Time Changes of the Brownian Motion: Poincare Inequality, Heat Kernel Estimate and Protodistance

Jun Kigami (Autor)

Buch | Softcover

118 Seiten

2019
American Mathematical Society (Verlag)
9781470436209 (ISBN)

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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 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.

Jun Kigami, Kyoto University, Japan.

Introduction
Generalized Sierpinski carpets
Standing assumptions and notations
Gauge function
The Brownian motion and the Green function
Time change of the Brownian motion
Scaling of the Green function
Resolvents
Poincare inequality
Heat kernel, existence and continuity
Measures having weak exponential decay
Protodistance and diagonal lower estimate of heat kernel
Proof of Theorem 1.1
Random measures having weak exponential decay
Volume doubling measure and sub-Gaussian heat kernel estimate
Examples
Construction of metrics from gauge function
Metrics and quasimetrics
Protodistance and the volume doubling property
Upper estimate of $p_/mu (t, x, y)$
Lower estimate of $p_/mu (t, x, y)$
Non existence of super-Gaussian heat kernel behavior
Bibliography
List of notations
Index

Erscheinungsdatum	21.07.2019
Reihe/Serie	Memoirs of the American Mathematical Society
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	192 g
Themenwelt	Mathematik / Informatik ► Mathematik
ISBN-13	9781470436209 / 9781470436209
Zustand	Neuware