Scaling Limit Of Random Walk Research Articles

Scaling limits of random walks, harmonic profiles, and stationary nonequilibrium states in Lipschitz domains

Scaling limits of random walks, harmonic profiles, and stationary nonequilibrium states in Lipschitz domains

Brownian motion on stable looptrees

In this article, we introduce Brownian motion on α-stable looptrees using resistance techniques, where α∈(1,2). We prove an invariance principle characterising it as the scaling limit of random walks on discrete looptrees, and prove precise local and global bounds on its heat kernel. We also conduct a detailed investigation of the volume growth properties of stable looptrees, and show that the random volume and heat kernel fluctuations are locally log-logarithmic, and globally logarithmic around leading terms of rα and t−αα+1 respectively. These volume fluctuations are the same order as for the Brownian continuum random tree, but the upper volume fluctuations (and corresponding lower heat kernel fluctuations) are different to those of stable trees.

Infinite stable looptrees

We give a construction of an infinite stable looptree, which we denote by $\mathcal{L}_{\alpha}^{\infty}$, and prove that it arises both as a local limit of the compact stable looptrees of Curien and Kortchemski (2015), and as a scaling limit of the infinite discrete looptrees of Richier (2017) and Bj\"ornberg and Stef\'ansson (2015). As a consequence, we are able to prove various convergence results for volumes of small balls in compact stable looptrees, explored more deeply in a companion paper. We also establish the spectral dimension of $\mathcal{L}_{\alpha}^{\infty}$, and show that it agrees with that of its discrete counterpart. Moreover, we show that Brownian motion on $\mathcal{L}_{\alpha}^{\infty}$ arises as a scaling limit of random walks on discrete looptrees, and as a local limit of Brownian motion on compact stable looptrees, which has similar consequences for the limit of the heat kernel.

Scaling Limits of Random Walk Bridges Conditioned to Avoid a Finite Set

This paper concerns a scaling limit of a one-dimensional random walk $$S^x_n$$ started from x on the integer lattice conditioned to avoid a non-empty finite set A, the random walk being assumed to be irreducible and have zero mean. Suppose the variance $$\sigma ^2$$ of the increment law is finite. Given positive constants b, c and T, we consider the scaled process $$S^{b_N}_{[tN]}/\sigma \sqrt{N}$$ , $$0\le t \le T$$ , started from a point $$b_N \approx b\sqrt{N}$$ conditioned to arrive at another point $$\approx -\,c\sqrt{N}$$ at $$t=T$$ and avoid A in between and discuss the functional limit of it as $$N\rightarrow \infty $$ . We show that it converges in law to a continuous process if $$E[|S_1|^3; S_1<0] <\infty $$ . If $$E[|S_1|^3; S_1<0] =\infty $$ , we suppose $$P[S_1<u]$$ to vary regularly as $$u\rightarrow -\,\infty $$ with exponent $$-\,\beta $$ , $$2\le \beta \le 3$$ and show that it converges to a process which has one downward jump that clears the origin if $$\beta <3$$ ; in case $$\beta =3$$ , there arises the same limit process as in case $$E[|S_1|^3; S_1<0] <\infty $$ . In case $$\sigma ^2=\infty $$ , we consider the special case when $$S_1$$ belongs to the domain of attraction of a stable law of index $$1<\alpha <2$$ having no negative jumps and obtain analogous results.

Gene flow across geographical barriers — scaling limits of random walks with obstacles

We study a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. We obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at zero reaches an exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at zero reaches another exponential level, etc. These random walks are used in population genetics to trace the position of ancestors in the past near geographical barriers.

High order heat-type equations and random walks on the complex plane

A probabilistic construction for the solution of a general class of high order heat-type equations is constructed in terms of the scaling limit of random walks in the complex plane.

