Let β_k(n) be the number of self-intersections of order k, appropriately renormalized, for a mean zero random walk X_n in Z^2 with 2+δmoments. On a suitable probability space we can construct X_n and a planar Brownian motion W_t such that for each k\geq 2, |β_k(n)-γ_k(n)|=O(n^{-a}), a.s. for some a>0 where γ_k(n) is the renormalized self-intersection local time of order k at time 1 for the Brownian motion W_{nt}/\sqrt n.
We study $γ_{k}(x_2,...,x_k;t)$, the k-fold renormalized self-intersection local time for Brownian motion in $R^1$. Our main result says that $γ_{k}(x_2,...,x_k;t)$ is continuously differentiable in the spatial variables, with probability 1.
In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the $d$% -dimensional fractional Brownian motion with Hurst parameter $H\in (0,1)$. As a consequence, we derive the existence of some exponential moments for this random variable.
In this article we calculate the third and fourth moment of the renormalized intersection local time of a planar Brownian motion. The third moment is calculated anlaytically, the fourth moment numerically. For the closed planar random walk the third moment of the distribution of the multiple point range is also calculated in leading order.
We study large deviations for the renormalized self-intersection local time of d-dimensional stable processes of index β\in (2d/3,d]. We find a difference between the upper and lower tail. In addition, we find that the behavior of the lower tail depends critically on whether β<d or β=d.
Let B_n be the number of self-intersections of a symmetric random walk with finite second moments in the integer planar lattice. We obtain moderate deviation estimates for B_n - E B_n and E B_n- B_n, which are given in terms of the best constant of a certain Gagliardo-Nirenberg inequality. We also prove the corresponding laws of the iterated logarithm.
We define renormalized intersection local times for random interlacements of Lévy processes in R^{d} and prove an isomorphism theorem relating renormalized intersection local times with associated Wick polynomials.
For a symmetric random walk in $Z^2$ with $2+δ$ moments, we represent $|\mathcal{R}(n)|$, the cardinality of the range, in terms of an expansion involving the renormalized intersection local times of a Brownian motion. We show that for each $k\geq 1$ \[ (\log n)^k [ \frac{1}{n} |\mathcal{R}(n)| +\sum_{j=1}^k (-1)^j (\textstyle{\frac1{2π}}\log n +c_X)^{-j} γ_{j,n}]\to 0, \qquad a.s. \] where $W_t$ is a Brownian motion, $W^{(n)}_t=W_{nt}/\sqrt n$, $γ_{j,n}$ is the renormalized intersection local time at time 1 for $W^{(n)}$, and $c_X$ is a constant depending on the distribution of the random wal…
In this article we study the distribution of the number of points of a simple random walk, visited a given number of times (the k-multiple point range). In a previous article we had developed a graph theoretical approach which is now extended from the closed to the the non restricted walk. Based on a study of the analytic properties of the first hit determinant a general method to define and calculate the generating functions of the moments of the distribution as analytical functions and express them as an absolutely converging series of graph contributions is given. So a method to calculate t…
Let B_t^H be a d-dimensional fractional Brownian motion with Hurst parameter H\in(0,1). Assume d\geq2. We prove that the renormalized self-intersection local time\ell=\int_0^T\int_0^tδ(B_t^H-B_s^H) ds dt -E\biggl(\int_0^T\int_0^tδ(B_t^H-B_s^H) ds dt\biggr) exists in L^2 if and only if H<3/(2d), which generalizes the Varadhan renormalization theorem to any dimension and with any Hurst parameter. Motivated by a result of Yor, we show that in the case 3/4>H\geq\frac{3}{2d}, r(ε)\ell_ε converges in distribution to a normal law N(0,Tσ^2), as εtends to zero, where \ell_ε is an approximation of \ell,…
The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-i…
We study the set of points $\mathcal{D}_{n,m}$ around which two independent Brownian motions wind at least $n$ (resp. $m$) times. We prove that its area is asymptotically equivalent, in $L^p$ and almost surely, to $\frac{\ell(\mathbb{R}^2)}{4π^2 n m}$, where $\ell$ is the intersection measure of the two trajectories. We also prove that the properly scaled Lebesgue measure carried by $\mathcal{D}_{n,m}$ converges almost surely weakly toward $\ell$.
The existence of self-intersection local time (SILT), when the time diagonal is intersected, of the $(α,d,β)$-superprocess is proved for $d/2<α$ and for a renormalized SILT when $d/(2+(1+β)^{-1})<α\leq d/2$. We also establish Tanaka-like formula for SILT.
