Random walk on the incipient infinite cluster for oriented percolation in high dimensions (2006)
Barlow, Martin T., Jarai, Antal A., Kumagai, Takashi, Slade, Gordon
We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on $Z^d \times Z_+$. In dimensions $d>6$, we obtain bounds on exit times, transition...
Stability of parabolic Harnack inequalities on metric measure spaces (2006)
BARLOW, Martin T., BASS, Richard F., KUMAGAI, Takashi
Let $(X,d,\mu)$ be a metric measure space with a local regular Dirichlet form. We give necessary and sufficient conditions for a parabolic Harnack inequality with global space-time scaling exponent...
Random walk on the incipient infinite cluster on trees (2005)
Barlow, Martin T., Kumagai, Takashi
Let ${\cal G}$ be the incipient infinite cluster (IIC) for percolation on a homogeneous tree of degree $n_0+1$. We obtain estimates for the transition density of the continuous time simple random...
Random walks on supercritical percolation clusters (2004)
We obtain Gaussian upper and lower bounds on the transition density qt(x,y) of the continuous time simple random walk on a supercritical percolation cluster ${\mathcal{C}}_{\infty}$ in the Euclidean...
Diffusion limited aggregation on a tree (2004)
Barlow, MArtin T., Pemantle, Robin, Perkins, Edwin A.
We study the following growth model on a regular d-ary tree. Points at distance n adjacent to the existing subtree are added with probabilities proportional to alpha^{-n}, where alpha=1. Results are...
Which values of the volume growth and escape time exponent are possible for a graph? (2004)
Let $\Gamma=(G,E)$ be an infinite weighted graph which is Ahlfors $\alpha$-regular, so that there exists a constant $c$ such that $c^{-1} r^\alpha\le V(x,r)\le c r^\alpha$, where $V(x,r)$ is the...
Some remarks on the elliptic Harnack inequality (2003)
This note gives three short results concerning the elliptic Harnack inequality (EHI), in the context of random walks on graphs. The first is that the EHI implies polynomial growth of the number of...
Contemporary Mathematics Heat kernels and sets with fractal structure (2003)
e vertices of the unit triangle in R (1.3) # i (x) = a i + a i ), 1 3. (So M = 3 and L = 2.) Then the fixed point F = FSG is called the Sierpinski gasket. If we take K 0 to be the closed convex hull...
Random walks on supercritical percolation clusters (2003)
We obtain Gaussian upper and lower bounds on the transition density q_t(x,y) of the continuous time simple random walk on a supercritical percolation cluster C_{\infty} in the Euclidean lattice. The...
Random walks on supercritical percolation clusters Martin T. Barlow (2003)
We obtain Gaussian upper and lower bounds on the transition density q t (x, y) of the continuous time simple random walk on a supercritical percolation cluster in the Euclidean lattice. The bounds,...
Which Values of the Volume Growth and Escape Time Exponent Are Possible for a Graph? (2001)
Let Gamma = (G; E) be an infinite weighted graph which is Ahlfors ff-regular, so that there exists a constant c such that c , where V (x; r) is the volume of the ball centre x and radius r. Define...
Markov Processes on Vermiculated Spaces (2001)
Martin T. Barlow, Steven N. Evans
A general technique is given for constructing new Markov processes from existing ones. The new process and its state space are both projective limits of sequences built by an iterative scheme. The...
Transition Density Asymptotics for Some Diffusion Processes with Multi-Fractal Structures (2001)
Barlow, Martin T.; University Of British Columbia; Barlow@math.ubc.ca, Kumagai, Takashi; Kyoto University; Kumagai@kurims.kyoto-u.ac.jp
We study the asymptotics as $t to 0$ of the transition density of a class of $mu$-symmetric diffusions in the case when the measure $mu$ has a multi-fractal structure. These diffusions include...
Coalescence of skew brownian motions (2001)
Barlow, Martin T., Burdzy, Krzysztof, Kaspi, Haya, Mandelbaum, Avi
Divergence Form Operators on Fractal-Like Domains (1999)
Martin T. Barlow, Richard F. Bass
We consider elliptic operators L in divergence form on certain domains in R d with fractal volume growth. The domains we look at are pre-Sierpinski carpets, which are derived from higher dimensional...
Diffusion-Limited Aggregation On A Tree (1999)
Martin T. Barlow, Robin Pemantle, Edwin A. Perkins, U. British, Columbia U. Wisconsin, U. British Columbia
this paper we treat the case ff ! 1; the case ff 1 has already been studied. Our main result, contained in Theorem 6.4 and Corollary 6.5, is as follows.
Random walks on graphical Sierpinski carpets (1998)
Martin T. Barlow, Richard F. Bass
We consider random walks on a class of graphs derived from Sierpinski carpets. We obtain upper and lower bounds (which are nonGaussian) on the transition probabilities which are, up to constants, the...
Martin T. Barlow, Robin Pemantle, Edwin A. Perkins, U. British, Columbia U. Wisconsin, U. British Columbia
this paper we treat the case ff ! 1; the case ff 1 has already been studied. Our main result, contained in Theorem 6.4 and Corollary 6.5, is as follows
Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes (1998)
Barlow, Martin T., Émery, Michel, Knight, Frank B., Song, Shiqi, Yor, Marc
Positivity Of Brownian Transition Densities (1997)
Martin T. Barlow, Richard F. Bass
Let B be a Borel subset of R d and let p(t; x; y) be the transition densities of Brownian motion killed on leaving B. Fix x and y in B. If p(t; x; y) is positive for one t, it is positive for every...
Weak homogenization of anisotropic diffusion on pre-Sierpi'nski carpets. (1997)
Martin T. Barlow, Kumiko Hattori, Tetsuya Hattori
We study a kind of `restoration of isotropy' on the pre-Sierpi'nski carpet. Let R x n (r) and R y n (r) be the effective resistances in the x and y directions, respectively, of the Sierpi'nski carpet...
Weak homogenization of anisotropic diffusion on pre-Sierpinski carpets. (1997)
Martin T. Barlow, Kumiko Hattori, Tetsuya Hattori
We study a kind of `restoration of isotropy' on the pre-Sierpi'nski carpet. Let R n (r) be the effective resistances in the x and y directions, respectively, of the Sierpi'nski carpet at the n-th...
Restoration of Isotropy on Fractals. (1997)
Martin T. Barlow, Kumiko Hattori, Tetsuya Hattori
We report a new type of restoration of macroscopic isotropy (homogenization) in fractals with microscopic anisotropy. The phenomenon is observed in various physical setups, including diffusions,...
Martin T. Barlow, Kumiko Hattori, Tetsuya Hattori
We report a new type of restoration of macroscopic isotropy (homogenization) in fractals with microscopic anisotropy. The phenomenon is observed in various physical setups, including diffusions,...
Martin T. Barlow, Kumiko Hattori, Tetsuya Hattori
We report a new type of restoration of macroscopic isotropy (homogenization) in fractals with microscopic anisotropy. The phenomenon is observed in various physical setups, including diffusions,...
Restoration of isotropy on fractals (1995)
Barlow, Martin T., Hattori, Kumiko, Hattori, Tetsuya, Watanabe, Hiroshi
We report a new type of restoration of macroscopic isotropy (homogenization) in fractals with microscopic anisotropy. The phenomenon is observed in various physical setups, including diffusions,...
Coupling and Harnack inequalities for Sierpinski carpets (1993)
Barlow, Martin T., Bass, Richard F.
Uniform Harnack inequalities for harmonic functions on the pre- and graphical Sierpinski carpets are proved using a probabilistic coupling argument. Various results follow from this, including the...