arXiv subjects
Publications and source records attributed to Atul Shekhar.
We give a full characterization of the fixed points of Branching Brownian motion with critical and supercritical drifts under no additional assumptions besides it being locally finite almost surely. In particular, we do not assume finite intensity (Kabluchko, 2012) or the finite top particle (Chen-Garban-Shekhar, 2023) conditions. We also give a full characterization of the domain of attraction of the fixed points of BBM.
Consider a sequence of random polynomials $P_n(z) = \prod_{k=1}^{n}(z - X_k)$, where $\{X_k\}_k$ are i.i.d. random variables distributed uniformly on the unit disc $\mathbb{D}$. Let $\Lambda_n = \{z \in \mathbb{C}: |P_n(z)| < 1\}$ be the lemniscate of $P_n$, and let $\mathscr{C}(\Lambda_n)$ be the number of connected components of $\Lambda_n$. In this paper, we prove that $\lim_{n\to\infty}\frac{\mathbb{E}[\mathscr{C}(\Lambda_n)]}{\sqrt{n}}= \gamma$, and identify the constant $\gamma$.
Complex solutions to squared Bessel SDEs appear naturally in relation to Schramm-Loewner evolutions. We prove a large deviation principle for such solutions as the dimension parameter tends to $-\infty$.
We give a complete characterisation of the domain of attraction of fixed points of branching Brownian motion (BBM) with critical drift. Prior to this classification, we introduce a suitable metric space of locally finite point measures on which we prove 1) that the BBM with critical drift is a well-defined Markov process and 2) that it satisfies the Feller property. Several applications of this characterisation are given.
We prove existence (and simpleness) of the trace for both forward and backward Loewner chains under fairly general conditions on semimartingale drivers. As an application, we show that stochastic Komatu-Loewner evolutions SKLE$_{\alpha,b}$ are generated by curves. As another application, motivated by a question of A. Sep\'{u}lveda, we show that for $\alpha >3/2$ and Brownian motion $B$, the driving function $|B_t|^\alpha$ generates a simple curve for small $t$. On a related note we also introduce a complex variant of Bessel-type SDEs and prove existence and uniqueness of strong solution. Such SDEs appear naturally while describing the trace of Loewner chains. In particular, we write SLE$_\kappa$, $\kappa <4$, in terms of stochastic flow of such SDEs.
In this note, we give a new proof of Liggett's theorem on the invariant measures of independent particle systems from [Lig78] in the particular case of independent drifted Brownian motions. This particular case has received a lot of attention recently due to its applications for the analysis of the local extrema of discrete Gaussian free field. The novelty of our proof is that it identifies directly the expected Poisson Point Process with exponential intensity without relying on the Choquet-Deny convolution equation $\mu * P=\mu$ ([ChoquetDeny60,Deny60]).
In this work, we characterize all the point processes $\theta=\sum_{i\in \mathbb{N}} \delta_{x_i}$ on $\mathbb{R}$ which are left invariant under branching Brownian motions with critical drift $-\sqrt{2}$. Our characterization holds under the only assumption that $\theta(\mathbb{R}_+)<\infty$ almost surely.
We consider a family of Bessel Processes that depend on the starting point $x$ and dimension $\delta$, but are driven by the same Brownian motion. Our main result is that almost surely the first time a process hits $0$ is jointly continuous in $x$ and $\delta$, provided $\delta\le 0$. As an application, we show that the SLE($\kappa$) welding homeomorphism is continuous in $\kappa$ for $\kappa\in [0,4]$. Our motivation behind this is to study the well known problem of the continuity of SLE$_\kappa$ in $\kappa$. The main tool in our proofs is random walks with increments distributed as infinite mean Inverse-Gamma laws.
We consider a variant of Bessel SDE by allowing the solution to be complex valued. Such SDEs appear naturally while studying the trace of Schramm-Loewner-Evolutions (SLE). We establish the existence and uniqueness of the strong solution to such SDEs when the dimension is negative. We also consider the stochastic flow associated to such SDEs and prove that it is almost surely continuous. Our proofs are based on an improvement of the derivative estimate of Rohde-Schramm \cite{RS05}. We finally show the connection between such stochastic flows and SLE$_{\kappa}$ for $\kappa <4$.
A quasislit is the image of a vertical line segment [0, iy], y > 0, under a quasiconformal homeomorphism of the upper half-plane fixing infinity. Quasislits correspond precisely to curves generated by the Loewner equation with a driving function in the Lip-1/2 class. It is known that a quasislit is contained in a cone depending only on its Loewner driving function Lip-1/2 seminorm, s. In this note we use the Loewner equation to give quantitative estimates on the opening angle of this cone in the full range s <4. The estimate is shown to be sharp for small s. As consequences, we derive explicit H\"older exponents for s < 4 as well as estimates on winding rates. We also relate quantitatively the Lip-1/2 seminorm with the quasiconformal dilatation and discuss the optimal regularity of quasislits achievable through reparametrization.
