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By using the analytic tools of Dirichlet forms, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along a fractal (which can be considered as a simplified rough porous medium). Unlike the regular spaces case, such equation involving a convection term must take a quite different form and the convection term must be singular to the "linear part" which is determined the heat semigroup. In order to study these parabolic type equations, a new kind of Sobolev inequalities for the Dirichlet form on the gasket will be established as an effective tool for our study. These Sobolev inequalities, which are interesting by their own and in contrast to the case of Euclidean spaces, involve two $L^p$ norms with respect two mutually singular measures. By examining properties of a singular convolution of the associated heat semigroup, we derive the space-time regularity of solutions to the parabolic equations under a few technical conditions. The Burgers equations on the Sierpinski gasket are also studied, for which a maximum principle for solutions is derived using techniques from backward stochastic differential equations, and the existence and uniqueness as well as the regularity of solutions are obtained., 30 pages

In this paper, we determine the Poisson boundary of the relativistic Brownian motion in two classes of Lorentzian manifolds, namely model manifolds of constant scalar curvature and Robertson--Walker space-times, the latter constituting a large family of curved manifolds. Our objective is two fold: on the one hand, to understand the interplay between the geometry at infinity of these manifolds and the asymptotics of random sample paths, in particular to compare the stochastic compactification given by the Poisson boundary to classical purely geometric compactifications such as the conformal or causal boundaries. On the other hand, we want to illustrate the power of the d\'evissage method introduced by the authors in [AT16], method which we show to be particularly well suited in the geometric contexts under consideration here., Comment: 32 pages, 8 figures

We prove a central limit theorem for the capacity of the range of a symmetric random walk on $\mathbb{Z}^{5}$, under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular, an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension $d\ge5$, for the probability that the ranges of two independent random walks intersect. The latter are then used for computing covariances of some intersection events at the leading order.

In this paper, we study a $\lambda(\mu)$-ground state of the Schrödinger operator $\mathcal{H}^{\mu}$ based on recurrent relativistic $\alpha$-stable processes. For a signed measure $\mu = \mu^+ - \mu^-$ being in a suitable Kato class, we construct the $\lambda(\mu)$-ground state which is bounded and continuous. Moreover, we prove that the $\lambda(\mu)$-ground state has the mean-value property. In particular, if $\lambda(\mu) = 1$, the mean-value property means the probabilistical harmonicity of $\mathcal{H}^{\mu}$. Finally, we show that if $\alpha > d = 1$, the relativistic $\alpha$-stable process is point recurrent.

We study gradient bounds and other functional inequalities for the diffusion semigroup generated by Kolmogorov type operators. The focus is on two different methods: coupling techniques and generalized $\Gamma$-calculus techniques. The advantages and drawbacks of each of these methods are discussed., Comment: 30 pages

We study internal diffusion-limited aggregation with random starting points on Z^d. In this model, each new particle starts from a vertex chosen uniformly at random on the existing aggregate. We prove that the limiting shape of the aggregate is a Euclidean ball., Comment: Corrected the proof of lemma 3.1 and a few typos

In this paper, we consider a large class of subordinate random walks $X$ on integer lattice $\mathbb{Z}^d$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying a certain lower scaling condition at zero. We establish estimates for one-step transition probabilities, the Green function and the Green function of a ball, and prove the Harnack inequality for non-negative harmonic functions., 31 pages

We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix $\gamma _{Z}$ of $Z := (u(s, y), u(t, x) - u(s, y))$, where $u$ is the solution to a system of $d$ non-linear stochastic heat equations in spatial dimension $k \geq 1$. We also obtain the optimal exponents for the $L^{p}$-modulus of continuity of the increments of the solution and of its Malliavin derivatives. These lead to optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty [ \times \mathbb{R} ^{k}\}$ in the non-Gaussian case in terms of Newtonian capacity, and improve a result in Dalang, Khoshnevisan and Nualart [Stoch PDE: Anal Comp 1 (2013) 94–151].

We consider a recurrent random walk of i.i.d. increments on the one-dimensional integer lattice and obtain a formula relating the hitting distribution of a half-line with the potential function, $a(x)$, of the random walk. Applying it, we derive an asymptotic estimate of $a(x)$ and thereby a criterion for $a(x)$ to be bounded on a half-line. The application is also made to estimate some hitting probabilities as well as to derive asymptotic behaviour for large times of the walk conditioned never to visit the origin.

