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Softcover ISBN:  9781470436209 
Product Code:  MEMO/259/1250 
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Softcover ISBN:  9781470436209 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 259; 2019; 118 ppMSC: 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 selfsimilar 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 offdiagonal subGaussian 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 subGaussian 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 superGaussian heat kernel behavior


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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 selfsimilar 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 offdiagonal subGaussian 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 subGaussian 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 superGaussian heat kernel behavior