We consider the “searching for a trail in a maze” composite hypothesis testing problem, in which one attempts to detect an anomalous directed path in a lattice 2D box of side $n$ based on observations on the nodes of the box. Under the signal hypothesis, one observes independent Gaussian variables of unit variance at all nodes, with zero mean off the anomalous path and mean $\mu_{n}$ on it. Under the null hypothesis, one observes i.i.d. standard Gaussians on all nodes. Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757] showed that if the unknown directed path under the signal hypothesis has known initial location, then detection is possible (in the minimax sense) if $\mu_{n}\gg1/\sqrt{\log n}$, while it is not possible if $\mu_{n}\ll1/\log n\sqrt{\log\log n}$. In this paper, we show that this result continues to hold even when the initial location of the unknown path is not known. As is the case with Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757], the upper bound here also applies when the path is undirected. The improvement is achieved by replacing the linear detection statistic used in Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757] with a polynomial statistic, which is obtained by employing a multiscale analysis on a quadratic statistic to bootstrap its performance. Our analysis is motivated by ideas developed in the context of the analysis of random polymers in Lacoin [Comm. Math. Phys. 294 (2010) 471–503].

We consider a model of linear market impact, and address the problem of replicating a contingent claim in this framework. We derive a nonlinear Black–Scholes equation that provides an exact replication strategy.

This equation is fully nonlinear and singular, but we show that it is well posed, and we prove existence of smooth solutions for a large class of final payoffs, both for constant and local volatility. To obtain regularity of the solutions, we develop an original method based on Legendre transforms.

The close connections with the problem of hedging with gamma constraints [SIAM J. Control Optim. 39 (2000) 73–96, Math. Finance 17 (2007) 59–80, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 633–666], with the problem of hedging under liquidity costs [Finance Stoch. 14 (2010) 317–341] are discussed. The optimal strategy and associated diffusion are related with the second-order target problems of [Ann. Appl. Probab. 23 (2013) 308–347], and with the solutions of optimal transport problems by diffusions of [Ann. Probab. 41 (2013) 3201–3240].

We also derive a modified Black–Scholes formula valid for asymptotically small impact parameter, and finally provide numerical simulations as an illustration.

In the diffusive hydrodynamic limit for a symmetric interacting particle system (such as the exclusion process, the zero range process, the stochastic Ginzburg–Landau model, the energy exchange model), a possibly nonlinear diffusion equation is derived as the hydrodynamic equation. The bulk diffusion coefficient of the limiting equation is given by the Green–Kubo formula and it can be characterized by a variational formula. In the case the system satisfies the gradient condition, the variational problem is explicitly solved and the diffusion coefficient is given from the Green–Kubo formula through a static average only. In other words, the contribution of the dynamical part of the Green–Kubo formula is $0$. In this paper, we consider the converse, namely if the contribution of the dynamical part of the Green–Kubo formula is $0$, does it imply the system satisfies the gradient condition or not. We show that if the equilibrium measure $\mu$ is product and $L^{2}$ space of its single site marginal is separable, then the converse also holds. The result gives a new physical interpretation of the gradient condition.

As an application of the result, we consider a class of stochastic models for energy transport studied by Gaspard and Gilbert in [J. Stat. Mech. Theory Exp. 2008 (2008) P11021; J. Stat. Mech. Theory Exp. 2009 (2009) P08020], where the exact problem is discussed for this specific model.

The persistent homology of a stationary point process on $\mathbf{R}^{N}$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.

Partial differential equations with random inputs have become popular models to characterize physical systems with uncertainty coming from imprecise measurement and intrinsic randomness. In this paper, we perform asymptotic rare-event analysis for such elliptic PDEs with random inputs. In particular, we consider the asymptotic regime that the noise level converges to zero suggesting that the system uncertainty is low, but does exist. We develop sharp approximations of the probability of a large class of rare events.

We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on $\mathbb{R}^{d}$ by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic topology. It then turns out that the growth rate of the Betti numbers and the properties of the limiting processes all depend on the distance of the region of interest from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes. We also derive the limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology.

We show that a large class of stochastic heat equations can be approximated by systems of interacting stochastic differential equations. As a consequence, we prove various comparison principles extending earlier works of [Stoch. Stoch. Rep. 37 (1991) 225–245] and [Ann. Probab. 45 (2017) 377–403] among others. Among other things, our results enable us to obtain sharp estimates on the moments of the solution. A main technical ingredient of our method is a local limit theorem which is of independent interest.

