18 citations · 57 across the 23 of their papers we have counts for
31 papers
Martingale and analytic dimensions
Sylvester Eriksson-Bique, Mathav Murugan
These notes introduce the martingale dimension of a strongly local regular Dirichlet form and the analytic dimension of a Lipschitz differentiability space. We explain why these di…
A simplified characterization of stable-like heat kernel estimates
Mathav Murugan
We study heat kernel estimates for symmetric pure jump processes on general metric measure spaces. Building on recent progress in the local setting due to S.~Eriksson-Bique, we dev…
Uniqueness of the maximal solution of the supercooled Stefan problem in 1D
Kai Hong Chau, Young-Heon Kim, Mathav Murugan
We prove uniqueness of the maximal weak solutions to the supercooled Stefan problem in 1 dimension. This follows by showing that in 1 dimension, the optimal solution of the corresp…
On the energy image density conjecture of Bouleau and Hirsch
Sylvester Eriksson-Bique, Mathav Murugan
We affirmatively resolve the energy image density conjecture of Bouleau and Hirsch (1986). Beyond the original framework of Dirichlet structures, we establish the energy image dens…
Martingale and analytic dimensions coincide under Gaussian heat kernel bounds
Mathav Murugan
Given a strongly local Dirichlet form on a metric measure space that satisfies Gaussian heat kernel bounds, we show that the martingale dimension of the associated diffusion proces…
Diffusions and random walks with prescribed sub-Gaussian heat kernel estimates
Mathav Murugan
Given suitable functions , we obtain necessary and sufficient conditions on for the existence of a metric measure space and a symmetric diffus…
Heat kernel estimates for boundary traces of reflected diffusions on uniform domains
Naotaka Kajino, Mathav Murugan
We study the boundary trace processes of reflected diffusions on uniform domains. We obtain stable-like heat kernel estimates for such a boundary trace process when the diffusion o…
First-order Sobolev spaces, self-similar energies and energy measures on the Sierpiński carpet
Mathav Murugan, Ryosuke Shimizu
We construct and investigate -Sobolev space, -energy, and the corresponding -energy measures on the planar Sierpiński carpet for all . Our method i…
Heat kernel for reflected diffusion and extension property on uniform domains
Mathav Murugan
We study reflected diffusion on uniform domains where the underlying space admits a symmetric diffusion that satisfies sub-Gaussian heat kernel estimates. A celebrated theorem of J…
Counterexamples to elliptic Harnack inequality for isotropic unimodal Lévy processes
Jens Malmquist, Mathav Murugan
Until now, it has been an open question whether every subordinated Brownian motion (SBM) satisfies the elliptic Harnack inequality (EHI). In this paper, we show that the answer is…
Conformal Assouad dimension as the critical exponent for combinatorial modulus
Mathav Murugan
The conformal Assouad dimension is the infimum of all possible values of Assouad dimension after a quasisymmetric change of metric. We show that the conformal Assouad dimension equ…
Helmholtz Solutions for the Fractional Laplacian and Other Related Operators
Vincent Guan, Mathav Murugan, Juncheng Wei
We show that the bounded solutions to the fractional Helmholtz equation, for in , are given by the bounded solutions to the classical Helmholtz…
On the comparison between jump processes and subordinated diffusions
Guanhua Liu, Mathav Murugan
Given a symmetric diffusion process and a jump process on the same underlying space, is there a subordinator such that the jump process and the subordinated diffusion processes are…
On the conformal walk dimension: Quasisymmetric uniformization for symmetric diffusions
Naotaka Kajino, Mathav Murugan
We introduce the notion of conformal walk dimension, which serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact…
Stability of EHI and regularity of MMD spaces
Martin T. Barlow, Zhen-Qing Chen, Mathav Murugan
We show that elliptic Harnack inequality is stable under form-bounded perturbations for strongly local Dirichlet forms on complete locally compact separable metric spaces that sati…
Geometric implications of fast volume growth and capacity estimates
Tim Jaschek, Mathav Murugan
We obtain connectivity of annuli for a volume doubling metric measure Dirichlet space which satisfies a Poincaré inequality, a capacity estimate and a fast volume growth condition.…
Parabolic Harnack inequality implies the existence of jump kernel
Guanhua Liu, Mathav Murugan
We prove that the parabolic Harnack inequality implies the existence of jump kernel for symmetric pure jump process. This allows us to remove a technical assumption on the jumping…
On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates
Naotaka Kajino, Mathav Murugan
We show that for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimens…
On the length of chains in a metric space
Mathav Murugan
We obtain an upper bound on the minimal number of points in an -chain joining two points in a metric space. This generalizes a bound due to Hambly and Kumagai (1999) for the cas…
A note on heat kernel estimates, resistance bounds and Poincaré inequality
Mathav Murugan
Sub-Gaussian heat kernel estimates are typical of fractal graphs. We show that sub-Gaussian estimates on graphs follow from a Poincaré inequality, capacity upper bound, and a slow…
Quasisymmetric uniformization and heat kernel estimates
Mathav Murugan
We show that the circle packing embedding in of a one-ended, planar triangulation with polynomial growth is quasisymmetric if and only if the simple random walk on t…
Boundary Harnack principle and elliptic Harnack inequality
Martin T. Barlow, Mathav Murugan
We prove a scale-invariant boundary Harnack principle for inner uniform domains over a large family of Dirichlet spaces. A novel feature of our work is that our assumptions are rob…
Stability of elliptic Harnack inequality
Martin T. Barlow, Mathav Murugan
We prove that the elliptic Harnack inequality (on a manifold, graph, or suitably regular metric measure space) is stable under bounded perturbations, as well as rough isometries.
Heat kernel estimates for anomalous heavy-tailed random walks
Mathav Murugan, Laurent Saloff-Coste
Sub-Gaussian estimates for the natural random walk is typical of many regular fractal graphs. Subordination shows that there exist heavy tailed jump processes whose jump indices ar…