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Phanuel Mariano

Publications and source records attributed to Phanuel Mariano.

16 recordsLinked to original sources
arXiv2026

Survival probability for jump processes in unbounded domains on metric measure spaces

We study the large time behavior of the survival probability $\mathbb{P}_x\left(τ_D>t\right)$ for symmetric jump processes in unbounded domains with a positive bottom of the spectrum. We prove asymptotic upper and lower bounds with explicit constants in terms of the bottom of the spectrum $λ(D)$. Our main result applies to symmetric jump processes in general metric measure spaces. For $α$-stable processes in unbounded uniformly $C^{1,1}$ domains, our results provide a probabilistic interpretation and an equivalent geometric condition for $λ(D)>0$. In the case of increasing horn-shaped domains, the exponential rate of decay for the survival probability is sharp. We also present examples of unbounded domains where our results apply.

math.PR↗
arXiv2026

Sharp Dirichlet eigenvalue inequalities on triangles

We prove sharp Dirichlet eigenvalue inequalities for planar triangles. We settle a conjecture of Laugesen and Siudeja by showing that the equilateral triangle uniquely minimizes a scale-invariant functional of the first Dirichlet eigenvalue, area, and perimeter. Consequences include an optimal two-term lower bound for the first Dirichlet eigenvalue in terms of area and perimeter. We also prove a Cheeger-type inequality with an explicit best constant considered by Parini. To prove these conjectures we propose a new method for proving Dirichlet eigenvalue inequalities on triangles. Our method is based on a new computable lower bound for second-order directional shape derivatives under vertex perturbations. It also uses validated finite-element error estimates and recently developed analytic estimates for eigenvalues of nearly degenerate triangles. The method is not specific to the functionals considered in this paper and it can be used to prove various other eigenvalue inequalities on triangles.

math.SP↗
arXiv2025

Geometric properties of optimizers for the maximum gradient of the torsion function

Consider $J(Ω):= \|\nabla u_Ω\|_\infty/\sqrt{|Ω|} $ and $J_P(Ω):= \|\nabla u_Ω\|_\infty/P(Ω) $, where $Ω$ is a planar convex domain, $u_Ω$ is the torsion function, $P(Ω)$ is the perimeter of $Ω$ and $|Ω|$ its area. We prove that there exist planar convex domains that maximize the functionals $J$ and $J_P$, and any maximizer has a $C^1$ boundary that contains a line segment on which $|\nabla u_Ω|$ attains its maximum.

math.AP↗
arXiv2025

Spectral bounds for exit times on metric measure Dirichlet spaces and applications

Assuming the heat kernel on a doubling Dirichlet metric measure space has a sub-Gaussian bound, we prove an asymptotically sharp spectral upper bound on the survival probability of the associated diffusion process. As a consequence, we can show that the supremum of the mean exit time over all starting points is finite if and only if the bottom of the spectrum is positive. Among several applications, we show that the spectral upper bound on the survival probability implies a bound for the Hot Spots constant for Riemannian manifolds. Our results apply to interesting geometric settings including sub-Riemannian manifolds and fractals.

math.PR↗
arXiv2024

A central limit theorem with explicit Lyapunov exponent and variance for products of $2\times2$ random non-invertible matrices

The theory of products of random matrices and Lyapunov exponents have been widely studied and applied in the fields of biology, dynamical systems, economics, engineering and statistical physics. We consider the product of an i.i.d. sequence of $2\times 2$ random non-invertible matrices with real entries. Given some mild moment assumptions we prove an explicit formula for the Lyapunov exponent and prove a central limit theorem with an explicit formula for the variance in terms of the entries of the matrices. We also give examples where exact values for the Lyapunov exponent and variance are computed. An important example where non-invertible matrices are essential is the random Hill's equation, which has numerous physical applications, including the astrophysical orbit problem.

math.PR↗
arXiv2024

On a conjecture of a Pólya functional for triangles and rectangles

We consider the functional given by the product of the first Dirichlet eigenvalue and the torsional rigidity of planar domains normalized by the area. This scale invariant functional was studied by Pólya and Szegő in 1951 who showed that it is bounded above by 1 for all domains. It has been conjectured that within the class of bounded convex planar domains the functional is bounded below by $π^{2}/24$ and above by $π^{2}/12$ and that these bounds are sharp. Remarkably, the conjecture remains open even within the class of triangles. The purpose of this paper is to prove the conjecture in this case. The conjecture is also proved for rectangles where a stronger monotonicity property is verified. Finally, the upper bound also holds for tangential quadrilateral.

math.AP↗
arXiv2023

Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian

For domains in $\mathbb{R}^d$, $d\geq 2$, we prove universal upper and lower bounds on the product of the bottom of the spectrum for the Laplacian to the power $p>0$ and the supremum over all starting points of the $p$-moments of the exit time of Brownian motion. It is shown that the lower bound is sharp for integer values of $p$ and that for $p \geq 1$, the upper bound is asymptotically sharp as $d\to\infty$. For all $p>0$, we prove the existence of an extremal domain among the class of domains that are convex and symmetric with respect to all coordinate axes. For this class of domains we conjecture that the cube is extremal.