We study an evolutionary game in which a producer at $x$ gives birth at rate 1 to an offspring sent to a randomly chosen point in $x+\mathcal{N}_{c}$, while a cheater at $x$ gives birth at rate $\lambda>1$ times the fraction of producers in $x+\mathcal{N}_{d}$ and sends its offspring to a randomly chosen point in $x+\mathcal{N}_{c}$. We first study this game on the $d$-dimensional torus $(\mathbb{Z}\bmod L)^{d}$ with $\mathcal{N}_{d}=(\mathbb{Z}\bmod L)^{d}$ and $\mathcal{N}_{c}$ = the $2d$ nearest neighbors. If we let $L\to\infty$ then $t\to\infty$ the fraction of producers converges to $1/\lambda$. In $d\ge3$ the limiting finite dimensional distributions converge as $t\to\infty$ to the voter model equilibrium with density $1/\lambda$. We next reformulate the system as an evolutionary game with “birth-death” updating and take $\mathcal{N}_{c}=\mathcal{N}_{d}=\mathcal{N}$. Using results for voter model perturbations we show that in $d=3$ with $\mathcal{N}=$ the six nearest neighbors, the density of producers converges to $(2/\lambda)-0.5$ for $4/3<\lambda<4$. Producers take over the system when $\lambda<4/3$ and die out when $\lambda>4$. In $d=2$ with $\mathcal{N}=[-c\sqrt{\log N},c\sqrt{\log N}]^{2}$ there are similar phase transitions, with coexistence occurring when $(1+2\theta)/(1+\theta)<\lambda<(1+2\theta)/\theta$ where $\theta=(e^{3/(\pi c^{2})}-1)/2$.

We consider particle systems with mean-field interactions whose distribution is invariant by translations. Under the assumption that the system seen from its centre of mass be reversible with respect to a Gibbs measure, we establish large deviation principles for its empirical measure at equilibrium. Our study covers the cases of McKean–Vlasov particle systems without external potential, and systems of rank-based interacting diffusions. Depending on the strength of the interaction, the large deviation principles are stated in the space of centered probability measures endowed with the Wasserstein topology of appropriate order, or in the orbit space of the action of translations on probability measures. An application to the study of atypical capital distribution is detailed.

We study the asymptotic behaviour of the Random Walk Metropolis algorithm on “ridged” probability densities where most of the probability mass is distributed along some key directions. Such class of probability measures arise in various applied contexts including for instance Bayesian inverse problems where the posterior measure concentrates on a manifold when the noise variance goes to zero. When the target measure concentrates on a linear manifold, we derive analytically a diffusion limit for the Random Walk Metropolis Markov chain as the scale parameter goes to zero. In contrast to the existing works on scaling limits, our limiting stochastic differential equation does not in general have a constant diffusion coefficient. Our results show that in some cases, the usual practice of adapting the step-size to control the acceptance probability might be sub-optimal as the optimal acceptance probability is zero (in the limit).

We consider the Degree-Corrected Stochastic Block Model (DC-SBM): a random graph on $n$ nodes, having i.i.d. weights $(\phi_{u})_{u=1}^{n}$ (possibly heavy-tailed), partitioned into $q\geq2$ asymptotically equal-sized clusters. The model parameters are two constants $a,b>0$ and the finite second moment of the weights $\Phi^{(2)}$. Vertices $u$ and $v$ are connected by an edge with probability $\frac{\phi_{u}\phi_{v}}{n}a$ when they are in the same class and with probability $\frac{\phi_{u}\phi_{v}}{n}b$ otherwise.

We prove that it is information-theoretically impossible to estimate the clusters in a way positively correlated with the true community structure when $(a-b)^{2}\Phi^{(2)}\leq q(a+b)$.

As by-products of our proof we obtain $(1)$ a precise coupling result for local neighbourhoods in DC-SBMs, that we use in Gulikers, Lelarge and Massoulié (2016) to establish a law of large numbers for local-functionals and $(2)$ that long-range interactions are weak in (power-law) DC-SBMs.

We consider stochastic differential systems driven by a Brownian motion and a Poisson point measure where the intensity measure of jumps depends on the solution. This behavior is natural for several physical models (such as Boltzmann equation, piecewise deterministic Markov processes, etc.). First, we give sufficient conditions guaranteeing that the semigroup associated with such an equation preserves regularity by mapping the space of $k$-times differentiable bounded functions into itself. Furthermore, we give an upper estimate of the operator norm. This is the key-ingredient in a quantitative Trotter–Kato-type stability result: it allows us to give an upper estimate of the distance between two semigroups associated with different sets of coefficients in terms of the difference between the corresponding infinitesimal operators. As an application, we present a method allowing to replace “small jumps” by a Brownian motion or by a drift component. The example of the 2D Boltzmann equation is also treated in all detail.

