Mostrando recursos 1 - 20 de 4.099

  1. Central limit theorems for supercritical branching nonsymmetric Markov processes

    Ren, Yan-Xia; Song, Renming; Zhang, Rui
    In this paper, we establish a spatial central limit theorem for a large class of supercritical branching, not necessarily symmetric, Markov processes with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem generalizes and unifies all the central limit theorems obtained recently in Ren, Song and Zhang [J. Funct. Anal. 266 (2014) 1716–1756] for supercritical branching symmetric Markov processes. To prove our central limit theorem, we have to carefully develop the spectral theory of nonsymmetric strongly continuous semigroups, which should be of independent interest.

  2. Inequalities for Hilbert operator and its extensions: The probabilistic approach

    Osȩkowski, Adam
    We present a probabilistic study of the Hilbert operator ¶ \[Tf(x)=\frac{1}{\pi}\int_{0}^{\infty}\frac{f(y)\,\mathrm{d}y}{x+y},\qquad x\geq0,\] defined on integrable functions $f$ on the positive halfline. Using appropriate novel estimates for orthogonal martingales satisfying the differential subordination, we establish sharp moment, weak-type and $\Phi$-inequalities for $T$. We also show similar estimates for higher dimensional analogues of the Hilbert operator, and by the further careful modification of martingale methods, we obtain related sharp localized inequalities for Hilbert and Riesz transforms.

  3. The determinant of the iterated Malliavin matrix and the density of a pair of multiple integrals

    Nualart, David; Tudor, Ciprian A.
    The aim of this paper is to show an estimate for the determinant of the covariance of a two-dimensional vector of multiple stochastic integrals of the same order in terms of a linear combination of the expectation of the determinant of its iterated Malliavin matrices. As an application, we show that the vector is not absolutely continuous if and only if its components are proportional.

  4. From random lines to metric spaces

    Kendall, Wilfrid S.
    Consider an improper Poisson line process, marked by positive speeds so as to satisfy a scale-invariance property (actually, scale-equivariance). The line process can be characterized by its intensity measure, which belongs to a one-parameter family if scale and Euclidean invariance are required. This paper investigates a proposal by Aldous, namely that the line process could be used to produce a scale-invariant random spatial network (SIRSN) by means of connecting up points using paths which follow segments from the line process at the stipulated speeds. It is shown that this does indeed produce a scale-invariant network, under suitable conditions on the...

  5. On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries

    Kang, Weining; Ramanan, Kavita
    Two frameworks that have been used to characterize reflected diffusions include stochastic differential equations with reflection (SDER) and the so-called submartingale problem. We consider a general formulation of the submartingale problem for (obliquely) reflected diffusions in domains with piecewise $\mathcal{C}^{2}$ boundaries and piecewise continuous reflection vector fields. Under suitable assumptions, we show that well-posedness of the submartingale problem is equivalent to existence and uniqueness in law of weak solutions to the corresponding SDER. The main step involves showing existence of a weak solution to the SDER given a solution to the submartingale problem. This generalizes the classical construction, due to...

  6. Strong invariance and noise-comparison principles for some parabolic stochastic PDEs

    Joseph, Mathew; Khoshnevisan, Davar; Mueller, Carl
    We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities.

  7. A criterion for convergence to super-Brownian motion on path space

    van der Hofstad, Remco; Holmes, Mark; Perkins, Edwin A.
    We give a sufficient condition for tightness for convergence of rescaled critical spatial structures to the canonical measure of super-Brownian motion. This condition is formulated in terms of the $r$-point functions for $r=2,\ldots,5$. The $r$-point functions describe the expected number of particles at given times and spatial locations, and have been investigated in the literature for many high-dimensional statistical physics models, such as oriented percolation and the contact process above 4 dimensions and lattice trees above 8 dimensions. In these settings, convergence of the finite-dimensional distributions is known through an analysis of the $r$-point functions, but the lack of tightness...

