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Ross G. Pinsky

Publications and source records attributed to Ross G. Pinsky.

51 records · Page 3Linked to original sources
arXiv2014

The Speed of a Random Walk Excited By Its Recent History

Let $N$ and $M$ be positive integers satisfying $1\le M\le N$, and let $0<p_0<p_1<1$. Define a process $\{X_n\}_{n=0}^\infty$ on $\mathbb{Z}$ as follows. At each step, the process jumps either one step to the right or one step to the left, according to the following mechanism. For the first $N$ steps, the process behaves like a random walk that jumps to the right with probability $p_0$ and to the left with probability $1-p_0$. At subsequent steps the jump mechanism is defined as follows: if at least $M$ out of the $N$ most recent jumps were to the right, then the probability of jumping to the right is $p_1$; however, if fewer than $M$ out of the $N$ most recent jumps were to the right, then the probability of jumping to the right is $p_0$. We calculate the speed of the process. Then we let $N\to\infty$ and $\frac MN\to r\in[0,1]$, and calculate the limiting speed. More generally, we consider the above questions for a random walk with a finite number $l$ of threshold levels, $(M_i,p_i)_{i=1}^l$, above the pre-threshold level $p_0$, as well as for one model with $l=N$ such thresholds.

math.PR↗
arXiv2012

Transience, recurrence and speed of diffusions with a non-Markovian two-phase "use it or lose it" drift

We investigate the transience/recurrence of a non-Markovian, one-dimensional diffusion process which consists of a Brownian motion with a non-anticipating drift that has two phases---a transient to $+\infty$ mode which is activated when the diffusion is sufficiently near its running maximum, and a recurrent mode which is activated otherwise. We also consider the speed of a diffusion with a two-phase drift, where the drift is equal to a certain positive constant when the diffusion is sufficiently near its running maximum, and is equal to another positive constant otherwise.

math.PR↗
arXiv2012

Detecting Tampering in a Random Hypercube

Consider the random hypercube $H_2^n(p_n)$ obtained from the hypercube $H_2^n$ by deleting any given edge with probabilty $1-p_n$, independently of all the other edges. A diameter path in $H_2^n$ is a longest geodesic path in $H_2^n$. Consider the following two ways of tampering with the random graph $H_2^n(p_n)$: (i) choose a diameter path at random and adjoin all of its edges to $H_2^n(p_n)$; (ii) choose a diameter path at random from among those that start at $0=(0,..., 0)$, and adjoin all of its edges to $H_2^n(p_n)$. We study the question of whether these tamperings are detectable asymptotically as $n\to\infty$.

math.PR↗
arXiv2012

Cyclic to Random Transposition Shuffles

Consider a permutation $σ\in S_n$ as a deck of cards numbered from 1 to $n$ and laid out in a row, where $σ_j$ denotes the number of the card that is in the $j$-th position from the left.\rm\ We define two cyclic to random transposition shuffles. The first one works as follows: for $j=1,..., n$, on the $j$-th step transpose the card that was \it originally\rm\ the $j$-th from the left with a random card (possibly itself). The second shuffle works as follows: on the $j$-th step, transpose the card that is \it currently\rm\ in the $j$-th position from the left with a random card (possibly itself). For these shuffles, for each $b\in[0,1]$, we calculate explicitly the limiting rescaled density function of $x,0\le x\le1$, for the probability that a card with a number around $bn$ ends up in a position around $xn$, and for each $x\in[0,1]$, we calculate the limiting rescaled density function of $b,0\le b\le 1$, for the probability that the card in a position around $xn$ will be a card with a number around $bn$. These density functions all have a discontinuity at $x=b$, and for each of them, the supremum of the density is obtained by approaching the discontinuity from one side, and, for certain values of the parameter, the infimum of the density is obtained by approaching the discontinuity from the other side.

math.PR↗
arXiv2012

Probabilistic and Combinatorial Aspects of the Card-Cyclic to Random Insertion Shuffle

