Abstract
The
4
N
{4N}
-carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a
4
N
{4N}
-carpet F, let
{
F
n
}
n
≥
0
{\{F_{n}\}_{n\geq 0}}
be the natural decreasing sequence of compact pre-fractal approximations with
⋂
n
F
n
=
F
{\bigcap_{n}F_{n}=F}
. On each
F
n
{F_{n}}
, let
ℰ
(
u
,
v
)
=
∫
F
N
∇
u
⋅
∇
v
d
x
{\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx}
be the classical Dirichlet form and
u
n
{u_{n}}
be the unique harmonic function on
F
n
{F_{n}}
satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass,
On the resistance of the Sierpiński carpet,
Proc. Roy. Soc. Lond. Ser. A
431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is
ρ
=
ρ
(
N
)
>
1
{\rho=\rho(N)>1}
such that
ℰ
(
u
n
,
u
n
)
ρ
n
{\mathcal{E}(u_{n},u_{n})\rho^{n}}
is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.
On the heat kernel and the Dirichlet form of Liouville Brownian motion
