Searcharxiv⌕ Search

arXiv subjects

Leonid Mytnik

Publications and source records attributed to Leonid Mytnik.

At least 37 records · Page 2Linked to original sources
arXiv2020

Absolute continuity of the Super-Brownian motion with infinite mean

In this work we prove that for any dimension $d\geq 1$ and any $γ\in (0,1)$ super-Brownian motion corresponding to the log-Laplace equation \begin{equation*} \begin{split} \frac{\partial v(t,x)}{\partial t } & = \frac{1}{2}\bigtriangleup v(t,x) + v^γ(t,x) ,\: (t,x) \in \mathbb{R}_+\times \mathbb{R}^d,\\ v(0,x)&= f(x) \end{split} \end{equation*} is absolutely continuous with respect to the Lebesgue measure at any fixed time $t>0$. Our proof is based on properties of solutions of the \LL\ equation. We also prove that when initial datum $v(0,\cdot)$ is a finite, non-zero measure, then the \LL\ equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.

math.PR↗
arXiv2020

The distance between the two BBM leaders

We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit $d_{12}$ of this random variable can be expressed in terms of PDEs related to the Fisher--KPP equation. We use such a representation to determine the sharp asymptotics of $\mathbb{P}(d_{12} > a)$ as $a\to+\infty$. These tail asymptotics were previously known to "exponential order;" we discover an algebraic correction to this behavior.

math.AP↗
arXiv2020

The Aldous chain on cladograms in the diffusion limit

In [Ald00], Aldous investigates a symmetric Markov chain on cladograms and gives bounds on its mixing and relaxation times. The latter bound was sharpened in [Sch02]. In the present paper we encode cladograms as binary, algebraic measure trees and show that this Markov chain on cladograms with fixed number of leaves converges in distribution as the number of leaves goes to infinity. We give a rigorous construction of the limit, whose existence was conjectured by Aldous and which we therefore refer to as Aldous diffusion, as a solution of a well-posed martingale problem. We show that the Aldous diffusion is a Feller process with continuous paths, and the algebraic measure Brownian CRT is its unique invariant distribution. Furthermore, we consider the vector of the masses of the three subtrees connected to a sampled branch point. In the Brownian CRT, its annealed law is known to be the Dirichlet distribution. Here, we give an explicit expression for the infinitesimal evolution of its quenched law under the Aldous diffusion.

math.PR↗
arXiv2019

Infinite rate symbiotic branching on the real line: The tired frogs model

Consider a population of infinitesimally small frogs on the real line. Initially the frogs on the positive half-line are dormant while those on the negative half-line are awake and move according to the heat flow. At the interface, the incoming wake frogs try to wake up the dormant frogs and succeed with a probability proportional to their amount among the total amount of involved frogs at the specific site. Otherwise, the incoming frogs also fall asleep. This frog model is a special case of the infinite rate symbiotic branching process on the real line with different motion speeds for the two types. We construct this frog model as the limit of approximating processes and compute the structure of jumps. We show that our frog model can be described by a stochastic partial differential equation on the real line with a jump type noise.

math.PR↗
arXiv2019

The speed of a random front for stochastic reaction-diffusion equations with strong noise

We study the asymptotic speed of a random front for solutions $u_t(x)$ to stochastic reaction-diffusion equations of the form \[ \partial_tu=\farc{1}{2}\partial_x^2u+f(u)+σ\sqrt{u(1-u)}\dot{W}(t,x),~t\ge 0,~x\in\Rm, \] arising in population genetics. Here, $f$ is a continuous function with $f(0)=f(1)=0$, and such that~$|f(u)|\le K|u(1-u)|^γ$ with~$γ\ge 1/2$, and $\dot{W}(t,x)$ is a space-time Gaussian white noise. We assume that the initial condition $u_0(x)$ satisfies $0\le u_0(x)\le 1$ for all $x\in\Rm$, $u_0(x)=1$ for~$x R_0$. We show that when $σ>0$, for each $t>0$ there exist~$R(u_t)<+\infty$ and~$L(u_t)<-\infty$ such that $u_t(x)=0$ for $x>R(u_t)$ and $u_t(x)=1$ for~$x 0$ there exists a finite deterministic speed~$V(σ)\in\Rm$ so that~$R(u_t)/t\to V(σ)$ as $t\to+\infty$, almost surely. This is in dramatic contrast with the deterministic case $σ=0$ for nonlinearities of the type $f(u)=u^m(1-u)$ with $0 1/2$ there exists $c_f\in\Rm$, so that~$σ^2V(σ)\to c_f$ as~$σ\to+\infty$ and give a characterization of $c_f$. The last result complements a lower bound obtained by Conlon and Doering \cite{cd05} for the special case of $f(u)=u(1-u)$ where a duality argument is available.

math.AP↗
arXiv2019

General Contact Process with Rapid Stirring

We study the limiting behavior of an interacting particle system evolving on the lattice $Z^{d}$ for $d\ge 3$. The model is known as the contact process with rapid stirring. The process starts with a single particle at the origin. Each particle may die, jump to a neighboring site if it is vacant or split. In the case of splitting, one of the offspring takes the place of the parent while the other, the newborn particle, is sent to another site in $Z^{d}$ according to a certain distribution; if the newborn particle lands on an occupied site, its birth is suppressed. We study the asymptotic behavior of the critical branching rate as the jumping rate (also known as the stirring rate) approaches infinity.

math.PR↗
arXiv2018

On the topological boundary of the range of super-Brownian motion-extended version

