About

Tai Melcher is an Associate Professor in the Department of Mathematics at the University of Virginia. His research focus is in the field of Probability. He is involved in various departmental activities and maintains a personal webpage with additional information about his work and interests.

Research topics

• Mathematics
• Pure mathematics
• Mathematical analysis
• Applied mathematics
• Statistical physics

Selected publications

• Functional inequalities for a family of infinite-dimensional diffusions with degenerate noise

  Journal of Functional Analysis · 2025-01-06 · 1 citations

  articleSenior authorCorresponding
  PublisherDOI

• Quasi-Invariance for Infinite-Dimensional Kolmogorov Diffusions

  Potential Analysis · 2023-02-25 · 5 citations

  articleOpen accessSenior author

  We prove Cameron-Martin type quasi-invariance for the heat kernel measure of infinite-dimensional Kolmogorov and similar degenerate diffusions. We first study quantitative functional inequalities, particularly Wang-type Harnack inequalities, for appropriate finite-dimensional approximations of these diffusions, and we prove that these inequalities hold with dimension-independent constants. Applying an approach developed in [7, 12, 13], these uniform bounds may then be used to prove that the heat kernel measure for these infinite-dimensional diffusions is quasi-invariant under changes of the initial state.

  PublisherOA PDFDOI

• Functional inequalities for a family of infinite-dimensional diffusions with degenerate noise

  arXiv (Cornell University) · 2023-11-02

  preprintOpen accessSenior author

  For a family of infinite-dimensional diffusions with degenerate noise, we develop a modified $Γ$ calculus on finite-dimensional projections of the equation in order to produce explicit functional inequalities that can be scaled to infinite dimensions. The choice of our $Γ$ operator appears canonical in our context, as the estimates depend only on the induced control distance. We apply the general analysis to a number of examples, exploring implications for quasi-invariance and uniqueness of stationary distributions.

  PublisherOA PDFDOI

• Large deviations principle for sub-Riemannian random walks

  arXiv (Cornell University) · 2022-10-11

  preprintOpen access

  We study large deviations for random walks on stratified (Carnot) Lie groups. For such groups, there is a natural collection of vectors which generates their Lie algebra, and we consider random walks with increments in only these directions. Under certain constraints on the distribution of the increments, we prove a large deviation principle for these random walks with a natural rate function adapted to the sub-Riemannian geometry of these spaces.

  PublisherOA PDFDOI

• Stochastic integrals and Brownian motion on abstract nilpotent Lie groups

  Journal of the Mathematical Society of Japan · 2021-06-22

  preprintOpen access1st authorCorresponding

  We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron–Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon–Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.

  PublisherOA PDFDOI

• Small-Time Asymptotics for Subelliptic Hermite Functions on SU(2) and the CR Sphere

  Potential Analysis · 2020-05-14

  preprintOpen accessSenior author
  PublisherOA PDFDOI

• Convergence of the empirical spectral measure of unitary Brownian motion

  Annales Henri Lebesgue · 2019-03-08 · 3 citations

  articleOpen accessSenior author

  Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>{</mml:mo> <mml:msubsup> <mml:mi>U</mml:mi> <mml:mi>t</mml:mi> <mml:mi>N</mml:mi> </mml:msubsup> <mml:mo>}</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> be a standard Brownian motion on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝕌</mml:mi> <mml:mfenced close=")" open="("> <mml:mi>N</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> . For fixed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℕ</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , we give explicit almost-sure bounds on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> -Wasserstein distance between the empirical spectral measure of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>U</mml:mi> <mml:mi>t</mml:mi> <mml:mi>N</mml:mi> </mml:msubsup> </mml:math> and the large- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> limiting measure. The bounds obtained are tight enough that we are able to use them to study the evolution of the eigenvalue process in time, bounding the rate of convergence of paths of the measures on compact time intervals. The proofs use tools developed by the first author to obtain rates of convergence of the empirical spectral measures in classical random matrix ensembles, as well as recent estimates for the rates of convergence of moments of the ensemble-averaged spectral distribution.

  PublisherOA PDFDOI

• Convergence of the empirical spectral measure of unitary Brownian motion

  arXiv (Cornell University) · 2017-05-08 · 1 citations

  preprintOpen accessSenior author

  Let $\{U^N_t\}_{t\ge 0}$ be a standard Brownian motion on $\mathbb{U}(N)$. For fixed $N\in\mathbb{N}$ and $t&gt;0$, we give explicit bounds on the $L_1$-Wasserstein distance of the empirical spectral measure of $U^N_t$ to both the ensemble-averaged spectral measure and to the large-$N$ limiting measure identified by Biane. We are then able to use these bounds to control the rate of convergence of paths of the measures on compact time intervals. The proofs use tools developed by the first author to study convergence rates of the classical random matrix ensembles, as well as recent estimates for the convergence of the moments of the ensemble-average spectral distribution.

  PublisherOA PDFDOI

• HYPOELLIPTIC HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS

  2016-08-14 · 9 citations

  articleSenior author

  Abstract. We study the law of a hypoelliptic Brownian motion on an infinite-dimensional Heisenberg group based on an abstract Wiener space. We show that the endpoint distribution, which can be seen as a heat kernel measure, is absolutely continuous with respect to a certain product of Gaussian and Lebesgue measures, that the heat kernel is quasi-invariant under translation by the Cameron–Martin subgroup, and that the Radon–Nikodym derivative is Malliavin smooth. Contents

  Publisher

• Small deviations for time-changed Brownian motions and applications to second-order chaos

  Electronic Journal of Probability · 2014-01-01 · 2 citations

  articleOpen accessSenior author

  We prove strong small deviations results for Brownian motion under independent time-changes satisfying their own asymptotic criteria. We then apply these results to certain stochastic integrals which are elements of second-order homogeneous chaos.

  PublisherOA PDFDOI

Recent grants

• CAREER: Heat kernel measures in infinite dimensions

  NSF · $450k · 2013–2022

• Stochastic Processes in non-Euclidean spaces

  NSF · $118k · 2009–2013

Frequent coauthors

• Maria Gordina

  7 shared

• Fabrice Baudoin

  Aarhus University

  4 shared

• Daniel Dobbs

  3 shared

• Elizabeth Meckes

  Case Western Reserve University

  2 shared

• Nathaniel Eldredge

  2 shared

• Joshua Campbell

  2 shared

• Bruce K. Driver

  University of California, San Diego

  2 shared

• David P. Herzog

  1 shared