math.PR↗
arXiv2021

Improved upper bounds for the Hot Spots constant of Lipschitz domains

The Hot Spots constant for bounded smooth domains was recently introduced by Steinerberger (2021) as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We generalize the Hot Spots constant to bounded Lipschitz domains and show that it leads to an if and only if condition for the weak Hot Spots conjecture HS2 from Bañuelos and Burdzy (1999). We also derive a new general formula for a dimension-dependent upper bound that can be tailored to any specific class of domains. This formula is then used to compute upper bounds for the Hot Spots constant of the class of all bounded Lipschitz domains in $\mathbb{R}^d$ for both small $d$ and for asymptotically large $d$ that significantly improve upon the existing results.

math.PR↗
arXiv2021

CLT with explicit variance for products of random singular matrices related to Hill's equation

We prove a central limit theorem (CLT) for the product of a class of random singular matrices related to a random Hill's equation studied by Adams$\unicode{x2013}$Bloch$\unicode{x2013}$Lagarias. The CLT features an explicit formula for the variance in terms of the distribution of the matrix entries and this allows for exact calculation in some examples. Our proof relies on a novel connection to the theory of $m$-dependent sequences which also leads to an interesting and precise nondegeneracy condition.

math.PR↗
arXiv2020

Conformal Skorokhod embeddings and related extremal problems

The conformal Skorokhod embedding problem (CSEP) is a planar variant of the classical problem where the solution is now a simply connected domain $D\subset\mathbb{C}$ whose exit time embeds a given probability distribution $μ$ by projecting the stopped Brownian motion onto the real axis. In this paper we explore two new research directions for the CSEP by proving general bounds on the principal Dirichlet eigenvalue of a solution domain in terms of the corresponding $μ$ and by proposing related extremal problems. Moreover, we give a new and nontrivial example of an extremal domain $\mathbb{U}$ that attains the lowest possible principal Dirichlet eigenvalue over all domains solving the CSEP for the uniform distribution on $[-1,1]$. Remarkably, the boundary of $\mathbb{U}$ is related to the Grim Reaper translating solution to the curve shortening flow in the plane. The novel tool used in the proof of the sharp lower bound is a precise relationship between the widths of the orthogonal projections of a simply connected planar domain and the support of its harmonic measure that we develop in the paper. The upper bound relies on spectral bounds for the torsion function which have recently appeared in the literature.

math.PR↗
arXiv2020

Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences

We consider three matrix models of order 2 with one random entry $ε$ and the other three entries being deterministic. In the first model, we let $ε\sim\textrm{Bernoulli}\left(\frac{1}{2}\right)$. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when $ε\sim\textrm{Bernoulli}\left(p\right)$ and $p\in [0,1]$ is a parameter. Both of these models are related to random Fibonacci sequences. In the last model, we compute the Lyapunov exponent exactly when the random entry is replaced with $ξε$ where $ε$ is a standard Cauchy random variable and $ξ$ is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.

math.PR↗
arXiv2019

Gradient Bounds for Kolmogorov Type Diffusions

We study gradient bounds and other functional inequalities for the diffusion semigroup generated by Kolmogorov type operators. The focus is on two different methods: coupling techniques and generalized $Γ$-calculus techniques. The advantages and drawbacks of each of these methods are discussed.

math.PR↗
arXiv2018

A derivation of the Black-Scholes option pricing model using a central limit theorem argument

The Black-Scholes model (sometimes known as the Black-Scholes-Merton model) gives a theoretical estimate for the price of European options. The price evolution under this model is described by the Black-Scholes formula, one of the most well-known formulas in mathematical finance. For their discovery, Merton and Scholes have been awarded the 1997 Nobel prize in Economics. The standard method of deriving the Black-Scholes European call option pricing formula involves stochastic differential equations. This approach is out of reach for most students learning the model for the first time. We provide an alternate derivation using the Lindeberg-Feller central limit theorem under suitable assumptions. Our approach is elementary and can be understood by undergraduates taking a standard undergraduate course in probability.

q-fin.GN↗
arXiv2018

The financial value of knowing the distribution of stock prices in discrete market models

An explicit formula is derived for the value of weak information in a discrete time model that works for a wide range of utility functions including the logarithmic and power utility. We assume a complete market with a finite number of assets and a finite number of possible outcomes. Explicit calculations are performed for a binomial model with two assets. The case of trinomial models is also discussed.

q-fin.MF↗
arXiv2017

Coupling in the Heisenberg group and its applications to gradient estimates

We construct a non-Markovian coupling for hypoelliptic diffusions which are Brownian motions in the three-dimensional Heisenberg group. We then derive properties of this coupling such as estimates on the coupling rate, and upper and lower bounds on the total variation distance between the laws of the Brownian motions. Finally we use these properties to prove gradient estimates for harmonic functions for the hypoelliptic Laplacian which is the generator of Brownian motion in the Heisenberg group.

math.PR↗