  8. Doob–Martin boundary of Rémy’s tree growth chain

    Evans, Steven N.; Grübel, Rudolf; Wakolbinger, Anton
    Rémy’s algorithm is a Markov chain that iteratively generates a sequence of random trees in such a way that the $n$th tree is uniformly distributed over the set of rooted, planar, binary trees with $2n+1$ vertices. We obtain a concrete characterization of the Doob–Martin boundary of this transient Markov chain and thereby delineate all the ways in which, loosely speaking, this process can be conditioned to “go to infinity” at large times. A (deterministic) sequence of finite rooted, planar, binary trees converges to a point in the boundary if for each $m$ the random rooted, planar, binary tree spanned by...

  9. Permanental vectors with nonsymmetric kernels

    Eisenbaum, Nathalie
    A permanental vector with a symmetric kernel and index $2$ is a squared Gaussian vector. The definition of permanental vectors is a natural extension of the definition of squared Gaussian vectors to nonsymmetric kernels and to positive indexes. The only known permanental vectors either have a positive definite kernel or are infinitely divisible. Are there some others? We present a partial answer to this question.

  10. The harmonic measure of balls in random trees

    Curien, Nicolas; Le Gall, Jean-François
    We study properties of the harmonic measure of balls in typical large discrete trees. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is of order $n$, most of the harmonic measure is supported on a boundary set of size approximately equal to $n^{\beta}$, where $\beta\approx0.78$ is a universal constant. To derive such results, we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree, and we first deal with the case of critical Galton–Watson trees conditioned to have height greater than $n$....

  11. Systems of interacting diffusions with partial annihilation through membranes

    Chen, Zhen-Qing; Fan, Wai-Tong (Louis)
    We introduce an interacting particle system in which two families of reflected diffusions interact in a singular manner near a deterministic interface $I$. This system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. A related interacting random walk model with discrete state spaces has recently been introduced and studied in Chen and Fan (2014). In this paper, we establish the functional law of large numbers for this new system, thereby extending the hydrodynamic limit in Chen and Fan (2014) to reflected diffusions...

  12. Dissipation and high disorder

    Chen, Le; Cranston, Michael; Khoshnevisan, Davar; Kim, Kunwoo
    Given a field $\{B(x)\}_{x\in\mathbf{Z}^{d}}$ of independent standard Brownian motions, indexed by $\mathbf{Z}^{d}$, the generator of a suitable Markov process on $\mathbf{Z}^{d},\mathcal{G}$, and sufficiently nice function $\sigma:[0,\infty)\mapsto [0,\infty)$, we consider the influence of the parameter $\lambda$ on the behavior of the system, \begin{eqnarray*}\mathrm{d}u_{t}(x)&=&(\mathcal{G}u_{t})(x)\,\mathrm{d}t+\lambda\sigma(u_{t}(x))\,\mathrm{d}B_{t}(x)\qquad [t>0,\ x\in\mathbf{Z}^{d}],\\u_{0}(x)&=&c_{0}\delta_{0}(x).\end{eqnarray*} We show that for any $\lambda>0$ in dimensions one and two the total mass $\sum_{x\in\mathbf{Z}^{d}}u_{t}(x)$ converges to zero as $t\to\infty$ while for dimensions greater than two there is a phase transition point $\lambda_{c}\in(0,\infty)$ such that for $\lambda>\lambda_{c},\sum_{x\in \mathbf{Z}^{d}}u_{t}(x)\to0$ as $t\to\infty$ while for $\lambda<\lambda_{c},\sum_{x\in \mathbf{Z}^{d}}u_{t}(x)\not\to0$ as $t\to\infty$.

  13. Stochastic analysis on sub-Riemannian manifolds with transverse symmetries

    Baudoin, Fabrice
    We prove a geometrically meaningful stochastic representation of the derivative of the heat semigroup on sub-Riemannian manifolds with tranverse symmetries. This representation is obtained from the study of Bochner–Weitzenböck type formulas for sub-Laplacians on 1-forms. As a consequence, we prove new hypoelliptic heat semigroup gradient bounds under natural global geometric conditions. The results are new even in the case of the Heisenberg group which is the simplest example of a sub-Riemannian manifold with transverse symmetries.