We consider random walks among random conductances on $\mathbb{Z}^2$ and establish precise asymptotics for the associated potential kernel and the Green's function of the walk killed upon exiting balls. The result is proven for random walks on i.i.d. supercritical percolation clusters among ergodic degenerate conductances satisfying a moment condition. We also provide a similar result for the time-dynamic random conductance model. As an application we present a scaling limit for the variances in the Ginzburg-Landau $\nabla \phi$-interface model., Comment: 17 pages; accepted version

We consider a stochastic flow $\phi_t(x,\omega)$ in $\mathbb{R}^n$ with initial point $\phi_0(x,\omega)=x$, driven by a single $n$-dimensional Brownian motion, and with an outward radial drift of magnitude $\frac{ F(\|\phi_t(x)\|)}{\|\phi_t(x)\|}$, with $F$ nonnegative, bounded and Lipschitz. We consider initial points $x$ lying in a set of positive distance from the origin. We show that there exist constants $C^*,c^*>0$ not depending on $n$, such that if $F>C^*n$ then the image of the initial set under the flow has probability 0 of hitting the origin. If $0\leq F \leq c^*n^{3/4}$, and if the initial set has nonempty interior, then the image of the set has positive probability of hitting the origin., Comment: 34 pages, 3 figures

We show that $f \in L^p (X ; m)$ implies $|f| dm \in S_{K}^{1}$ for $p \gt D$ with $D \gt 0$, where $S_{K}^{1}$ is a subfamily of Kato class measures relative to a semigroup kernel $p_t (x, y)$ of a Markov process associated with a (non-symmetric) Dirichlet form on $L^2 (X ; m)$. We only assume that $p_t (x, y)$ satisfies the Nash type estimate of small time depending on $D$. No concrete expression of $p_t (x,y)$ is needed for the result.

It is well known that there are close connections between non-intersecting processes in one dimension and random matrices, based on the reflection principle. There is a generalisation of the reflection principle for more general (e.g. planar) processes, due to Fomin, in which the non-intersection condition is replaced by a condition involving loop-erased paths. In the context of independent Brownian motions in suitable planar domains, this also has close connections to random matrices. An example of this was first observed by Sato and Katori (Phys Rev E 83:041127, 2011). We present further examples which give rise to various Cauchy-type ensembles. We also extend Fomin’s identity to the affine setting and show that in this case, by considering independent Brownian motions in an annulus, one obtains a novel interpretation of the circular orthogonal ensemble.

On etablit une methode generale pour elargir l’espace d’etats d’un processus de Markov, de telle facon que l’ensemble des points ajoutes est polaire. Cette extension est une amelioration de l’extension triviale, dans quel cas le processus est bloque dans les points ajoutes, et elle produit une technique nouvelle de construction des solutions etendues pour des ED(P) stochastiques, a partir de tous les points de depart, telle qu’elles soient des solutions au moins apres tout moment de temps strictement positif. Concretement, on adopte cette strategie pour etudier des ED stochastiques avec des coefficients singuliers, sur un espace d’etats de dimension infinie (par example des EDP stochastiques de type evolution), pour lesquelles on rencontre des situations ou tous les points de l’espace ne sont pas des conditions initiales autorisees. La meme chose peut se passer dans la construction des solutions pour des problemes de martingales ou pour des processus de Markov a partir de formes de Dirichlet (generalisees), pour lesquelles notre nouvelle technique s’applique aussi.

In this article, we study a class of reflected backward stochastic differential equations (introduced in El Karoui et al. [Ann. Probab. 25 (1997) 702–737], RBSDE for short) with nonlinear resistance by means of Skorohod’s equation. The advantage of this approach lies in its pathwise nature and, therefore, provides additional information about solutions of RBSDE. As an application of our approach, we will consider reflected backward problems with resistance as well. This class of RBSDEs possess significance in the super-hedging with wealth constraint.

