My research interests are in geometric analysis and
probability. I have been especially interested in the overlap of these two
fields, using Brownian motion and diffusions as tools in geometry. Particular
topics of interest include classical minimal surfaces, more general minimal
submanifolds, mean curvature flow and Ricci flow for surfaces, potential theory
on manifolds, random walks on Riemannian and sub-Riemannian manifolds,
small-time heat kernel asymptotics on Riemannian and sub-Riemannian manifolds,
martingale methods in analysis, and couplings of semi-martingales.
Below is a list of publications and preprints, most with
links to the paper in some form.
Non-Markovian maximal couplings and a vertical reflection
principle on a class of sub-Riemannian manifolds (with L. Luo),
arXiv:2402.13976, preprint.
Localized bounds on log-derivatives of the heat kernel
on incomplete Riemannian manifolds (with L. Sacchelli), arXiv:2212.09559,
to appear in Ann. Inst. Henri Poincaré Probab. Stat.
Variations of the sub-Riemannian distance on Sasakian
manifolds with applications to coupling (with F. Baudoin, E. Grong, and
A. Thalmaier), arXiv:2212.07715, preprint.
Uniform, localized asymptotics for sub-Riemannian heat
kernels and diffusions (with L. Sacchelli), arXiv:2012.12888, preprint.
Geometric and Martin boundaries of a Cartan-Hadamard
surface, ALEA Lat. Am. J. Probab. Math. Stat.
18
(2021), no. 2, 1669-1687.
A pseudo-probabilistic approach to the dilation equation
for wavelets (with S. Dumnich), arXiv:1710.01364, unpublished preprint.
Brownian motion and the parabolicity of minimal
graphs, arXiv:0810.0669, unpublished preprint.
Radial processes for sub-Riemannian Brownian motions
and applications (with F. Baudoin, E. Grong, K. Kuwada, A. Thalmaier),
Electron. J. Probab.
25 (2020), Paper No. 97, 17 pp.
Extensions of Brownian motion to a family of
Grushin-type singularities (with U. Boscain),
Electron. Commun. Probab.
25 (2020), Paper No. 29, 12 pp.
Heat kernel asymptotics on sub-Riemannian manifolds with
symmetries and applications to the bi-Heisenberg group (with D. Barilari
and U. Boscain), Ann. Fac. Sci. Toulouse Math. (6)
28
(2019), no. 4, 707-732.
Intrinsic random walks in Riemannian and sub-Riemannian
geometry via volume sampling (with A. Agrachev, U. Boscain, and L.
Rizzi), ESAIM Control Optim. Calc. Var.
24 (2018),
no. 3, 1075-1105.
Intrinsic random walks and sub-Laplacians in sub-Riemannian
geometry (with U. Boscain and L. Rizzi), Adv. Math.
314 (2017), 124-184.
On the heat diffusion for generic Riemannian and
sub-Riemannian structures (with D. Barilari, U. Boscain, and G. Charlot),
Int. Math. Res. Not. IMRN (2017), no. 15, 4639-4672.
A stochastic target approach to Ricci flow on
surfaces (with I. Popescu), Ann. Probab.
44
(2016), no. 2, 1341-1425.
e-print from
Project Euclid (subscription required), or the
preprint from arXiv.
Martingales arising from minimal submanifolds and other
geometric contexts, Illinois J. Math.
58
(2014), no. 2, 323-357. [actually published in 2015]
e-print from
Project Euclid (subscription required), or the
preprint from arXiv.
Brownian motion and the Dirichlet problem at infinity on
two-dimensional Cartan-Hadamard manifolds, Potential Anal.
41 (2014), no. 2, 443-462.
On parabolicity and area growth of minimal surfaces,
J. Geom. Anal.
23 (2013), no. 3, 1173-1188.
Small time heat kernel asymptotics at the sub-Riemannian
cut locus (with D. Barilari and U. Boscain), J. Differential Geom.
92 (2012), no. 3, 373-416.
Stochastic methods for minimal surfaces, Geometric
analysis: partial differential equations and surfaces, Contemp.
Math., vol. 570, Amer. Math. Soc., Providence, RI (2012), 111-136.
A martingale approach to minimal surfaces,
J. Funct. Anal.
256 (2009), no. 8, 2440-2472.
The small-time asymptotics of the heat kernel
at the cut locus, Comm. Anal. Geom.
15
(2007), no. 4, 845-890.
Analysis of the cut locus via the heat kernel
(with D. Stroock), Surveys in Differential Geometry, Vol. 9,
International Press, Boston (2004), 337-349.
Equilibrium configurations for a floating drop
(with A. Elcrat and D. Siegel), J. Math. Fluid Mech.
6 (2004), no. 4, 405-429.
C-singular solutions of the capillary problem
(with R. Finn), J. Reine Angew. Math.
512 (1999),
1-25.