The eigenvalues of the Hatano–Nelson non-Hermitian Anderson matrices, in the spectral regions in which the Lyapunov exponent exceeds the non-Hermiticity parameter, are shown to be real and exponentially close to the Hermitian eigenvalues. This complements previous results, according to which the eigenvalues in the spectral regions in which the non-Hermiticity parameter exceeds the Lyapunov exponent are aligned on curves in the complex plane.

Let $(\boldsymbol{\mathsf{X}},\boldsymbol{\mathsf{Y}})$ denote $n$ independent, identically distributed copies of two arbitrarily correlated Rademacher random variables $(\mathsf{X},\mathsf{Y})$. We prove that the inequality $\mathrm{I}(f(\boldsymbol{\mathsf{X}});g(\boldsymbol{\mathsf{Y}}))\le \mathrm{I}(\mathsf{X};\mathsf{Y})$ holds for any two Boolean functions: $f,g\colon \{-1,1\}^{n}\to \{-1,1\}$ [$\mathrm{I}(\cdot ;\cdot)$ denotes mutual information]. We further show that equality in general is achieved only by the dictator functions $f(\boldsymbol{x})=\pm g(\boldsymbol{x})=\pm x_{i}$, $i\in \{1,2,\ldots,n\}$.

We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf [Probab. Theory Related Fields 164 (2016) 1027–1069] as the minimal solution of an appropriate Cauchy problem under more stringent conditions. A particular limit of these transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well known for not having unique solutions even when one restricts solutions to have linear growth. Using an appropriate diffusion transformation, we show that the aforementioned stochastic solution can be written in terms of the unique classical solution of an alternative Cauchy problem with suitable boundary conditions. This in particular resolves the long-standing issue of non-uniqueness with the Black–Scholes equations in derivative pricing in the presence of bubbles. Finally, we use these path transformations to propose a unified framework for solving explicitly the optimal stopping problem for one-dimensional diffusions with discounting, which in particular is relevant for the pricing and the computation of optimal exercise boundaries of perpetual American options.

We study the long-time behavior of the dynamics of interacting planar Brownian particles, confined by an external field and subject to a singular pair repulsion. The invariant law is an exchangeable Boltzmann–Gibbs measure. For a special inverse temperature, it matches the Coulomb gas known as the complex Ginibre ensemble. The difficulty comes from the interaction which is not convex, in contrast with the case of one-dimensional log-gases associated with the Dyson Brownian motion. Despite the fact that the invariant law is neither product nor log-concave, we show that the system is well-posed for any inverse temperature and that Poincaré inequalities are available. Moreover, the second moment dynamics turns out to be a nice Cox–Ingersoll–Ross process, in which the dependency over the number of particles leads to identify two natural regimes related to the behavior of the noise and the speed of the dynamics.

First-passage percolation is a random growth model defined using i.i.d. edge-weights $(t_{e})$ on the nearest-neighbor edges of $\mathbb{Z}^{d}$. An initial infection occupies the origin and spreads along the edges, taking time $t_{e}$ to cross the edge $e$. In this paper, we study the size of the boundary of the infected (“wet”) region at time $t$, $B(t)$. It is known that $B(t)$ grows linearly, so its boundary $\partial B(t)$ has size between $ct^{d-1}$ and $Ct^{d}$. Under a weak moment condition on the weights, we show that for most times, $\partial B(t)$ has size of order $t^{d-1}$ (smooth). On the other hand, for heavy-tailed distributions, $B(t)$ contains many small holes, and consequently we show that $\partial B(t)$ has size of order $t^{d-1+\alpha }$ for some $\alpha >0$ depending on the distribution. In all cases, we show that the exterior boundary of $B(t)$ [edges touching the unbounded component of the complement of $B(t)$] is smooth for most times. Under the unproven assumption of uniformly positive curvature on the limit shape for $B(t)$, we show the inequality $\#\partial B(t)\leq (\log t)^{C}t^{d-1}$ for all large $t$.

Using the geodesic distance on the $n$-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. We find that the expectations are essentially the same as for the Poisson–Delaunay mosaic in $n$-dimensional Euclidean space. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in $\mathbb{R}^{n+1}$, so we also get the expected number of faces of a random inscribed polytope. As proved in Antonelli et al. [Adv. in Appl. Probab. 9–12 (1977–1980)], an orthant section of the $n$-sphere is isometric to the standard $n$-simplex equipped with the Fisher information metric. It follows that the latter space has similar stochastic properties as the $n$-dimensional Euclidean space. Our results are therefore relevant in information geometry and in population genetics.

In this paper we revisit and generalize various stochastic models extending the deterministic Cucker–Smale model for self-organization. We study flocking and swarming properties. We show how these properties strongly depend on the structure and on the variance of the noise.