We show that if $\partial\mathcal{R}$ is the boundary of the range of super-Brownian motion and dim denotes Hausdorff dimension, then with probability one, for any open set $U$, $\partial\mathcal{R}\cap U\neq\emptyset$ implies $$\text{dim}(U\cap\partial\mathcal{R})=\begin{cases} 4-2\sqrt2\approx1.17&\text{ if }d=2\\ \frac{9-\sqrt{17}}{2}\approx 2.44&\text{ if }d=3. \end{cases}$$ This improves recent results of the last two authors (arxiv:1711.03486) by working with the actual topological boundary, rather than the boundary of the zero set of the local time, and establishing a local result for the dimension.

math.PR↗
arXiv2018

Strong existence and uniqueness for stable stochastic differential equations with distributional drift

We consider the stochastic differential equation $$ dX_t = b(X_t) dt + dL_t,$$ where the drift $b$ is a generalized function and $L$ is a symmetric one dimensional $α$-stable Lévy processes, $α\in (1, 2)$. We define the notion of solution to this equation and establish strong existence and uniqueness whenever $b$ belongs to the Besov--Hölder space $\mathcal{C}^β$ for $β>1/2-α/2$.

math.PR↗
arXiv2017

Pathwise Uniqueness for the Stochastic Heat Equation with Hölder Continuous Drift and Noise Coefficients

We study the solutions of the stochastic heat equation with multiplicative space-time white noise. We prove a comparison theorem between the solutions of stochastic heat equations with the same noise coefficient which is Hölder continuous of index $γ>3/4$, and drift coefficients that are Lipschitz continuous. Later we use the comparison theorem to get sufficient conditions for the pathwise uniqueness for solutions of the stochastic heat equation, when both the white noise and the drift coefficients are Hölder continuous.

math.PR↗
arXiv2017

Finite system scheme for infinite rate mutually catalytic branching with infinite branching rate

For many stochastic diffusion processes with mean field interaction, convergence of the rescaled total mass processes towards a diffusion process is known. Here we show convergence of the so-called finite system scheme for interacting jump-type processes known as mutually catalytic branching processes with infinite branching rate. Due to the lack of second moments the rescaling of time is different from the finite rate mutually catalytic case. The limit of rescaled total mass processes is identified as the finite rate mutually catalytic branching diffusion. The convergence of rescaled processes holds jointly with convergence of coordinate processes, whereas the latter converge at a different time scale.

math.PR↗
arXiv2016

Regularization by noise and flows of solutions for a stochastic heat equation

Motivated by the regularization by noise phenomenon for SDEs we prove existence and uniqueness of the flow of solutions for the non-Lipschitz stochastic heat equation $$\frac{\partial u}{\partial t}=\frac12\frac{\partial^2 u}{\partial z^2} + b(u(t,z)) + \dot{W}(t,z), $$ where $\dot W$ is a space-time white noise on $\mathbb{R}_+\times\mathbb{R}$ and $b$ is a bounded measurable function on $\mathbb{R}$. As a byproduct of our proof we also establish the so-called path--by--path uniqueness for any initial condition in a certain class on the same set of probability one. This extends recent results of Davie (2007) to the context of stochastic partial differential equations.

math.PR↗
arXiv2016

Singularities of stable super-Brownian motion

We investigate in this work the spectrum of singularities of super-Brownian motion with stable branching. The main purpose is to provide a uniform description of the latter in high dimension $d\geq\tfrac{2}{γ-1}$, presenting the singularities existing at every time t and characterising as well the set of random times at which singularities of higher order appear. In lower dimensions, we give a partial description of the singularities which complement the recent study of the density by Mytnik and Wachtel.

math.PR↗
arXiv2015

On the boundary of the support of super-Brownian motion: with appendices

We study the density X(t,x) of one-dimensional super-Brownian motion and find the asymptotic behaviour of P(0<X(t,x)<a) as a approaches 0, as well as the Hausdorff dimension of the boundary of the support of X(t). The answers are in terms of the lead eigenvalue of the Ornstein-Uhlenbeck generator with a particular killing term. This work is motivated in part by questions of pathwise uniqueness for associated stochastic partial differential equations.

math.PR↗
arXiv2015

Regularity and irregularity of superprocesses with $(1+β)$-stable branching mechanism

We would like to give an overview of results on regularity, or better to say "irregularity", properties of densities at fixed times of super-Brownian motion with $(1+β)$-stable branching for $β<1$. First, the following dichotomy for the density is shown: it is continuous in the dimension $d=1$ and locally unbounded in all higher dimensions where it exists. Then in $d=1$ we determine pointwise and local Hölder exponents of the density, and calculate the multifractal spectrum corresponding to pointwise Hölder exponents.

math.PR↗
arXiv2015

Stochastic Differential Equation for Brox Diffusion

This paper studies the weak and strong solutions to the stochastic differential equation $ dX(t)=-\frac12 \dot W(X(t))dt+d\mathcal{B}(t)$, where $(\mathcal{B}(t), t\ge 0)$ is a standard Brownian motion and $W(x)$ is a two sided Brownian motion, independent of $\mathcal{B}$. It is shown that the Itô-McKean representation associated with any Brownian motion (independent of $W$) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. Itô calculus for the solution is developed. For dealing with the singularity of drift term $\int_0^T \dot W(X(t))dt$, the main idea is to use the concept of local time together with the polygonal approximation $W_π$. Some new results on the local time of Brownian motion needed in our proof are established.

math.PR↗
arXiv2015

Uniqueness for Volterra-type stochastic integral equations

We study uniqueness for a class of Volterra-type stochastic integral equations. We focus on the case of non-Lipschitz noise coefficients. The connection of these equations to certain degenerate stochastic partial differential equations plays a key role.

math.PR↗