## Transition density estimates for Brownian motion on scale irregular Sierpinski gaskets (1997)

Venue: | Commun. Math. Phys |

Citations: | 38 - 12 self |

### BibTeX

@ARTICLE{Barlow97transitiondensity,

author = {M. T. Barlow and B. M. Hambly},

title = {Transition density estimates for Brownian motion on scale irregular Sierpinski gaskets},

journal = {Commun. Math. Phys},

year = {1997},

volume = {33},

pages = {595--620}

}

### Years of Citing Articles

### OpenURL

### Abstract

We construct Brownian motion on a class of fractals which are spatially homogeneous but which do not have any exact self-similarity. We obtain transition density estimates for this process which are up to constants best possible. 1 Introduction There is now a fairly extensive literature on the heat equation on fractal spaces, and on the spectral properties of such spaces. Most of these papers treat sets F which have exact selfsimilarity, so that there exist 1-1 contractions / i : F ! F such that / i (F ) " / j (F ) is (in some sense) small when i 6= j, and F = [ i / i (F ): (1.1) In the simplest cases, such as the nested fractals of Lindstrøm [18], F ae R d , the / i are linear, and / i (F ) " / j (F ) is finite when i 6= j. For very regular fractals such as nested fractals, or Sierpinski carpets, it is possible to construct a diffusion X t with a semigroup P t which is symmetric with respect to ¯, the Hausdorff measure on F , and to obtain estimates on the density p t (x; y) of P ...

### Citations

565 |
Techniques in Fractal Geometry
- Falconer
- 1997
(Show Context)
Citation Context ...(:) and dim P;d (:) for Hausdorff and packing dimension with respect to the metric d. The following result follows easily from (2.12) and the density theorems for Hausdorff and packing measure -- see =-=[6]-=-. Lemma 2.3 (a) dimH;d (F ) = lim inf n!1 d f (n), (b) dim P;d (F ) = lim sup n!1 d f (n). For some simple fractals the distance d is equivalent to Euclidean distance. We just prove this for the examp... |

265 |
Dirichlet forms and symmetric Markov processes
- Fukushima, Oshima, et al.
- 1994
(Show Context)
Citation Context ...fi /w ): Letting m !1 the result follows. 2 4 Transition density estimates: upper bounds Let P t be the semigroup of positive operators associated with the Dirichlet form (E; F) on L 2 (F; ��) -- =-=see [9]-=-. As (E; F) is regular and local, there exists a Feller diffusion (X t ; ts0; P x ; x 2 F ) with semigroup P t , which we will call Brownian motion on F . As in [8] we deduce from Theorem 3.3 that Gs=... |

95 |
Brownian motion on the Sierpinski gasket
- Barlow, Perkins
- 1988
(Show Context)
Citation Context ...ote that if �� n j a (where a = 2 or 3) then dw (n) j log t a = log l a , d s (n) j 2 log m a = log t a , and we recover the estimates for the heat kernels on the fractals SG(2) and SG(3) obtained=-= in [5, 15]. For non-co-=-nstant ��, dw (n) and d s (n) are the `effective walk and spectral dimensions at level n'. For given t, x, y, let m;n be as in the Theorem, so that T \Gamma1 n �� t and L \Gamma1 m �� jx \... |

61 | A harmonic calculus for p.c.f. self-similar sets - Kigami - 1993 |

55 | Random recursive constructions: asymptotic geometric and topological properties - Mauldin, Williams - 1986 |

51 |
Construction of Brownian motion on the Sierpinski carpet
- Barlow, Bass
- 1989
(Show Context)
Citation Context ...(1 \Gamma c \Gamma1 4 ), and as SB = W n 1 P z -a.s., we deduce there exist c 5 ? 0, c 6 2 (0; 1) such that P z (W n 1st)sc 5 T n t + c 6 ; ts0: (4.7) This bound is quite crude, but we can now, as in =-=[2]-=-, use it to derive a much better estimate on P z (W n 1st). We first define k = k(m; n) = inffjs0 : Tm+j Bm+j T n Bm g: (4.8) As the function k(m; n) plays a crucial role in our bounds, we need to spe... |

32 |
Estimates of transition densities for Brownian motion on nested fractals, Probab. Theory Relat
- Kumagai
- 1993
(Show Context)
Citation Context ...diffusion X t with a semigroup P t which is symmetric with respect to ��, the Hausdorff measure on F , and to obtain estimates on the density p t (x; y) of P t with respect to ��. In these cas=-=es (see [3, 15]-=-) there exist constants dw , d s (called, following the physics literature, the walk and spectral dimensions of F ) such that p t (x; y)sc 1 t \Gammad s =2 exp(\Gammac 2 ( jx \Gamma yj dw t ) 1=(dw \G... |

31 |
X.Y.: Dirichlet forms on fractals: Poincare constant and resistance. Probab. Theory Related Fields 93
- Kusuoka, Zhou
- 1992
(Show Context)
Citation Context ...ct to ��, and that p t (x; y) satisfies the Chapman-Kolmogorov equations. We now obtain upper bounds on p t (x; y), beginning with the on-diagonal upper bound, where we follow closely the argument=-= of [17]. Lemma 4.1 Ther-=-e is a constant c 1 such that if T \Gamma1 nstsT \Gamma1 n\Gamma1 then jjP t jj 1!1sc 1 M n : (4.1) Proof. For w 2 W n write f w = f ffi /w and �� f w = Z F (` n ��) f w (x)�� (` n ��)... |