We develop stochastic calculus for symmetric Markov processes in terms of time reversal operators. For this, we introduce the notion of the progressively additive functional in the strong sense with time-reversible defining sets. Most additive functionals can be regarded as such functionals. We obtain a refined formula between stochastic integrals by martingale additive functionals and those by Nakao's divergence-like continuous additive functionals of zero energy. As an application, we give a stochastic characterization of harmonic functions on a domain with respect to the infinitesimal generator of semigroup on L2-space obtained by lower-order perturbations.

An $\mathbb{R}^d$-valued Markov process $X^{(x)}_t=(X^{1,x_1}_t,\dots,X^{d,x_d}_t)$, $t\ge0,x\in\mathbb{R}^d$ is said to be multi-self-similar with index $(��_1,\dots,��_d)\in[0,\infty)^d$ if the identity in law \[(c_iX_t^{i,x_i/c_i};i=1,\dots,d)_{t\ge0}\ed(X_{ct}^{(x)})_{t\ge0}\,,\] where $c=\prod_{i=1}^dc_i^{��_i}$, is satisfied for all $c_1,\dots,c_d>0$ and all starting point $x$. Multi-self-similar Markov processes were introduced by Jacobsen and Yor \cite{jy} in the aim of extending the Lamperti transformation of positive self-similar Markov processes to $\mathbb{R}^d_+$-valued processes. This paper aims at giving a complete description of all $\mathbb{R}^d$-valued multi-self-similar Markov processes. We show that their state space is always a union of open orthants with 0 as the only absorbing state and that there is no finite entrance law at 0 for these processes. We give conditions for these processes to satisfy the Feller property. Then we show that a Lamperti-type representation is also valid for $\mathbb{R}^d$-valued multi-self-similar Markov processes. In particular, we obtain a one-to-one relationship between this set of processes and the set of Markov additive processes with values in $\{-1,1\}^d\times\mathbb{R}^d$. We then apply this representation to study the almost sure asymptotic behavior of multi-self-similar Markov processes.

The stiff problem is concerned with a thermal conduction model with a singular barrier of zero volume. In this paper, we shall build the phase transitions for the stiff problems in one-dimensional space. It turns out that every phase transition definitely depends on the total thermal resistance of the barrier, and the three phases correspond to the so-called impermeable pattern, semi-permeable pattern and permeable pattern of thermal conduction respectively. For each pattern, the related boundary condition of the flux at the barrier is also derived. Mathematically, we shall introduce and explore the so-called snapping out Markov process, which is the probabilistic counterpart of semi-permeable pattern in the stiff problem.

We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in Z^d. This endeavor solves a problem of Barlow and Taylor (1991)., Comment: 37 pages, 5 figures

We study the stochastic wave equation with multiplicative noise and singular drift: \[ \partial_tu(t,x)=\Delta u(t,x)+u^{-\alpha}(t,x)+g(u(t,x))\dot{W}(t,x) \] where $x$ lies in the circle $\mathbf{R}/J\mathbf{Z}$ and $u(0,x)>0$. We show that (i) If $03$ then with probability one, $u(t,x)\ne0$ for all $(t,x)$., Comment: 35 pages

Since the seminal work of Lamperti there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than 0 which subsequently was extended to zero initial condition. For real self-similar Markov processes (rssMps) there is a generalization of Lamperti's representation giving a one-to-one correspondence between Markov additive processes and rssMps with initial condition different from the origin. We develop fluctuation theory for Markov additive processes and use Kuznetsov measures to construct the law of transient real self-similar Markov processes issued from the origin. The construction gives a pathwise representation through two-sided Markov additive processes extending the Lamperti-Kiu representation to the origin., Comment: 38 pages

We give a precise behavior of spectral functions for symmet- ric stable processes applying the asymptotic expansion of resolvent kernels. the transition density function of {Xt}t≥0. We write simply p(t;x;y) for p 0 (t;x;y). In (12), we established a necessary and sufficient condition for p (t;x;y) to satisfy c1p(t;x;y) ≤ p (t;x;y) ≤ c2p(t;x;y) for some positive constants c1 and c2; we call this property stability of fundamental solution. To show this, we define the bottom of the spectrum of the operator (1=V )H by

Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

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 $n$. An important ingredient of our approach is the analogous continuous model (related to Aldous' continuum random tree), where the dimension of harmonic measure of a level set of the tree is equal to $\beta$, whereas the dimension of the level set itself is equal to $1$. The constant $\beta$ is expressed in terms of the asymptotic distribution of the conductance of large critical Galton-Watson trees., Comment: Published at http://dx.doi.org/10.1214/15-AOP1050 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Let $Z$ be a subordinate Brownian motion in ${\mathbb R}^d$, $d\ge 2$, via a subordinator with Laplace exponent $��$. We kill the process $Z$ upon exiting a bounded open set $D\subset {\mathbb R}^d$ to obtain the killed process $Z^D$, and then we subordinate the process $Z^D$ by a subordinator with Laplace exponent $��$. The resulting process is denoted by $Y^D$. Both $��$ and $��$ are assumed to satisfy certain weak scaling conditions at infinity. We study the potential theory of $Y^D$, in particular the boundary theory. First, in case that $D$ is a $��$-fat bounded open set, we show that the Harnack inequality holds. If, in addition, $D$ satisfies the local exterior volume condition, then we prove the Carleson estimate. In case $D$ is a smooth open set and the lower weak scaling index of $��$ is strictly larger than $1/2$, we establish the boundary Harnack principle with explicit decay rate near the boundary of $D$. On the other hand, when $��(��)=��^��$ with $��\in (0,1/2]$, we show that the boundary Harnack principle near the boundary of $D$ fails for any bounded $C^{1,1}$ open set $D$. Our results give the first example where the Carleson estimate holds true, but the boundary Harnack principle does not. One of the main ingredients in the proofs is the sharp two-sided estimates of the Green function of $Y^D$. Under an additional condition on $��$, we establish sharp two-sided estimates of the jumping kernel of $Y^D$ which exhibit some unexpected boundary behavior. We also prove a boundary Harnack principle for non-negative functions harmonic in a smooth open set $E$ strictly contained in $D$, showing that the behavior of $Y^D$ in the interior of $D$ is determined by the composition $��\circ ��$., A few typos corrected. Accepted for publication in Potential Analysis (55 pp). arXiv admin note: text overlap with arXiv:1610.00872

Convergence is proved for solutions of Dirichlet problems in regions with many small excluded sets (holes), as the holes become smaller and more numerous. The problem is formulated in the context of Markov processes associated with general Dirichlet forms, for random and nonrandom excluded sets. Sufficient conditions are given under which the sequence of entrance times or hitting times of the excluded sets converges in the stable topology. Convergence in the stable topology is a strengthened form of convergence in distribution, introduced by Renyi. Stable convergence of the entrance times implies convergence of the solutions of the corresponding Dirichlet problems. Some additional results are given in a supplement on random center models., Comment: Fixed some typos which were introduced while "de-macroing"

We prove the existence of limiting laws for symmetric stable-like processes penalized by generalized Feynman-Kac functionals and characterize them by the gauge functions and the ground states of Schrödinger type operators.

In the present paper, we prove that with probability one, the Stratonovich signatures of a multidimensional diffusion process (possibly degenerate) over $[0,1]$, which is the collection of all iterated Stratonovich’s integrals of the diffusion process over $[0,1]$, determine the diffusion sample paths.

In this paper, we use the theory of symmetric Dirichlet forms to derive Feynman–Kac formulae for the forward problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities corresponding to different electrode models on bounded Lipschitz domains. Subsequently, we employ these Feynman–Kac formulae to rigorously justify stochastic homogenization in the case of a stochastic boundary value problem arising from an inverse anomaly detection problem. Motivated by this theoretical result, we prove an estimate for the speed of convergence of the projected mean-square displacement of the underlying process which may serve as the theoretical foundation for the development of new scalable stochastic numerical homogenization schemes.

An old argument of Royden and Tsuji is modified to give a necessary and sufficient condition for a reversible countable state Markov chain to be transient. This Royden criterion is quite convenient and can, on occasion, be used as a substitute for the criterion of Nash-Williams [6]. The result we give here yields a very simple proof that the Nash-Williams criterion implies recurrence. The Royden criterion also yields as a trivial corollary that a recurrent reversible random walk on a state space $X$ remains recurrent when it is constrained to run on a subset $X'$ of $X$. An apparently weaker criterion for transience is also given. As an application, we discuss the transience of a random walk on a horn shaped subset of $\mathbb{Z}^d$.

