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In this article we study the scaling limit of the interface model on Z d where the Hamiltonian is given by a mixed gradient and Laplacian interaction. We show that in any dimension the scaling limit is given by the Gaussian free field. We discuss the appropriate spaces in which the convergence takes place. While in infinite volume the proof is based on Fourier analytic methods, in finite volume we rely on some discrete PDE techniques involving finite-difference approximation of elliptic boundary value problems. [ABSTRACT FROM AUTHOR]

We study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided. [ABSTRACT FROM AUTHOR]

Given two distinct subsets A, B in the state space of some dynamical system, transition path theory (TPT) was successfully used to describe the statistical behavior of transitions from A to B in the ergodic limit of the stationary system. We derive generalizations of TPT that remove the requirements of stationarity and of the ergodic limit and provide this powerful tool for the analysis of other dynamical scenarios: periodically forced dynamics and time-dependent finite-time systems. This is partially motivated by studying applications such as climate, ocean, and social dynamics. On simple model examples, we show how the new tools are able to deliver quantitative understanding about the statistical behavior of such systems. We also point out explicit cases where the more general dynamical regimes show different behaviors to their stationary counterparts, linking these tools directly to bifurcations in non-deterministic systems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability of functionals and trajectories of two different, independent, time-inhomogeneous, discrete-time Markov chains on a general state space. We obtain various stability estimates such as an estimate for a difference in expectations of functionals, L 2 {L_{2}} stability, and a probability of large deviations. The key condition that is used is the minorization condition on the whole space. We consider different limitations on the functional and on the proximity of two chains. We use the coupling method as a primary technique in our proofs. [ABSTRACT FROM AUTHOR]

In this paper we investigate scaling limits of the odometer in divisible sandpiles on d-dimensional tori following up the works of Chiarini et al. (Odometer of long-range sandpiles in the torus: mean behaviour and scaling limits, 2018), Cipriani et al. (Probab Theory Relat Fields 172:829–868, 2017; Stoch Process Appl 128(9):3054–3081, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalized Gaussian fields in the limit by specifying the Fourier multiplier of their covariance kernel. In particular, using a Fourier multiplier approach, we can recover fractional Gaussian fields of the form (- Δ) - s / 2 W for s > 2 and W a spatial white noise on the d-dimensional unit torus. [ABSTRACT FROM AUTHOR]

We study the h-transform of Doob for nonlocal branching processes, we show that the branching property is preserved provided that h is a coherent state, and we emphasize the probabilistic representation of the solution to the associated nonlinear evolution equation. The tools are from the analytic and probabilistic potential theory. We also investigate the h-transform of a subordinate C 0 -semigroup of sub-Markovian operators on an L p space. [ABSTRACT FROM AUTHOR]

Motivated by the study of the metastable stochastic Ising model at subcritical temperature and in the limit of a vanishing magnetic field, we extend the notion of (κ , λ) -capacities between sets, as well as the associated notion of soft-measures, to the case of overlapping sets. We recover their essential properties, sometimes in a stronger form or in a simpler way, relying on weaker hypotheses. These properties allow to write the main quantities associated with reversible metastable dynamics, e.g. asymptotic transition and relaxation times, in terms of objects that are associated with two-sided variational principles. We also clarify the connection with the classical "pathwise approach" by referring to temporal means on the appropriate time scale. [ABSTRACT FROM AUTHOR]

We study harmonic functions with respect to the Riemannian metric d s 2 = d x 1 2 + ⋯ + d x n 2 x n 2 α n - 2 in the upper half space R + n = { x 1 , … , x n ∈ R n : x n > 0 } . They are called α -hyperbolic harmonic. An important result is that a function f is α -hyperbolic harmonic íf and only if the function g x = x n - 2 - n + α 2 f x is the eigenfunction of the hyperbolic Laplace operator △ h = x n 2 ▵ - n - 2 x n ∂ ∂ x n corresponding to the eigenvalue 1 4 α + 1 2 - n - 1 2 = 0 . This means that in case α = n - 2 , the n - 2 -hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of α -hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion. [ABSTRACT FROM AUTHOR]

Let X be a Hunt process on a locally compact space X such that the set E X of its Borel measurable excessive functions separates points, every function in E X is the supremum of its continuous minorants in E X , and there are strictly positive continuous functions v , w ∈ E X such that v / w vanishes at infinity. A numerical function u ≥ 0 on X is said to be nearly hyperharmonic, if ∫ ∗ u ∘ X τ V d P x ≤ u (x) for every x ∈ X and every relatively compact open neighborhood V of x, where τ V denotes the exit time of V. For every such function u, its lower semicontinuous regularization u ^ is excessive. The main purpose of the paper is to give a short, complete and understandable proof for the statement that u = inf { w ∈ E X : w ≥ u } for every Borel measurable nearly hyperharmonic function on X. Principal novelties of our approach are the following: A quick reduction to the special case, where starting at x ∈ X with u (x) < ∞ the expected number of times the process X visits the set of points y ∈ X , where u ^ (y) : = lim inf z → y u (z) < u (y) , is finite. The consequent use of (only) the strong Markov property. The proof of the equality ∫ u d μ = inf { ∫ w d μ : w ∈ E X , w ≥ u } not only for measures μ satisfying ∫ w d μ < ∞ for some excessive majorant w of u, but also for all finite measures. At the end, the measurability assumption on u is weakened considerably. [ABSTRACT FROM AUTHOR]

$\mathbb{Z}^{d}$ -extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green–Kubo's formula is invariant under induction. This allows us to relate the hitting probability of sites with the symmetrized potential kernel, giving an alternative proof and generalizing a theorem of Spitzer. Finally, this relation is used to improve, in turn, the assumptions of the generalized central limit theorem. Applications to Lorentz gases in finite horizon and to the geodesic flow on Abelian covers of compact manifolds of negative curvature are discussed. [ABSTRACT FROM AUTHOR]