  14. Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree

    Barlow, M. T.; Croydon, D. A.; Kumagai, T.
    The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of the intrinsic metrics, measures and embeddings of the subsequential limits in this space are obtained, with it being proved in particular that the Hausdorff dimension of any limit in its intrinsic metric is almost surely equal to $8/5$. In addition, the tightness result is applied to deduce that the annealed law of the simple random walk on the two-dimensional uniform spanning tree...

  15. Introduction to memorial issue for Donald Burkholder (1927–2013)


  16. A Gaussian upper bound for martingale small-ball probabilities

    Lee, James R.; Peres, Yuval; Smart, Charles K.
    Consider a discrete-time martingale $\{X_{t}\}$ taking values in a Hilbert space $\mathcal{H}$. We show that if for some $L\geq1$, the bounds $\mathbb{E}[\|X_{t+1}-X_{t}\|_{\mathcal{H}}^{2}\vert X_{t}]=1$ and $\|X_{t+1}-X_{t}\|_{\mathcal{H}}\leq L$ are satisfied for all times $t\geq0$, then there is a constant $c=c(L)$ such that for $1\leq R\leq\sqrt{t}$, ¶ \[\mathbb{P}(\|X_{t}-X_{0}\|_{\mathcal{H}}\leq R)\leq c\frac{R}{\sqrt{t}}.\] Following Lee and Peres [Ann. Probab. 41 (2013) 3392–3419], this estimate has applications to small-ball estimates for random walks on vertex-transitive graphs: We show that for every infinite, connected, vertex-transitive graph $G$ with bounded degree, there is a constant $C_{G}>0$ such that if $\{Z_{t}\}$ is the simple random walk on $G$, then for every $\varepsilon>0$...

  17. Random walks and isoperimetric profiles under moment conditions

    Saloff-Coste, Laurent; Zheng, Tianyi
    Let $G$ be a finitely generated group equipped with a finite symmetric generating set and the associated word length function $\vert \cdot\vert $. We study the behavior of the probability of return for random walks driven by symmetric measures $\mu$ that are such that $\sum\rho(\vert x\vert )\mu(x)<\infty$ for increasing regularly varying or slowly varying functions $\rho$, for instance, $s\mapsto(1+s)^{\alpha}$, $\alpha\in(0,2]$, or $s\mapsto(1+\log(1+s))^{\varepsilon}$, $\varepsilon>0$. For this purpose, we develop new relations between the isoperimetric profiles associated with different symmetric probability measures. These techniques allow us to obtain a sharp $L^{2}$-version of Erschler’s inequality concerning the Følner functions of wreath products. Examples...

  18. Transience and recurrence of a Brownian path with limited local time

    Kolb, Martin; Savov, Mladen
    In this paper, we study the behavior of Brownian motion conditioned on the event that its local time at zero stays below a given increasing function $f$ up to time $T$. For a class of nonincreasing functions $f$, we show that the conditioned process converges, as $T\rightarrow\infty$, to a limit process and we derive necessary and sufficient conditions for the limit to be transient. In the transient case, the limit process is described explicitly, and in the recurrent case we quantify the entropic repulsion phenomenon by describing the repulsion envelope, stating how much slower than $f$ the local time of...

  19. Characteristic functions of measures on geometric rough paths

    Chevyrev, Ilya; Lyons, Terry
    We define a characteristic function for probability measures on the signatures of geometric rough paths. We determine sufficient conditions under which a random variable is uniquely determined by its expected signature, thus partially solving the analogue of the moment problem. We furthermore study analyticity properties of the characteristic function and prove a method of moments for weak convergence of random variables. We apply our results to signature arising from Lévy, Gaussian and Markovian rough paths.

  20. High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder

    Dey, Partha S.; Zygouras, Nikos
    The directed polymer model at intermediate disorder regime was introduced by Alberts–Khanin–Quastel [Ann. Probab. 42 (2014) 1212–1256]. It was proved that at inverse temperature $\beta n^{-\gamma}$ with $\gamma=1/4$ the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case...

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