Large deviation principles of occupation distribution for generalized Feyn-man-Kac functionals are presented in the framework of symmetric Markov processes having doubly Feller or strong Feller property. As a consequence, we obtain the $L^p$-independence of spectral radius of our generalized Feynman-Kac functionals. We also prove Fukushima's decomposition in the strict sense for functions locally in the domain of Dirichlet form having energy measure of Dynkin class without assuming no inside killing.

We study infinite tree and ultrametric matrices, and their action on the boundary of the tree. For each tree matrix we show the existence of a symmetric random walk associated to it and we study its Green potential. We provide a representation theorem for harmonic functions that includes simple expressions for any increasing harmonic function and the Martin kernel. In the boundary, we construct the Markov kernel whose Green function is the extension of the matrix and we simulate it by using a cascade of killing independent exponential random variables and conditionally independent uniform variables. For ultrametric matrices we supply probabilistic conditions to study its potential properties when immersed in its minimal tree matrix extension., Comment: 4 figures

In this paper, we study properties of the dual process and Schrodinger-type operators of a non-symmetric diffusion with measure-valued drift. Let mu=(mu^1,..., mu^d) be such that each mu^i is a signed measure on R^d belonging to the Kato class K_{d, 1}. We show that a killed diffusion process with measure-valued drift in any bounded domain has a dual process with respect to a certain reference measure. For an arbitrary bounded domain, we show that a scale invariant Harnack inequality is true for the dual process. We also show that, if the domain is bounded C^{1,1}, the boundary Harnack principle for the dual process is true and the (minimal) Martin boundary for the dual process can be identified with the Euclidean boundary. It is also shown that the harmonic measure for the dual process is locally comparable to that of the h-conditioned Brownian motion with h being the ground state. Under the gaugeability assumption, if the domain is bounded Lipschitz, the (minimal) Martin boundary for the Schrodinger operator obtained from the diffusion with measure-value drift can be identified with the Euclidean boundary., 32 page

We prove the Harnack inequality and boundary Harnack principle for the absolute value of a one-dimensional recurrent subordinate Brownian motion killed upon hitting 0, when 0 is regular for itself and the Laplace exponent of the subordinator satisfies certain global scaling conditions. Using the conditional gauge theorem for symmetric Hunt processes we prove that the Green function of this process killed outside of some interval $(a, b)$ is comparable to the Green function of the corresponding killed subordinate Brownian motion. We also consider several properties of the compensated resolvent kernel $h$, which is harmonic for our process on $(0, \infty)$.

We prove Fatou’s theorem for nonnegative harmonic functions with respect to killed subordinate Brownian motions with Gaussian components on bounded $C^{1,1}$ open sets $D$. We prove that nonnegative harmonic functions with respect to such processes on $D$ converge nontangentially almost everywhere with respect to the surface measure as well as the harmonic measure restricted to the boundary of the domain. In order to prove this, we first prove that the harmonic measure restricted to $\partial D$ is mutually absolutely continuous with respect to the surface measure. We also show that tangential convergence fails on the unit ball.

In a recent work Levine et al. (2015) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus., Comment: 29 pages. Minor corrections in the proof of the scaling limit for general weights made

Let $u$ be a harmonic function on a complete simply connected manifold $M$ whose sectional curvatures are bounded between two negative constants. It is proved here a pointwise criterion of non-tangential convergence for points of the geometric boundary: the finiteness of the density of energy, which is the geometric analogue of the density of the area integral in the Euclidean half-space.

Let $W$ denote $d$-dimensional Brownian motion. We find an explicit formula for the essential supremum of Hausdorff dimension of $W(E)\cap F$, where $E\subset(0,\infty)$ and $F\subset \mathbf {R}^d$ are arbitrary nonrandom compact sets. Our formula is related intimately to the thermal capacity of Watson [Proc. Lond. Math. Soc. (3) 37 (1978) 342-362]. We prove also that when $d\ge2$, our formula can be described in terms of the Hausdorff dimension of $E\times F$, where $E\times F$ is viewed as a subspace of space time., Published in at http://dx.doi.org/10.1214/14-AOP910 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This paper concerns the number $Z_n$ of sites visited up to time $n$ by a random walk $S_n$ having zero mean and moving on the two dimensional square lattice ${\bf Z}^2$. Asymptotic evaluation of the conditional expectation of $Z_n$ for large $n$ given that $S_n=x$ is carried out under some exponential moment condition. It gives an explicit form of the leading term valid uniformly in $(x, n)$, $|x|< cn$ for each $c>0$.