Let $\{B_{t}\}_{t\geq0}$ be a fractional Brownian motion with Hurst parameter $\frac{2}{3}<H<1$. We prove that the approximation of the derivative of self-intersection local time, defined as \begin{align*} α_{\varepsilon} &= \int_{0}^{T}\int_{0}^{t}p'_{\varepsilon}(B_{t}-B_{s})\text{d}s\text{d}t, \end{align*} where $p_\varepsilon(x)$ is the heat kernel, satisfies a central limit theorem when renormalized by $\varepsilon^{\frac{3}{2}-\frac{1}{H}}$. We prove as well that for $q\geq2$, the $q$-th chaotic component of $α_{\varepsilon}$ converges in $L^{2}$ when $\frac{2}{3}<H<\frac{3}{4}$, and sat…
In this work, we study the optimal discretization error of stochastic integrals, in the context of the hedging error in a multidimensional Itô model when the discrete rebalancing dates are stopping times. We investigate the convergence, in an almost sure sense, of the renormalized quadratic variation of the hedging error, for which we exhibit an asymptotic lower bound for a large class of stopping time strategies. Moreover, we make explicit a strategy which asymptotically attains this lower bound a.s. Remarkably, the results hold under great generality on the payoff and the model. Our analysis…
Let $B_s$ be a $d$-dimensional Brownian motion and $ω(dx)$ be an independent Poisson field on $\mathbb{R}^d$. The almost sure asymptotics for the logarithmic moment generating function [\log\math bb{E}_0\exp\biggl{\pmθ\int_0^t\bar{V}(B_s) ds\biggr}\qquad (t\to\infty)] are investigated in connection with the renormalized Poisson potential of the form [\bar{V}(x)=\int_{\mathbb{R}^d}{\frac{1}{|y-x|^p}}[ω(dy)-dy],\qquad x\in\mathbb{R}^d.] The investigation is motivated by some practical problems arising from the models of Brownian motion in random media and from the parabolic Anderson models.
In this article we establish some estimates related to the Gaussian densities and to Hermite polynomials in order to obtain an almost sure estimate for each term of the Itô-Wiener expansion of the self-intersection local times of the Brownian motion. In dimension $d\geqslant 4$ the self-intersection local times of the Brownian motion can be considered as a family of measures on the classical Wiener space. We provide some asymptotics relative to these measures. Finally, we try to estimate the quadratic Wasserstein distance between these measures and the Wiener measure.
If β_t is renormalized self-intersection local time for planar Brownian motion, we characterize when Ee^{γβ_1} is finite or infinite in terms of the best constant of a Gagliardo-Nirenberg inequality. We prove large deviation estimates for β_1 and -β_1. We establish lim sup and lim inf laws of the iterated logarithm for β_t as t\to\infty.
Let $G$ be a Cayley graph of a nonamenable group with spectral radius $ρ< 1$. It is known that branching random walk on $G$ with offspring distribution $μ$ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring $\bar μ$ satisfies $\bar μ\leq ρ^{-1}$. Benjamini and Müller (2010) conjectured that throughout the transient supercritical phase $1<\barμ \leq ρ^{-1}$, and in particular at the recurrence threshold $\bar μ= ρ^{-1}$, the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost sur…
We consider chordal SLE(kappa) curves for kappa > 4, where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension min {2-8/kappa,1}. We study the random sets of points at which the curve collides with the real line at a specified "angle" and compute an almost sure dimension spectrum describing the metric size of these sets. We work with the forward SLE flow and a key tool in the analysis is Girsanov's theorem, which is used to study events on which moments concentrate. The two-point correlation estimates are proved using the direct method.
We define and study a family of generalized non-intersection exponents for planar Brownian motions that is indexed by subsets of the complex plane: For each $A\subset\CC$, we define an exponent $ξ(A)$ that describes the decay of certain non-intersection probabilities. To each of these exponents, we associate a conformally invariant subset of the planar Brownian path, of Hausdorff dimension $2-ξ(A)$. A consequence of this and continuity of $ξ(A)$ as a function of $A$ is the almost sure existence of pivoting points of any sufficiently small angle on a planar Brownian path.
