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Register a new userOn the Limit Law of a Random Walk Conditioned to Reach a High Level
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We consider a random walk with a negative drift and with a jump distribution which under Cramers change of measure belongs to the domain of attraction of a spectrally positive stable law. If conditioned to reach a high level and suitably scaled, this random walk converges in law to a nondecreasing Markov process which can be interpreted as a spectrally-positive Levy %-Khinchin process conditioned not to overshoot level one.
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We consider a simple random walk on $mathbb{Z}^d$ started at the origin and stopped on its first exit time from $(-L,L)^d cap mathbb{Z}^d$. Write $L$ in the form $L = m N$ with $m = m(N)$ and $N$ an integer going to infinity in such a way that $L^2 sim A N^d$ for some real constant $A > 0$. Our main result is that for $d ge 3$, the projection of the stopped trajectory to the $N$-torus locally converges, away from the origin, to an interlacement process at level $A d sigma_1$, where $sigma_1$ is the exit time of a Brownian motion from the unit cube $(-1,1)^d$ that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) and Sznitman (2009).
We consider the limit behavior of a one-dimensional random walk with unit jumps whose transition probabilities are modified every time the walk hits zero. The invariance principle is proved in the scheme of series where the size of modifications depends on the number of series. For the natural scaling of time and space arguments the limit process is (i) a Brownian motion if modifications are small, (ii) a linear motion with a random slope if modifications are large, and (iii) the limit process satisfies an SDE with a local time of unknown process in a drift if modifications are moderate.
We consider a random walk $tilde S$ which has different increment distributions in positive and negative half-planes. In the upper half-plane the increments are mean-zero i.i.d. with finite variance. In the lower half-plane we consider two cases: increments are positive i.i.d. random variables with either a slowly varying tail or with a finite expectation. For the distributions with a slowly varying tails, we show that ${frac{1}{sqrt n} tilde S(nt)}$ has no weak limit in $De$; alternatively, the weak limit is a reflected Brownian motion.
Place an obstacle with probability $1-p$ independently at each vertex of $mathbb Z^d$ and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. For $d geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment such that the origin is contained in an infinite connected component free of obstacles. It has previously been shown that with high probability, the random walk conditioned on survival up to time $n$ will be localized in a ball of volume asymptotically $dlog_{1/p}n$. In this work, we prove that this ball is free of obstacles, and we derive the limiting one-time distributions of the random walk conditioned on survival. Our proof is based on obstacle modifications and estimates on how such modifications affect the probability of the obstacle configurations as well as their associated Dirichlet eigenvalues, which is of independent interest.
We consider a discrete time simple symmetric random walk among Bernoulli obstacles on $mathbb{Z}^d$, $dgeq 2$, where the walk is killed when it hits an obstacle. It is known that conditioned on survival up to time $N$, the random walk range is asymptotically contained in a ball of radius $varrho_N=C N^{1/(d+2)}$ for any $dgeq 2$. For $d=2$, it is also known that the range asymptotically contains a ball of radius $(1-epsilon)varrho_N$ for any $epsilon>0$, while the case $dgeq 3$ remains open. We complete the picture by showing that for any $dgeq 2$, the random walk range asymptotically contains a ball of radius $varrho_N-varrho_N^epsilon$ for some $epsilon in (0,1)$. Furthermore, we show that its boundary is of size at most $varrho_N^{d-1}(log varrho_N)^a$ for some $a>0$.
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