Consider a permutation $σ\in S_n$ as a deck of cards numbered from 1 to $n$ and laid out in a row, where $σ_j$ denotes the number of the card that is in the $j$-th position from the left.\rm\ We study some probabilistic and combinatorial aspects of the shuffle on $S_n$ defined by removing and then randomly reinserting each of the $n$ cards once, with the removal and reinsertion being performed according to the original left to right order of the cards. The novelty here in this nonstandard shuffle is that every card is removed and reinserted exactly once. The bias that remains turns out to be quite strong and possesses some surprising features.

math.PR↗
arXiv2011

Asymptotic Behavior of the Principal Eigenvalue for a Class of Non-Local Elliptic Operators Related to Brownian Motion with Spatially Dependent Random Jumps

Let $D\subset R^d$ be a bounded domain and let $\mathcal P(D)$ denote the space of probability measures on $D$. Consider a Brownian motion in $D$ which is killed at the boundary and which, while alive, jumps instantaneously according to a spatially dependent exponential clock with intensity $γV$ to a new point, according to a distribution $μ\in\mathcal P(D)$. From its new position after the jump, the process repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator $-L_{γ,μ}$, defined by L_{γ,μ}u\equiv -\frac12Δu+γV C_μ(u), with the Dirichlet boundary condition, where $C_μ$ is the "$μ$-centering" operator defined by C_μ(u)=u-\int_Du dμ. The principal eigenvalue, $λ_0(γ,μ)$, of $L_{γ,μ}$ governs the exponential rate of decay of the probability of not exiting $D$ for large time. We study the asymptotic behavior of $λ_0(γ,μ)$ as $γ\to\infty$. In particular, if $μ$ possesses a density in a neighborhood of the boundary, which we call $μ$, then \lim_{γ\to\infty}γ^{-\frac12}λ_0(γ,μ)=\frac{\int_{\partial D}\fracμ{\sqrt {V}}dσ}{\sqrt2\int_D\frac1{V}dμ}. If $μ$ and all its derivatives up to order $k-1$ vanish on the boundary, but the $k$-th derivative does not vanish identically on the boundary, then $λ_0(γ,μ)$ behaves asymptotically like $c_kγ^{\frac{1-k}2}$, for an explicit constant $c_k$.

math.PR↗
arXiv2011

Asymptotics for Exit Problem and Principal Eigenvalue for a Class of Non-Local Elliptic Operators Related to Diffusion Processes with Random Jumps and Vanishing Diffusion

Let $D\subset R^d$ be a bounded domain and denote by $\mathcal P(D)$ the space of probability measures on $D$. Let \begin{equation*} L=\frac12\nabla\cdot a\nabla +b\nabla \end{equation*} be a second order elliptic operator. Let $μ\in\mathcal P(D)$ and $δ>0$. Consider a Markov process $X(t)$ in $D$ which performs diffusion in $D$ generated by the operator $δL$ and is stopped at the boundary, and which while running, jumps instantaneously, according to an exponential clock with spatially dependent intensity $V>0$, to a new point, according to the distribution $μ$. The Markov process is generated by the operator $L_{δ,μ, V}$ defined by \begin{equation*} L_{δ,μ, V}ϕ\equiv δL ϕ+V(\int_Dϕdμ-ϕ). \end{equation*} %where $C_μ$ is the % "$μ$-centering" operator defined by %\begin{equation*} %C_μ(ϕ)=ϕ-\int_Dϕdμ. %\end{equation*} Let $ϕ_{δ,μ,V}$ denote the solution to the Dirichlet problem \begin{equation*}\label{Dirprob} \begin{aligned} &L_{δ,μ,V}ϕ=0\ \text{in}\ D;\\ &ϕ=f\ \text{on}\ \partial D, \end{aligned} \end{equation*} where $f$ is continuous. The solution has the stochastic representation \begin{equation*} ϕ_{δ,μ,V}(x)=E_xf(X(τ_D)). \end{equation*} One has that $ϕ_{0,μ,V}(f)\equiv\lim_{δ\to0}ϕ_{δ,μ,V}(x)$ is independent of $x\in D$. We evaluate this constant in the case that $μ$ has a density in a neighborhood of $\partial D$. We also study the asymptotic behavior as $δ\to0$ of the principal eigenvalue $λ_0(δ,μ,V)$ for the operator $L_{δ,μ, V}$, which generalizes previously obtained results for the case $L=\frac12 Δ$.