Small-space and large-time estimates and asymptotic expansion of the distribution function and (the derivatives of) the density function of hitting times of points for symmetric L\'evy processes are studied. The L\'evy measure is assumed to have completely monotone density function, and a scaling-type condition inf z Psi"(z) / Psi'(z) > 0 is imposed on the L\'evy-Khintchine exponent Psi. Proofs are based on generalised eigenfunction expansion for processes killed upon hitting the origin., Comment: 24 pages

We prove the sets of polynomials on configuration spaces are cores of Dirichlet forms describing interacting Brownian motion in infinite dimensions. Typical examples of these stochastic dynamics are Dyson's Brownian motion and Airy interacting Brownian motion. Both particle systems have logarithmic interaction potentials, and naturally arise from random matrix theory. The results of the present paper will be used in a forth coming paper to prove the identity of the infinite-dimensional stochastic dynamics related to the random matrix theories constructed by apparently different methods: the method of space-time correlation functions and that of stochastic analysis., Comment: 6 pages, revised version, published in PJA in 2014

We obtain an asymptotic estimate of the Green function of a random walk on $\boldsymbol{Z}^2$ having zero mean and killed when it exits from the upper half plane. A little more than the second moment condition is assumed. The estimate obtained is used to derive an exact asymptotic form of the hitting distribution of the lower half plane of the walk. The higher dimensional walks are dealt with in the same way.

We study the biased random walk in positive random conductances on $\mathbb {Z}^d$. This walk is transient in the direction of the bias. Our main result is that the random walk is ballistic if, and only if, the conductances have finite mean. Moreover, in the sub-ballistic regime we find the polynomial order of the distance moved by the particle. This extends results obtained by Shen [Ann. Appl. Probab. 12 (2002) 477-510], who proved positivity of the speed in the uniformly elliptic setting., Published in at http://dx.doi.org/10.1214/13-AOP835 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Soit une superdiffusion $X$ sur $\mathbb{R}^{d}$ correspondant a l’operateur semi-lineaire $\mathcal{A}(u)=Lu+\beta u-ku^{2}$, ou $L$ est lui-meme un operateur eliptique du second ordre, $\beta(\cdot)$ est dans la classe de Kato, et $k(\cdot)\ge0$ est borne sur les compacts de $\mathbb{R}^{d}$ et est positif sur un ensemble de mesure de Lebesgue positive. L’objectif principal de cet article est de completer les resultats obtenus dans (Ann. Probab. 32 (2004) 78–99), dans le sens suivant. Soit $\lambda_{\infty}$ la borne $L^{\infty}$ de croissance du semigroupe correspondant a l’operateur $L+\beta$ de type Schrodinger. Si $\lambda_{\infty}\neq0$, nous prouvons alors que – dans un certain sens – le taux exponentiel de croissance/decroissance de la masse totale $\|X_{t}\|$, est $\lambda_{\infty}$. Nous decrivons egalement le comportement limite de $\exp(-\lambda_{\infty}t)\|X_{t}\|$, quand $t\to\infty$, sous cette meme hypothese. Ces resultats sont a comparer avec ceux obtenus dans (Ann. Probab. 32 (2004) 78–99), ou il est demontre que la valeur propre principale generalisee $\lambda_{2}$ de l’operateur donne le taux de croissance locale quand elle est positive et qu’il y a extinction locale quand ce n’est pas le cas. Il est aise de montrer que $\lambda_{\infty}\ge\lambda_{2}$, et nous discutons les cas $\lambda_{\infty}>\lambda_{2}$ et $\lambda_{\infty}=\lambda_{2}$. Quand $\lambda_{\infty}=0$, et sous certaines conditions portant sur $\beta$, nous obtenons une condition necessaire et suffisante pour que la superdiffusion $X$ s’eteigne faiblement. Nous montrons que l’intensite de branchement $k$ affecte l’extinction faible; alors qu’il est connu que $k$ n’affecte pas l’extinction faible locale. (Celle-ci dependant uniquement du signe de $\lambda_{2}$ et est equivalente a l’extinction locale.)

