mfenced separators="" open="(" close=")"><msubsup><mi>X</mi><mi>t</mi><mi>H</mi></msubsup><mo>−</mo><msubsup><mi>X</mi><mrow><mo>∞</mo></mrow><mi>H</mi></msubsup></mfenced></mrow></semantics></math></inline-formula> converges in distribution to a normal random variable, as <i>t</i> tends to infinity, and obtain two strong laws of large numbers associated with the solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>X</mi><mi>H</mi></msup></semantics></math></inline-formula>.
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