Workshop "Persistence probabilities and related fields" |
Programme
Abstracts
Sebastian Andres (Bonn)Continuity of the Heat Kernel and Spectral Dimension for Liouville Brownian MotionThe Liouville Brownian motion, recently introduced by Garban, Rhodes and Vargas, is a diffusion process evolving in a planar random geometry induced by the Liouville measure $M_\gamma$, formally written as $M_\gamma(dz)=e^{\gamma X(z)-{\gamma^2} \mathbb{E}[X(z)^2]/2} dz$, $\gamma\in(0,2)$, for a Gaussian free field $X$. It is an $M_\gamma$-symmetric diffusion defined as the time-change of the standard two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure $M_\gamma$. In this talk we present some new results on the heat kernel $p_t(x,y)$. Specifically, we discuss its continuity, an on-diagonal upper bound of the form $p_t(x,x)\leq C t^{-1} \log(1/t)$ for small $t$ and that the spectral dimension equals $2$. This is joint work with Naotaka Kajino. Jerry Buckley (Tel Aviv)Hyperbolic hole probabilitiesThe hyperbolic Gaussian analytic function is a random holomorphic function on the unit disc, whose zero set is invariant (in distribution) under automorphisms of the disc. A hole is the event that there are no zeroes in a given hyperbolic disc. I will discuss the asymptotic decay of the probability of this event, under various regimes. Joint work with A. Nishry, R. Peled and M. Sodin. Amir Dembo (Stanford)Persistence ProbabilitiesPersistence probabilities concern how likely it is that a stochastic process has a long excursion above fixed level and of what are the relevant scenarios for this behavior. Power law decay is expected in many cases of physical significance and the issue is to determine its power exponent parameter. I will survey recent progress in this direction (jointly with Sumit Mukherjee), dealing with stationary Gaussian processes that arise from random algebraic polynomials of independent coefficients and from the solution to heat equation initiated by white noise. If time permits, I will allude to related joint works with Jian Ding and Fuchang Gao, about persistence for iterated partial sums and other auto-regressive sequences, and with Sumit Mukherjee and Hironobu Sakagawa on persistence probabilities for height of certain dynamical random interface models. Steffen Dereich (Münster)Persistence probabilities for integrated random walksWe study the one-sided exit problem, also known as one-sided barrier problem for integrated random walks and L\'evy processes. The probability of having a large excursion in one direction is typically of polynomial order and the leading exponent (the persistence exponent) is quite robust under the specification of the driving process. We will provide a simple coupling argument that allows to show universality of the persistence exponent for integrated walks. Further, we will provide the polynomial decay of the probability that an integrated random walk bridge that is pinned to zero on two sides stays positive for long time. The talk is based on joint work with Frank Aurzada (Darmstadt) and Misha Lifshits (St. Petersburg). Jean-Dominique Deuschel (Berlin)Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interactionWe consider a random field $\Phi:\{1, \ldots,N \}\to\mathbb{R}$ as a model for a linear chain attracted to the defect line $\Phi=0$, that is, the x-axis. The free law of the field is specified by the density $\exp(-\sum iV(\triangle \Phi i)) $ with respect to the Lebesgue measure on $\mathbb{R}^N$, where $ \triangle$ is the discrete Laplacian and we allow for a very large class of potentials $V(\cdot)$. The interaction with the defect line is introduced by giving the field a reward $\varepsilon \geq 0 $ each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity $\varepsilon $ of the pinning reward varies: both in the pinning ($a=p$) and in the wetting ($a=w$) case, there exists a critical value $ \varepsilon_{ca}$ such that when $\varepsilon >\varepsilon_{ca}$ the field touches the defect line a positive fraction of times (localization), while this does not happen for $ \varepsilon < \varepsilon_{ca} $ (delocalization). The two critical values are nontrivial and distinct: $ 0 < \varepsilon_{cp} < \varepsilon_{cw} < \infty $, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at $\varepsilon = \varepsilon_{cp}$ is delocalized. On the other hand, the transition in the wetting model is of first order and for $\varepsilon = \varepsilon_{cw}$ the field is localized. The core of our approach is a Markov renewal theory description of the field. This is joint work with F. Caravenna. Alexis Devulder (Versailles)Persistence of some additive functionals of Sinai's walkWe are interested in Sinai's walk $(S_n)_{n\in N}$. We prove that the annealed probability that $sum_{k=0}^n f(S_k)$ is strictly positive for all $nin[1,N]$ is equal to $1/(log N)^{frac{3-sqrt{5}}{2}+o(1)}$, for a large class of functions $f$, and in particular for $f(x)=x$. The persistence exponent $frac{3-sqrt{5}}{2}$ first appears in a physics paper of Le Doussal, Monthus and Fischer. The proof relies on techniques of localization for Sinai's walk and uses results of Cheliotis about the sign changes of the bottom of valleys of a two-sided Brownian motion. Leif Döring (Zürich)Self-Similarity, Kuznetsov Mesasures and some Statistical MechanicsThe study of self-similar Markov processes has been developped extensively through the last 10 years. In this talk we discuss connections to probabilistic potential theory, in particular Kuznetsov measures and quasi-processes, and give examples how such constructions appear in models from statistical mechanics. Alexander Drewitz (Columbia)Random walk among a Poisson system of moving trapsBrownian motion among immobile Poisson traps as well as their discrete space counterpart has received significant attention during the last decades. In particular, the asymptotics of the annealed and quenched survival probabilities have been investigated, and even a certain understanding of the path measure has been obtained. We consider the discrete setting of simple random walk where the traps now are not immobile anymore but perform independent simple random walks themselves. Our particular interest is in the understanding of the asymptotics of the survival probabilities. Based on joint work with J. G\"artner, A.F. Ram\'irez, and R. Sun. Peter Eichelsbacher (Bochum)Large deviation probabilities and moderate deviations in random matrix theoryWe present some deviation results for certain statistics in random matrix theory. We consider the number of eigenvalues in an interval, in the bulk and close to the edge of the spectrum of a Wigner matrix as well as the $i$th largest eigenvalue. We are able to prove a moderate deviations principle for these statistics, which relies on fine asymptotics of the variance of the eigenvalue counting function of GUE matrices and moreover is based on the Tao and Vu Four Moment Theorem. We prove a moderate deviations principle for the $\log$-determinant of a Wigner matrix and establish Cram\'er-type large deviation probablities and Berry-Esseen bounds for the $\log$-determinant for the GUE and for non-Hermitian Gaussian random matrices. Finally we discuss some connections to mod-$\varphi$-convergence. Naomi Feldheim (Tel Aviv)Persistence of Gaussian stationary processesWe study the probability $p_N$ of a real-valued Gaussian stationary process to be positive on a large interval $[0,N]$. It is natural to expect this quantity to decay exponentially with $N$, as such behavior is often demonstrated when correlations at large distances are low (ex., the persistence probability of i.i.d. discrete random variables, or the probability of a gap in Poisson or Cox point processes). It was predicted by physicists (in works by Bray, Ehrhardt, Majumdar) that also in the Gaussian stationary case, "typically" $p_N$ should have an asymptotic exponential behavior. However, the only rigorous results so far were upper and lower exponential bounds for particular cases, such as a recent result of Antezana, Buckley, Marzo, Olsen for the cardinal sine kernel. In a joint work with Ohad Feldheim, we give upper and lower exponential bounds on $p_N$ for a large family of processes, using spectral analysis. More precisely, we show that such bounds hold if, in some neighborhood of the origin the spectral measure of the process has density which is bounded away from zero and from infinity. Ohad N. Feldheim (Tel Aviv)Local Dependence and Persistence in Discrete Sliding Window ProcessesLet $\{X_i\}_i\in\mathbb{N}$ be a stationary stochastic process on the integers which takes values in a set of size $m$. We consider the probability of the event $X_1=X_2=X_3=\dots=X_t$. This is the discrete counterpart of persistence probability. (Consider, for example, the sign function of some real valued process in discrete time). A natural notion of local dependence for such processes is being a $k$-block factor, which means that there exists a function $f:R^k\to R$ such that $\{X_i\}_i\in\mathbb{N}$ is equal in law to $\{f(Z_{i},Z_{i+1},...,Z_{i+k- 1}\}_i\in\mathbb{N}$ where $Z_i$ are uniform $(0,1)$ i.i.d. random variables. It is very common for local