Stochastic and variational approach to the Lax-Friedrichs scheme

We present a stochastic and variational aspect of the Lax-Friedrichs scheme applied to hyperbolic scalar conservation laws. This is a finite difference version of Fleming’s results (’69) that the vanishing viscosity method is characterized by stochastic processes and calculus of variations. We convert the difference equation into that of the Hamilton-Jacobi type and introduce corresponding calculus of variations with random walks. The stability of the scheme is obtained through the calculus of variations. The convergence of approximation is derived from the law of large numbers in hyperbolic scaling limit of random walks. The main advantages due to our approach are the following: Our framework is basically pointwise convergence, not L1 as usual, which yields uniform convergence except “small” neighborhoods of shocks; The convergence proof is verified for arbitrarily large time interval, which is hard to obtain in the case of flux functions of general types depending on both space and time; The approximation of characteristics curves is available as well as that of PDE-solutions, which is particularly important for applications of the Lax-Friedrichs scheme to the weak KAM theory.

Liouville Brownian motion

We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric $e^{\gamma X(z)}\,dz^{2}$, $\gamma<\gamma_{c}=2$ and $X$ is a Gaussian Free Field. Such a process is conjectured to be related to the scaling limit of random walks on large planar maps eventually weighted by a model of statistical physics which are embedded in the Euclidean plane or in the sphere in a conformal manner. The construction amounts to changing the speed of a standard two-dimensional Brownian motion $B_{t}$ depending on the local behavior of the Liouville measure “$M_{\gamma}(dz)=e^{\gamma X(z)}\,dz$”. We prove that the associated Markov process is a Feller diffusion for all $\gamma<\gamma_{c}=2$ and that for all $\gamma<\gamma_{c}$, the Liouville measure $M_{\gamma}$ is invariant under $P_{\mathbf{t}}$. This Liouville Brownian motion enables us to introduce a whole set of tools of stochastic analysis in Liouville quantum gravity, which will be hopefully useful in analyzing the geometry of Liouville quantum gravity.

Discrete approximation of symmetric jump processes on metric measure spaces

In this paper we give general criteria on tightness and weak convergence of discrete Markov chains to symmetric jump processes on metric measure spaces under mild conditions. As an application, we investigate discrete approximation for a large class of symmetric jump processes. We also discuss some application of our results to the scaling limit of random walk in random conductance.

Loop-erased random walk and Poisson kernel on planar graphs

Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on ℤ2 is SLE2. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into ℂ so that edges do not cross one another). We show that if the scaling limit of the random walk is planar Brownian motion, then the scaling limit of its loop-erasure is SLE2. Our main contribution is showing that for such graphs, the discrete Poisson kernel can be approximated by the continuous one. One example is the infinite component of super-critical percolation on ℤ2. Berger and Biskup showed that the scaling limit of the random walk on this graph is planar Brownian motion. Our results imply that the scaling limit of the loop-erased random walk on the super-critical percolation cluster is SLE2.

Effective resistances for harmonic structures on p.c.f. self-similar sets

In mathematics, analysis on fractals was originated by the works of Kusuoka [17] and Goldstein[8]. They constructed the ‘Brownian motion on the Sierpinski gasket’ as a scaling limit of random walks on the pre-gaskets. Since then, analytical structures such as diffusion processes, Laplacians and Dirichlet forms on self-similar sets have been studied from both probabilistic and analytical viewpoints by many authors, see [4], [20], [10], [22] and [7]. As far as finitely ramified fractals, represented by the Sierpinski gasket, are concerned, we now know how to construct analytical structures on them due to the results in [20], [18] and [11]. In particular, for the nested fractals introduced by Lindstrøm [20], one can study detailed features of analytical structures such as the spectral dimensions and various exponents of heat kernels by virtue of the strong symmetry of nested fractals, cf. [6] and [15]. Furthermore in [11], Kigami proposed a notion of post critically finite (p.c.f. for short) self-similar sets, which was a pure topological description of finitely ramified self-similar sets. Also it was shown that we can construct Dirichlet forms and Laplacians on a p.c.f. self-similar set if there exists a difference operator that is invariant under a kind of renormalization. This invariant difference operator was called a harmonic structure. In Section 2, we will give a review of the results in [11].