21 |
Transition densities for Brownian motion on the Sierpinski carpet. Probab. Theory Related Fields 91
- Barlow, Bass
- 1992
(Show Context)
Citation Context ...diffusion X t with a semigroup P t which is symmetric with respect to ��, the Hausdorff measure on F , and to obtain estimates on the density p t (x; y) of P t with respect to ��. In these cas=-=es (see [3, 15]-=-) there exist constants dw , d s (called, following the physics literature, the walk and spectral dimensions of F ) such that p t (x; y)sc 1 t \Gammad s =2 exp(\Gammac 2 ( jx \Gamma yj dw t ) 1=(dw \G... |

18 |
Harmonic calculus on limits of networks and its application to dendrites
- Kigami
- 1995
(Show Context)
Citation Context ...e will construct a ��-symmetric diffusion X t , with semigroup P t , on F . We do this analytically, by constructing a regular local Dirichlet form E on L 2 (F; ��). Here we follow the ideas o=-=f [16], [14]-=-, [8]; though the arguments of these papers do not directly cover the case treated here, they can be adapted without difficulty to our situation. Once we have constructed P t , we can prove the existe... |

16 | Dirichlet Forms, Diffusion Processes and Spectral Dimensions for Nested Fractals - Fukushima - 1988 |

16 |
Brownian motion on a random recursive Sierpinski gasket
- Hambly
- 1997
(Show Context)
Citation Context ... since Theorem 5.4 shows that the bounds in Theorem 4.5 are, up to constants, the best possible). Note, however, that for the on diagonal bounds there is less oscillation in the random recursive case =-=[11]-=- than that observed here. 7 Spectral results Write L for the infinitesimal generator of the semigroup (P t ): we call L the Laplacian on the fractal F . The uniform continuity of p t (see Lemma 5.2) i... |

16 | Di↵usion processes on nested fractals - Kusuoka - 1993 |

16 |
Brownian motion on nested fractals
- Lindstrm
- 1990
(Show Context)
Citation Context ...ere exist 1-1 contractions / i : F ! F such that / i (F ) " / j (F ) is (in some sense) small when i 6= j, and F = [ i / i (F ): (1.1) In the simplest cases, such as the nested fractals of Lindst=-=r��m [18], F a-=-e R d , the / i are linear, and / i (F ) " / j (F ) is finite when i 6= j. For very regular fractals such as nested fractals, or Sierpinski carpets, it is possible to construct a diffusion X t wi... |

14 |
Espaces de Dirichlet I, le cas Elementaire
- Beurling, Deny
- 1958
(Show Context)
Citation Context ... We now construct a Dirichlet form E on L 2 (F; ��), following the ideas of [8, 14, 10]. It will be useful to keep in mind the interpretation of Dirichlet forms in terms of electrical networks -- =-=see [4, 14]-=-. Note that as F n is a discrete set, the space C(F n ) of continuous functions on F n is just the space of all functions on F n . For f 2 C(F 0 ) define E 0 (f; g) = 1 2 X x;y2F 0 (f(x) \Gamma f(y))(... |

13 |
Brownian motion on a homogeneous random fractal, Probab. Theory Related Fields 94
- Hambly
- 1992
(Show Context)
Citation Context ..., which are locally spatially homogeneous, but which do not satisfy any exact scaling relation of the form (1.1). To give the essential flavour of our results we consider a fractal first discussed in =-=[10]. Con-=-sider two regular fractals, the standard Sierpinski gasket SG(2) and a variant SG(3) - see Figure 1. Each of these sets may be defined by F = " 1 n=0 F n where (for a = 2 or 3) F n is obtained fr... |

8 | Random walks, electrical resistance, and nested fractals - Barlow - 1993 |

1 |
Transition density estimates for Brownain motion on affine nested fractals
- Fitzsimmons, Hambly, et al.
- 1994
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Citation Context ...we have constructed P t , we can prove the existence of a density p t (x; y) with respect to ��, and obtain bounds on p t , by using similar techniques to those developed for regular fractals in [=-=3], [7]-=-. To maintain consistency with notation for more general SGs introduced later, set B n = L n and let dw (n) = log T n = log B n ; d s (n) = 2 log M n = log T n ; (1.4) and for n; ms0 set k = k(m; n) =... |

1 |
Self-similar sets
- Hutchinson
- 1981
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Citation Context ...ap and x 0 2 R D . Let \Psi = f/ 1 ; : : : ; /m g be a finite family of ff-similitudes. For B ae R D , define \Phi(B) = [ m i=1 / i (B); and let \Phi n (B) = \Phi ffi : : : ffi \Phi(B): By Hutchinson =-=[12]-=-, the map \Phi on the set of compact subsets of R D has a unique fixed point F , which is a self-similar set satisfying F = \Phi(F ). As each / i is a contraction, it has a unique fixed point. Let F 0... |