dependence to imply an exponential upper bound on persistence, but can it also imply a lower bound? Somewhat surprisingly, the answer is yes. In a joint work with Noga Alon, we use combinatorial tools to give essentially tight asymptotic lower bounds on the persistence of $k$-block factors which take values in a set of size $m$. We also mention a different, more general notion of local dependence, which is called k-dependence, and advertise a recent work of Holroyd and Liggett where they show that k-dependence does not imply any non-trivial lower bound on persistence. We relate this result to works from the 80's by Aaronson, Gilat, Keane, Linial, Tsirelson, and de Valk, about the difference between those two notions of local dependence. Subro Ghosh (Princeton University)Rigidity phenomena in random point sets and applicationsIn several naturally occurring (infinite) point processes, we show that the number (and other statistical properties) of the points inside a finite domain are determined, almost surely, by the point configuration outside the domain. This curious phenomenon we refer to as "rigidity". We will discuss rigidity phenomena in point processes and their applications. Depending on time, we will talk about applications to stochastic geometry and to random instances of some classical questions in Fourier analysis. Nadine Guillotin-Plantard (Lyon)Persistence exponent for random processes in Brownian sceneryIn this talk we will consider the one-sided exit problem for random processes in random scenery, that is the asymptotic behaviour for large $T$, of the probability $$\PP\Big[ \sup_{t\in[0,T]} \Delta_t \leq 1\Big] $$ where $$\Delta_t = \int_{\mathbb{R}} L_t(x) \, dW(x).$$ Here $W=\acc{W(x); x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and $\acc{L_t(x); x\in\mathbb{R},t\geq 0}$ is the local time of some self-similar random process with stationary increments, independent from the process $W$. This is a joint work with F. Castell and F. Watbled Michael Högele (Potsdam)A solution selection problem with alpha stable perturbationsIn 1981 Bafico and Baldi studied a scalar ordinary differential equation with non-unique solution perturbed additive Brownian motion in the limit of small noise intensity. They showed that the corresponding laws are tight and converge towards a mixture of the two extremal solutions. Inspired by their result we investigate the analogous problem in the case of alpha stable perturbations. This is joint work with F. Flandoli (U Pisa). Alexander Iksanov (Kiev)Moments of first passage times and related quantities for ordinary and perturbed random walksLet $(\xi_1, \eta_1), (\xi_2, \eta_2), \ldots$ be a sequence of i.i.d.\ two-dimensional random vectors with generic copy $(\xi,\eta)$. No condition is imposed on the dependence structure between $\xi$ and $\eta$. Let $(S_n)_{n \in\mathbb{N}_0}$ be the zero-delayed ordinary random walk with increments $\xi_n$ for $n\in\mathbb{N}$, \textit{i.e.}, $S_0 = 0$ and $S_n = \xi_1+\ldots+\xi_n$, $n \in \mathbb{N}$. Then define its perturbed variant $(T_n)_{n\in\mathbb{N}}$, called \emph{perturbed random walk}, by \begin{equation} \label{eq:PRW} T_n := S_{n-1} + \eta_n, \quad n \in \mathbb{N}. \end{equation} For the ordinary random walk $(S_n)$, quantities of traditional interest are (a) the first passage time into the interval $(x,\infty)$; (b) the number of visits to the interval $(-\infty,x]$ and (c) the last exit time from $(-\infty,x]$. I shall discuss ultimate criteria for the finiteness of exponential moments of these quantities and compare them with previously known criteria (due to Kesten and Maller \cite{Kesten+Maller:1996}) for the finiteness of power moments. Further I shall point out the asymptotics of the exponential moments, as $x \to \infty$. Finally, extensions of the finiteness results to the perturbed random walk $(T_n)$ will also be given. Emile Le Page (Univ. Bretagne-Sud)A conditional limit theorem for products of random matricesLet $\mu$ be a probability measure on the group $GL( d , \mathbb{R} )$ of d by d invertible real matrices. We consider a sequence $(g(n))\ n \geq 1$ of i.i.d random matrices of law $\mu$. We suppose that the first Lyapunov exponent of this sequence is zero.