math.PR↗
arXiv2009

One-Dimensional Diffusions That Eventually Stop Down-Crossing

Consider a diffusion process corresponding to the operator $L=\frac12a\frac{d^2}{dx^2}+b\frac d{dx}$ and which is transient to $+\infty$. For $c>0$, we give an explicit criterion in terms of the coefficients $a$ and $b$ which determines whether or not the diffusion almost surely eventually stops making down-crossings of length $c$. As a particular case, we show that if $a=1$, then the diffusion almost surely stops making down-crossings of length $c$ if $b(x)\ge\frac1{2c}\log x+\fracγc\log\log x$, for some $γ>1$ and for large $x$, but makes down-crossings of length $c$ at arbitrarily large times if $b(x)\le\frac1{2c}\log x+\frac1c\log\log x$, for large $x$.

math.PR↗
arXiv2009

Transcience/recurrence for normally reflected Brownian motion in unbounded domains

Let $D\subsetneq R^d$ be an unbounded domain and let $B(t)$ be a Brownian motion in $D$ with normal reflection at the boundary. We study the transcience/recurrence dichotomy, focusing mainly on domains of the form $D=\{(x,z)\in R^{l+m}:|z|<H(|x|)\}$, where $d=l+m$ and $H$ is a sufficiently regular function. This class of domains includes various horn-shaped domains and generalized slab domains.

math.PR↗
arXiv2008

Explicit and Almost Explicit Spectral Calculations for Diffusion Operators

The diffusion operator $$ H_D=-\frac12\frac d{dx}a\frac d{dx}-b\frac d{dx}=-\frac12\exp(-2B)\frac d{dx}a\exp(2B)\frac d{dx}, $$ where $B(x)=\int_0^x\frac ba(y)dy$, defined either on $R^+=(0,\infty)$ with the Dirichlet boundary condition at $x=0$, or on $R$, can be realized as a self-adjoint operator with respect to the density $\exp(2Q(x))dx$. The operator is unitarily equivalent to the Schrödinger-type operator $H_S=-\frac12\frac d{dx}a\frac d{dx}+V_{b,a}$, where $V_{b,a}=\frac12(\frac{b^2}a+b')$. We obtain an explicit criterion for the existence of a compact resolvent and explicit formulas up to the multiplicative constant 4 for the infimum of the spectrum and for the infimum of the essential spectrum for these operators. We give some applications which show in particular how $\infσ(H_D)$ scales when $a=νa_0$ and $b=γb_0$, where $ν$ and $γ$ are parameters, and $a_0$ and $b_0$ are chosen from certain classes of functions. We also give applications to self-adjoint, multi-dimensional diffusion operators.

math.SP↗
arXiv2008

The Fujita Exponent for Semilinear Heat Equations with Quadratically Decaying Potential or in an Exterior Domain

Consider the equation u_t=Δu-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=ϕ(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\fracω{|x|^2}$ as $|x|\to\infty$, for some $ω\neq0$, and $a(x)$ is on the order $|x|^m$ as $|x|\to\infty$, for some $m\in (-\infty,\infty)$. A solution to the above equation is called global if $T=\infty$. Under some additional technical conditions, we calculate a critical exponent $p^*$ such that global solutions exist for $p>p^*$, while for $1<p\le p^*$, all solutions blow up in finite time. We also show that when $V\equiv0$, the blow-up/global solution dichotomy for \eqref{abstract} coincides with that for the corresponding problem in an exterior domain with the Dirichlet boundary condition, including the case in which $p$ is equal to the critical exponent.

math.AP↗
arXiv2007

When the law of large numbers fails for increasing subsequences of random permutations