Let $L_n^{X}(x)$ denote the number of visits to $x \in {\bf Z}^2$ of the simple planar random walk $X$, up till step $n$. Let $X'$ be another simple planar random walk independent of $X$. We show that for any $0<b<1/(2 π)$, there are $n^{1-2πb+o(1)}$ points $x \in {\bf Z}^2$ for which $L_n^{X}(x)L_n^{X'}(x)\geq b^2 (\log n)^4$. This is the discrete counterpart of our main result, that for any $a<1$, the Hausdorff dimension of the set of {\it thick intersection points} $x$ for which $\limsup_{r \to 0} {\mathcal I}(x,r)/(r^2|\log r|^4)=a^2$, is almost surely $2-2a$. Here ${\mathcal I}(x,r)$ is t…
We study the three-dimensional cubic nonlinear wave equation (NLW) with random initial data below $L^2(\mathbb{T}^3)$. By considering the second order expansion in terms of the random linear solution, we prove almost sure local well-posedness of the renormalized NLW in negative Sobolev spaces. We also prove a new instability result for the defocusing cubic NLW without renormalization in negative Sobolev spaces, which is in the spirit of the so-called triviality in the study of stochastic partial differential equations. More precisely, by studying (un-renormalized) NLW with given smooth determi…
In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst index 1/2<H<1, which is analogous to the linear self-attracting diffusion. For 1-dimensional process we study its convergence and the corresponding weighted local time. For 2-dimensional process, as a related problem, we show that the renormalized self-intersection local time exists in L^2 if $\frac12<H<\frac3{4}$.
We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.
We prove that every random walk in i.i.d. environment in dimension greater than or equal to 2 that has an almost sure positive speed in a certain direction, an annealed invariance principle and some mild integrability condition for regeneration times also satisfies a quenched invariance principle. The argument is based on intersection estimates and a theorem of Bolthausen and Sznitman.
We show that the intersection exponents for planar Brownian motions are analytic. More precisely, let $B$ and $B'$ be independent planar Brownian motions started from distinct points, and define the exponent $ξ(1, λ)$ by $$ E[P[B[0,t] \cap B'[0,t] = \emptyset | B[0,t]]^λ] \approx t^{-ξ(1, λ)/2}, t \to \infty. $$ Then the mapping $λ\mapsto ξ(1, λ)$ is real analytic in $(0,\infty)$. The same result is proved for the exponents $ξ(k, λ)$ where $k$ is a positive integer. In combination with the determination of $ξ(k, λ)$ for integer $k \ge 1$ and real $λ\ge 1$ in our previous papers, this gives the…
We investigate absorption, i.e., almost sure convergence to an absorbing state, in time-varying (non-homogeneous) discrete-time Markov chains with finite state space. We consider systems that can switch among a finite set of transition matrices, which we call the modes. Our analysis is focused on two properties: 1) almost sure convergence to an absorbing state under any switching, and 2) almost sure convergence to a desired set of absorbing states via a proper switching policy. We derive necessary and sufficient conditions based on the structures of the transition graphs of modes. More specifi…
We derive the exact value of intersection exponents between planar Brownian motions or random walks, confirming predictions from theoretical physics by Duplantier and Kwon. Let B and B' be independent Brownian motions (or simple random walks) in the plane, started from distinct points. We prove that the probability that the paths B[0,t] and B'[0,t] do not intersect decays like t^{-5/8}. More precisely, there is a constant c>0 such that if |B(0) - B'(0)| =1, for all t \ge 1, c^{-1} t^{-5/8} \le ¶[ B[0,t] \cap B'[0,t] = \emptyset ] \le c t^{-5/8}. One consequence is that the set of cut-points of…
We show that with probability 1, the trace B[0,1] of Brownian motion in space, has positive capacity with respect to exactly the same kernels as the unit square. More precisely, the energy of occupation measure on B[0,1] in the kernel f(|x-y|), is bounded above and below by constant multiples of the energy of Lebesgue measure on the unit square. (The constants are random, but do not depend on the kernel.) As an application, we give almost-sure asymptotics for the probability that an alpha-stable process approaches within epsilon of B[0,1], conditional on B[0,1]. The upper bound on energy is ba…
The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $γ$-Liouville quantum gravity (LQG) surface, $γ\in (0,2)$, decorated with a space-filling form of Schramm's SLE$_κ$, $κ= 16/γ^2 \in (4,\infty)$, $η$ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $η$ which can be defined as a function of $η$ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $η^{-1}(…
Let $(Z_t)_{t\geq 0}$ denote the derivative martingale of branching Brownian motion, i.e.\@ the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke [\textit{Ann. Probab.}, 15(3):1052--1061, 1987] says that this martingale converges almost surely to a limit $Z_\infty$, positive on the event of survival. In this paper, our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture b…