\\ We denote by $|\ |$ the Euclidean norm on the vectorial space $V=\mathbb{R}^{d}$ and put $S(n)=g(n)g(n-1)...g(1)\ n \geq 1$. For $r>0$ and v in V such that |v| > r we define $T(v,r)$ by $T(v,r) = \inf \{ n; |S(n) v| \leq r\}$. Under moment conditions for $\mu$ and geometric properties on the support of $\mu$ we give as n tends to infinity the asymptotic of the probability of the event $\{T(v,r)>n\}$ and we prove the limit law of $(\frac{1}{\sqrt{n}})ln|S(n)v|$ conditioned on $\{T(v,r )>n\}$. Mikhail Lifshits (St. Petersburg and Lindköping)Least Energy Functions Accompanying Wiener ProcessWe investigate how well can one approximate Wiener process by smooth functions on long intervals of time. Consider uniform norms \[ ||h||_T:= \sup_{0\le t\le T} |h(t)|, \qquad h\in \CC[0,T], \] and Sobolev-type energy norms \[ |h|^2_T:= \int_0^T h'(t)^2 dt, \qquad h\in AC[0,T]. \] Let $W$ be a Wiener process. We are mostly interested in its approximation characteristic \[ \iwz(T,r):=\inf \big\{|h|_T; h\in AC[0,T], ||h-W||_T\le r, h(0)=0, h(T)=W(T) \big\}. \] The unique function at which the infimum is attained is called {\it taut string} with fixed end. Similar quantity plays important role in the setting of the famous Strassen's functional law of the iterated logarithm. Quite recently, it re-appeared in some applied problems. Our first result is as follows. Theorem. There exists a constant $\C>0$ such that for any fixed $r>0$ we have \[ \frac{r} { T^{1/2}}\ \iwz(T,r) \toalsur \C, \qquad \textrm{as } T\to\infty. \] There is also the mean convergence to $\C$ of any order. We give analytical and numerical bounds for $\C$ and solve an adapted version of our problem. The latter can be described as ''How to lead a Brownian dog on a leash in a sustainable way?'' Furthermore, we evaluate average least energy for fixed $T$ and replace the rigid restriction $||h-W||_T\le r$ by introducing a penalty function. Work supported by grants RFBR 13-01-00172 and SPbSU 6.38.672.2013. This is a joint work with Z. Kabluchko and E. Setterqvist. Satya Majumdar (Paris-Sud XI - Orsay)First-passage properties in diffusion with stochastic resettingWe study a simple model of search where the searcher undergoes normal diffusion, and occasionally resets to its initial starting point stochastically with rate $r$. The effect of a finite resetting rate r turns out to be rather drastic. It leads to finite mean search time which, as a function of r, has a minimum at an optimal resetting rate $r^*$. This makes the search process efficient. We then generalize this model to study multiple searchers. Resetting alters fundamentally the late time decay of the survival probability of a stationary target when there are multiple searchers: while the {\em typical} survival probability decays exponentially with time, the {\em average} decays as a power law with an exponent depending continuously on the density of searchers. We also consider various generalisations of this simple model. M.R. Evans and S.N. Majumdar, ``Diffusion with Stochastic Resetting", Phys. Rev. Lett. 106, 160601 (2011). M.R. Evans and S.N. Majumdar, ``Diffusion with Optimal Resetting", J. Phys. A-Math. \& Theor. 44, 435001 (2011). M.R. Evans and S.N. Majumdar, ``Diffusion with resetting in arbitrary spatial dimension", arXiv: 1404:4574, to appear in J. Phys. A: Math. \& Theo (2014). Alon Nishry (IAS Princeton)Hole probabilities for Random Taylor series.Suppose that f is a random entire function, which is given by a Taylor series with independent complex Gaussian coefficients. A 'hole' is the event where the function f has no zeros in the disk of radius r centered at zero. I will consider the problem of finding logarithmic asymptotics for the rate of decay of the hole probability for large values of r. If time allows, I may discuss a recent large deviations approach to problems of this type (for a special choice of the Gaussian coefficients). Christophe Profeta (Évry)Persistence of integrated stable processesIn this talk, we shall compute the persistence exponent of the integral of a stable L\'evy process in terms of its self-similarity and positivity parameters. The method relies on the study of the law of the stable process $L$ evaluated at the first time its integral $X$ hits zero, when the bivariate process $(X,L)$ starts from a coordinate axis. Such results extend classical formul\ae\,\! by McKean (1963) and Gor'kov (1975) for integrated Brownian motion. Kilian Raschel (Tours)Random walks and Brownian motion with drift in conesIn this talk I will review some recent results on first exit times from cones for Brownian motion and random walks with drift. For Brownian motion, I will explain how to obtain exact and asymptotic expressions for the probability to stay in a given cone up to some prescribed time, by solving the heat equation (results due to DeBlassie, and Bañuelos and Smits). Then we will see how to obtain analogous results for Brownian motion with drift (Garbit and