Let the random variable $Z_{n,k}$ denote the number of increasing subsequences of length $k$ in a random permutation from $S_n$, the symmetric group of permutations of $\{1,...,n\}$. In a recent paper [Random Structures Algorithms 29 (2006) 277--295] we showed that the weak law of large numbers holds for $Z_{n,k_n}$ if $k_n=o(n^{2/5})$; that is, \[\lim_{n\to\infty}\frac{Z_{n,k_n}}{EZ_{n,k_n}}=1\qquad in probability.\] The method of proof employed there used the second moment method and demonstrated that this method cannot work if the condition $k_n=o(n^{2/5})$ does not hold. It follows from results concerning the longest increasing subsequence of a random permutation that the law of large numbers cannot hold for $Z_{n,k_n}$ if $k_n\ge cn^{1/2}$, with $c>2$. Presumably there is a critical exponent $l_0$ such that the law of large numbers holds if $k_n=O(n^l)$, with $l 0$, for some $l>l_0$. Several phase transitions concerning increasing subsequences occur at $l=1/2$, and these would suggest that $l_0={1/2}$. However, in this paper, we show that the law of large numbers fails for $Z_{n,k_n}$ if $\limsup_{n\to\infty}\frac{k_n}{n^{4/9}}=\infty$. Thus, the critical exponent, if it exists, must satisfy $l_0\in[{2/5},{4/9}]$.

math.PR↗
arXiv2005

The compact support property for measure-valued processes

The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued processes corresponding to semi-linear equations of the form \[ \begin{aligned}& u_t=Lu+βu-αu^p \text{in} R^d\times (0,\infty), p\in(1,2]; &u(x,0)=f(x) \text{in} R^d; &u(x,t)\ge0 \text{in} R^d\times[0,\infty). \end{aligned} \] In particular, we shall investigate how the interplay between the underlying motion (the diffusion process corresponding to $L$) and the branching affects the compact support property. In \cite{EP99}, the compact support property was shown to be equivalent to a certain analytic criterion concerning uniqueness of the Cauchy problem for the semi-linear parabolic equation related to the measured valued process. In a subsequent paper \cite{EP03}, this analytic property was investigated purely from the point of view of partial differential equations. Some of the results obtained in this latter paper yield interesting results concerning the compact support property. In this paper, the results from \cite{EP03} that are relevant to the compact support property are presented, sometimes with extensions. These results are interwoven with new results and some informal heuristics. Taken together, they yield a rather comprehensive picture of the compact support property. \it Inter alia\rm, we show that the concept of a measure-valued process \it hitting\rm a point can be investigated via the compact support property, and suggest an alternate proof of a result concerning the hitting of points by super-Brownian motion.

math.PR↗
arXiv2004

Uniqueness/nonuniqueness for nonnegative solutions of the Cauchy problem for $u_t=Δu-u^p$ in a punctured space

Consider classical solutions to the following Cauchy problem in a punctured space: $ &u_t=Δu -u^p \text{in} (R^n-\{0\})\times(0,\infty); & u(x,0)=g(x)\ge0 \text{in} R^n-\{0\}; &u\ge0 \text{in} (R^n-\{0\})\times[0,\infty). $ We prove that if $p\ge\frac n{n-2}$, then the solution to \eqref{abstract} is unique for each $g$. On the other hand, if $p<\frac n{n-2}$, then uniqueness does not hold when $g=0$; that is, there exists a nontrivial solution with vanishing initial data.

math.AP↗
arXiv2004

The compact support property for measure-valued diffusions

The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued diffusion processes corresponding to semi-linear equations of the form \[& u_t=Lu+βu-αu^p \text{in} R^d\times (0,\infty), p\in(1,2]; &u(x,0)=f(x) \text{in} R^d; &u(x,t)\ge0 \text{in} R^d\times[0,\infty). \] In particular, we shall investigate how the interplay between the underlying motion (the diffusion process corresponding to $L$) and the branching affects the compact support property. In \cite{EP99}, the compact support property was shown to be equivalent to a certain analytic criterion concerning uniqueness of the Cauchy problem for the semilinear parabolic equation related to the measured valued diffusion. In a subsequent paper \cite{EP03}, this analytic property was investigated purely from the point of view of partial differential equations. Some of the results obtained in this latter paper yield interesting results concerning the compact support property. In this paper, the results from \cite{EP03} that are relevant to the compact support property are presented, sometimes with extensions. These results are interwoven with new results and some informal heuristics. Taken together, they yield a fairly comprehensive picture of the compact support property. \it Inter alia\rm, we show that the concept of a measure-valued diffusion \it hitting\rm a point can be investigated via the compact support property, and suggest an alternate proof of a result concerning the hitting of points by super-Brownian motion.

math.PR↗