Motivated by the study of trace for Schramm-Loewner evolutions, we consider evolutions of planar domains governed by ordinary differential equations with holomorphic vector fields $F$ defined on the upper half plane $\mathbb{H}$. We show a smoothing effect of the presence of noise on the boundary behaviour of associated conformal maps. More precisely, if $F$ is H\"older, we show that evolving domains vary continuously in uniform topology and their boundaries are continuously differentiable Jordan arcs. This is in contrast with examples from deterministic setting where the corner points on the boundary of domain $F(\mathbb{H})$ may give rise to corner points on the boundaries of corresponding evolving domains.
To explore the relation between properties of Loewner chains and properties of their driving functions, we study Loewner chains driven by functions $U$ of finite total variation. Under some appropriate conditions, we show existence of the simple trace $\gamma$ and establish continuity of the map $U$ to $\gamma$ with respect to uniform topology on $\gamma$ and the total variation topology on $U$. In the spirit of work of Wong and Tran-Lind, we also obtain conditions on the driving function that ensures the trace to be continuously differentiable.
We consider the stochastic differential equation $$ X_t = x_0 + \int_0^t f(X_s)ds + \int_0^tσ(X_s)dB^{H}_s,$$ with $x_0 \in \mathbb{R}^d$, $d \geq 1$, $f: \mathbb{R}^d \rightarrow \mathbb{R}^d$ is bounded continuous, $σ: \mathbb{R}^d \rightarrow \mathbb{R}^{d\times d}$ is a uniformly elliptic, bounded, twice continuously differentiable conservative vector field and $B^H$ is fractional Brownian motion with $H \in (\frac{1}{3}, \frac{1}{2}]$. When $d=1$, $H= \frac{1}{2}$, and $f$ is Hölder continuous, in the spirit of Davie [D07], we establish the existence of a null set $\mathcal{N}$ depending only on $f, σ$ such that for all $x_0\in \mathbb{R}$ and $ω\in Ω\setminus \mathcal{N}$, the above equation admits a path-by-path unique solution. Our proof is based on establishing the uniform continuous differentiability of the flow associated with the equation. We also establish the path-by-path uniqueness for $d \geq 1$ and $H \in (\frac{1}{3}, \frac{1}{2}]$, but the null set may depend on $x_0$, thus extending a result of Catellier-Gubinelli [CG12].
Existence of Loewner trace is revisited. We identify finite energy paths (the "skeleton of Wiener measure") as natural class of regular drivers for which we find simple and natural estimates in terms of their (Cameron--Martin) norm. Secondly, now dealing with potentially rough drivers, a representation of the derivative of the (inverse of the) Loewner flow is given in terms of a rough- and then pathwise Föllmer integral. Assuming the driver within a class of Itô-processes, an exponential martingale argument implies existence of trace. In contrast to classical (exact) SLE computations, our arguments are well adapted to perturbations, such as non-constant $κ$ (assuming $<2$ for technical reasons) and additional finite-energy drift terms.
We consider rough paths with jumps. In particular, the analogue of Lyons' extension theorem and rough integration are established in a jump setting, offering a pathwise view on stochastic integration against cadlag processes. A class of Levy rough paths is introduced and characterized by a sub-ellipticity condition on the left-invariant diffusion vector fields and and a certain integrability property of the Carnot--Caratheodory norm with respect to the Levy measure on the group, using Hunt's framework of Lie group valued Levy processes. Examples of Levy rough paths include standard multi-dimensional Levy process enhanced with stochastic area as constructed by D. Williams, the pure area Poisson process and Brownian motion in a magnetic field. An explicit formula for the expected signature is given.
Recently, Hairer--Pillai proposed the notion of $θ$-roughness of a path which leads to a deterministic Norris lemma. In the Gubinelli framework (Hoelder, level 2) of rough paths, they were then able to prove a Hoermander type result (SDEs driven by fractional Brownian motion, $H>1/3$). We take a step back and propose a natural "roughness" condition relative to a given $p$-rough path in the sense of Lyons; the aim being a Doob-Meyer result for rough integrals in the sense of Lyons. The interest in our (weaker) condition is that it is immediately verified for large classes of Gaussian processes, also in infinite dimensions. We conclude with an application to non-Markovian system under Hoermander's condition.