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. Here, we focus on the regularizing effect of a large class of memoryful high-order or time-fractional PDEs—through their fundamental solutions—on stochastic integral equations (SIEs) driven by space–time white noise. Surprisingly, we show that maximum spatial regularity is achieved in the fourth-order-bi-Laplacian case; and any further increase in the spatial-Laplacian order is entirely translated into additional temporal regularization of the SIE. We started this program in [Discrete Contin. Dyn. Syst. Ser. A 33 (2013) 413–463, Stoch. Dyn. 6 (2006) 521–534], where we introduced two different stochastic versions of the fourth order memoryful PDE associated with the Brownian-time Brownian motion (BTBM): (1) the BTBM SIE and (2) the BTBM SPDE, both driven by space–time white noise. Under wide conditions, we showed the existence of random field locally-Hölder solutions to the BTBM SIE with striking and unprecedented time-space Hölder exponents, in spatial dimensions $d=1,2,3$. In particular, we proved that the spatial regularity of such solutions is nearly locally Lipschitz in $d=1,2$. This gave, for the first time, an example of a space–time white noise driven equation whose solutions are smoother than the corresponding Brownian sheet in either time or space. ¶ In this paper, we introduce the $2\beta^{-1}$-order $\beta$-inverse-stable-Lévy-time Brownian motion ($\beta$-ISLTBM) SIEs, $\beta\in \{1/2^{k};k\in\mathbb{N}\}$, driven by space–time white noise. Based on the dramatic regularizing effect of the BTBM density ($\beta=1/2$), and since the kernels in these $\beta$-ISLTBM SIEs are fundamental solutions to higher order Laplacian PDEs; one may suspect that we get even more dramatic spatial regularity than the BTBM SIE case. We show, however, that the BTBM SIE spatial regularity and its random field third spatial dimension limit are maximal among all $\beta$-ISLTBM SIEs—no matter how high we take the order $1/\beta$ of the Laplacian. This gives a limit as to how far we can push the SIEs spatial regularity when driven by the rough white noise. Furthermore, we show that increasing the order of the Laplacian $\beta^{-1}$ beyond the BTBM bi-Laplacian manifests entirely as increased temporal regularity of our random field solutions that asymptotically approaches that of the Brownian sheet as $\beta\searrow0$. Our solutions are both direct and lattice limit solutions. We treat many stochastic fractional PDEs and their corresponding higher order SPDEs, including BTBM and $\beta$-inverse-stable-Lévy-time Brownian motion SPDEs, in separate articles.

We examine diffusion-limited aggregation for a one-dimensional random walk with long jumps. We achieve upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has. In this paper we handle the case of transient walks., Comment: 30 pages, 3 phase transitions

Nous presentons une methode de renormalisation pour construire certains modeles de potentiels aleatoires dans un nuage Poissonnien qui sont physiquement plus realistes. Nous obtenons le mouvement brownien dans un potentiel aleatoire renormalise et les modeles d’Anderson paraboliques associes. Par exemple, avec cette renormalisation, nous pouvons construire rigoureusement des modeles consistants avec la loi de la gravitation de Newton.

In this paper we will give a proof of the McMillan twist point theorem using geometry, potential theory and Ito's formula but not the Riemann mapping theorem.

We consider a subordinate Brownian motion $X$ in $\mathbb{; ; ; R}; ; ; ^d$, $d \ge 1$, where the Laplace exponent $\phi$ of the corresponding subordinator satisfies some mild conditions. The scale invariant Harnack inequality is proved for $X$.We first give new forms of asymptotical properties of the L\' evy and potential density of the subordinator near zero. Using these results we find asymptotics of the L\' evy density and potential density of $X$ near the origin, which is essential to our approach. The examples which are covered by our results include geometric stable processes and relativistic geometric stable processes, i.e. the cases when the subordinator has the Laplace exponent \[ \phi(\lambda)=\log(1+\lambda^{; ; ; \alpha/2}; ; ; ) (00)\, . \]}; ;

This erratum corrects the article arXiv:0806.2044 published in Ann. Probab. 36 (2008) 931--970, Comment: Published in at http://dx.doi.org/10.1214/11-AOP684 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)