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Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance
 
Jun Kigami Kyoto University, Kyoto, Japan
Time Changes of the Brownian Motion: Poincare Inequality, Heat Kernel Estimate and Protodistance
Softcover ISBN:  978-1-4704-3620-9
Product Code:  MEMO/259/1250
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5255-1
Product Code:  MEMO/259/1250.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3620-9
eBook: ISBN:  978-1-4704-5255-1
Product Code:  MEMO/259/1250.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
Time Changes of the Brownian Motion: Poincare Inequality, Heat Kernel Estimate and Protodistance
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Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance
Jun Kigami Kyoto University, Kyoto, Japan
Softcover ISBN:  978-1-4704-3620-9
Product Code:  MEMO/259/1250
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
eBook ISBN:  978-1-4704-5255-1
Product Code:  MEMO/259/1250.E
List Price: $81.00
MAA Member Price: $72.90
AMS Member Price: $48.60
Softcover ISBN:  978-1-4704-3620-9
eBook ISBN:  978-1-4704-5255-1
Product Code:  MEMO/259/1250.B
List Price: $162.00 $121.50
MAA Member Price: $145.80 $109.35
AMS Member Price: $97.20 $72.90
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2592019; 118 pp
    MSC: Primary 31; 60; Secondary 28; 30; 43; 80

    In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including \(n\)-dimensional cube \([0, 1]^n\) are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on \([0, 1]^n\), density of the medium is homogeneous and represented by the Lebesgue measure. The author's study includes densities which are singular to the homogeneous one. He establishes a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on \([0, 1]^2\) and self-similar measures.

    The author shows the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, he obtains diagonal lower and upper estimates of the heat kernel as time tends to \(0\). In particular, to express the principal part of the lower diagonal heat kernel estimate, he introduces “protodistance” associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal sub-Gaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Generalized Sierpinski carpets
    • 3. Standing assumptions and notations
    • 4. Gauge function
    • 5. The Brownian motion and the Green function
    • 6. Time change of the Brownian motion
    • 7. Scaling of the Green function
    • 8. Resolvents
    • 9. Poincaré inequality
    • 10. Heat kernel, existence and continuity
    • 11. Measures having weak exponential decay
    • 12. Protodistance and diagonal lower estimateof heat kernel
    • 13. Proof of Theorem 1.1
    • 14. Random measures having weak exponential decay
    • 15. Volume doubling measure and sub-Gaussian heat kernel estimate
    • 16. Examples
    • 17. Construction of metrics from gauge function
    • 18. Metrics and quasimetrics
    • 19. Protodistance and the volume doubling property
    • 20. Upper estimate of $p_{\mu }(t, x, y)$
    • 21. Lower estimate of $p_{\mu }(t, x, y)$
    • 22. Non existence of super-Gaussian heat kernel behavior
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2592019; 118 pp
MSC: Primary 31; 60; Secondary 28; 30; 43; 80

In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including \(n\)-dimensional cube \([0, 1]^n\) are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on \([0, 1]^n\), density of the medium is homogeneous and represented by the Lebesgue measure. The author's study includes densities which are singular to the homogeneous one. He establishes a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on \([0, 1]^2\) and self-similar measures.

The author shows the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, he obtains diagonal lower and upper estimates of the heat kernel as time tends to \(0\). In particular, to express the principal part of the lower diagonal heat kernel estimate, he introduces “protodistance” associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal sub-Gaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown.

  • Chapters
  • 1. Introduction
  • 2. Generalized Sierpinski carpets
  • 3. Standing assumptions and notations
  • 4. Gauge function
  • 5. The Brownian motion and the Green function
  • 6. Time change of the Brownian motion
  • 7. Scaling of the Green function
  • 8. Resolvents
  • 9. Poincaré inequality
  • 10. Heat kernel, existence and continuity
  • 11. Measures having weak exponential decay
  • 12. Protodistance and diagonal lower estimateof heat kernel
  • 13. Proof of Theorem 1.1
  • 14. Random measures having weak exponential decay
  • 15. Volume doubling measure and sub-Gaussian heat kernel estimate
  • 16. Examples
  • 17. Construction of metrics from gauge function
  • 18. Metrics and quasimetrics
  • 19. Protodistance and the volume doubling property
  • 20. Upper estimate of $p_{\mu }(t, x, y)$
  • 21. Lower estimate of $p_{\mu }(t, x, y)$
  • 22. Non existence of super-Gaussian heat kernel behavior
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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