R.). For random walks, we shall first present the results of Denisov and Wachtel on the asymptotic behavior of the probability to stay in a given cone for random walks without drift. Finally, we will come to random walks with drift, and we shall speak on recent asymptotic results of Duraj, and Garbit and R. Mladen Savov (Reading)Brownian motion with limited local timeIn this talk we discuss the problem of studying the limit behaviour of Brownian motion with limited local time. That is given that $L(t)$ is the local time at zero of the standard Brownian motion $B=(B_s)_{s\geq 0}$, we study the possible limits, for a class of deterministic functions $f$, of \[\Pbb{B\in A\big|L(s)\leq f(s),s\leq t}.\] If the limit exists and is denoted by $\Qb$ we discuss the properties of $\Qb$. In this work we are able to deduce a number of conjectures set out in [I. Benjamini and N. Berestycki. An integral test for the transience of a Brownian path with limited local time. Ann. Inst. H. Poincar\'e Probab. Statist. \textbf{47} (2) (2011) 539--558]. Thus when \[I(f)=\int_{1}^{\infty}\frac{f(t)}{t^{\frac{3}{2}}}dt<\infty\] we show that $\Qb$ exists and under it the limiting process is transcient. When $I(f)$ just fails to be finite we show that $\Qb$ exists, the limiting process under $\Qb$ is recurrent and we completely describe the repulsion envelop, namely all the functions $w\downarrow 0$ such that \[\lim\ttinf{t}\Qbb{L(t)\leq f(t)w(t)}=1.\] The results eventually hinge on very precise estimates of the probability of stable subordinators to stay \textbf{above} a curve namely $f^{-1}(s)$. To obtain this asymptotic results we use a simple ODE. It seems the method could be generalized to more general L\'evy processes but we have not pursued this in our work. Joint work with Martin Kolb. Grégory Schehr (Paris-Sud XI - Orsay)Real roots of random polynomials and zero crossing properties of diffusion equationI will present a study of various statistical properties of real roots of three different classes of random polynomials, of degree n: (generalized) Kac polynomials, Weyl and Binomial polynomials. I will show in particular that the gap probability on the real axis, i.e. the probability that these polynomials have no real root in a given interval, is related, when the degree n is large, to the persistence properties of the heat equation with random initial conditions. I will also present results for the probability that such polynomials have exactly k roots in a given interval. Finally I will discuss the fluctuations of the largest real roots, for such polynomials of large degree. Thomas Simon (Lille)Windings of the stable Kolmogorov processWe investigate the windings around the origin of the two-dimensional Markov process $(X,L)$ having the stable L\'evy process $L$ and its primitive process $X$ as coordinates, in the non-trivial case when $\vert L\vert$ is not a subordinator. Extending McKean's result in the Brownian case, we show that these windings have an almost sure limit velocity, which is computed in terms of the self-similarity and the positivity parameters. If time permits, I will also discuss the upper tails of the distribution of the half-winding times. This is joint work with Christophe Profeta. Vlad Vysotsky (Arizona State Univ. and St. Petersburg)Random walks that avoid bounded sets, and applications to the largest gap problemConsider a centred random walk with a finite variance in dimension one. The hitting times for half-lines are fully understood, and we are interested in the tail behavior of the hitting times of a bounded set $B$. Our result is the exact asymptotic, where the order of decay comes from persistence probabilities and the constant is given rather explicitly in terms of the set. As the main application, we prove a limit theorem for the size of the largest gap (maximal spacing) within the range of the first $n$ steps of the random walk. Vitali Wachtel (München)One-sided boundary crossing for random walksIn this talk we shall discuss the tail-behaviour of first passage times over different boundaries for one-dimensional random walks. We shall start with classical results for constant boundaries and describe sufficient conditions on boundary functions, for which the exit time has the same asymptotics as for constant boundaries. Dmitry Zaporozhets (St. Petersburg)Persistence and Spherical Intrinsic VolumesWe will discuss the connection between the Gaussian persistence probabilities and the spherical intrinsic volumes. The multidimensional generalisation will be also considered. Based on the joint work with Zakhar Kabluchko. |