Firstly, we shall introduce the so-called snapping out Walsh's Brownian motion and present its relation with Walsh's Brownian motion. Then the stiff problem related to Walsh's Brownian motion will be described and we shall build a phase transition for it. The snapping out Walsh's Brownian motion corresponds to the so-called semi-permeable pattern of this stiff problem. [ABSTRACT FROM AUTHOR]

We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry–Émery curvature type condition, this BV class is identified with the heat semigroup based Besov class B 1 , 1 / 2 (X) that was introduced in our previous paper. Assuming furthermore a quasi Bakry–Émery curvature type condition, we identify the Sobolev class W 1 , p (X) with B p , 1 / 2 (X) for p > 1 . Consequences of those identifications in terms of isoperimetric and Sobolev inequalities with sharp exponents are given. [ABSTRACT FROM AUTHOR]

We consider a Glauber dynamics associated with the Ising model on a large two-dimensional box with minus boundary conditions and in the limit of a vanishing positive external magnetic field. The volume of this box increases quadratically in the inverse of the magnetic field. We show that at subcritical temperature and for a large class of starting measures, including measures that are supported by configurations with macroscopic plus-spin droplets, the system rapidly relaxes to some metastable equilibrium—with typical configurations made of microscopic plus-phase droplets in a sea of minus spins—before making a transition at an asymptotically exponential random time towards equilibrium—with typical configurations made of microscopic minus-phase droplets in a sea of plus spins inside a large contour that separates this plus phase from the boundary. We get this result by bounding from above the local relaxation times towards metastable and stable equilibria. This makes possible to give a pathwise description of such a transition, to control the asymptotic behaviour of the mixing time in terms of soft capacities and to give estimates of these capacities. [ABSTRACT FROM AUTHOR]

For spectrally negative Lévy processes, adapting an approach from Li and Palmowski (Stoch. Process. Appl. 128(10), 3273–3299 2018) we identify joint Laplace transforms involving local times evaluated at either the first passage times, or independent exponential times, or inverse local times. The Laplace transforms are expressed in terms of the associated scale functions. Connections are made with the permanental process and the Markovian loop soup measure. [ABSTRACT FROM AUTHOR]

A new physico-mathematical approach is developed for modeling oil saturation distribution within a plane polluted aquifer subjected to a flow field governed by a doublet system. This system consists of extraction and injection (E/I) wells, located in a contaminated aquifer, which are used for groundwater remediation applications. The development of the model is done by considering a two-dimensional, nonlinear diffusion-convection transport equation, applying a self-similar scheme. A closed-form analytical solution is obtained for the steady oil saturation distribution in the aquifer for the above-mentioned flow field. The mathematical solution developed here is designed to simulate hydraulic interactions of E/I wells with the oil polluted aquifer. As such, the model was calibrated and its applicability has been shown. [ABSTRACT FROM AUTHOR]

A new approach to the problem of finding the distribution of integral functionals under the excursion measure is presented. It is based on the technique of excursion straddling a time, stochastic analysis, and calculus on local time, and it is done for Brownian motion with drift reflecting at 0, and under some additional assumptions for some class of Itó diffusions. The new method is an alternative to the classical potential-theoretic approach and gives new specific formulas for distributions under the excursion measure. [ABSTRACT FROM AUTHOR]

In Cohen et al. (Potential Anal. 44(4), 745–766, 2016), we introduced several classes of Carleson-type measures with respect to a radial reference measure σ on a homogeneous tree T, equipped with the nearest-neighbor transition operator and studied their relationships under certain assumptions on σ. We defined two classes of measures σ we called good and optimal and showed that if σ is optimal and μ is a σ-Carleson measure on T in the sense that there is a constant C such that the μ measure of every sector is bounded by C times the σ measure of the sector, then there exists Cμ > 0 such that ∑ f (v) μ (v) ≤ C μ ∑ f (v) σ (v) for every non-negative subharmonic function f on T, and we conjectured that this holds if and only if σ is good. In this paper we develop tools for studying the above conjecture and identify conditions on a class of non-negative subharmonic functions for which we can prove the conjecture for all functions in such a class. We show that these conditions hold for the set of all non-negative subharmonic functions which are generated by eigenfunctions of the Laplacian on T. [ABSTRACT FROM AUTHOR]

By using analytic tools from stochastic analysis, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along fractals (which can be considered as models of simplified rough porous media). Unlike the regular space case, such parabolic type equations involving non-linear convection terms must take a different form, due to the fact that convection terms must be singular to the "linear part" which defines 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. These Sobolev inequalities, which are interesting on their own and in contrast to the case of Euclidean spaces, involve two L p norms with respect to two mutually singular measures. By examining properties of singular convolutions of the associated heat semigroup, we derive the space-time regularity of solutions to these 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, uniqueness, and regularity of its solutions are obtained. [ABSTRACT FROM AUTHOR]

We consider a family { L t , t ∈ [ 0 , T ] } of closed operators generated by a family of regular (non-symmetric) Dirichlet forms { (B (t) , V) , t ∈ [ 0 , T ] } on L 2 (E ; m) . We show that a bounded (signed) measure μ on (0 , T) × E is smooth, i.e. charges no set of zero parabolic capacity associated with ∂ ∂ t + L t , if and only if μ is of the form μ = f · m 1 + g 1 + ∂ t g 2 with f ∈ L 1 ((0 , T) × E ; d t ⊗ m) , g 1 ∈ L 2 (0 , T ; V ′) , g 2 ∈ L 2 (0 , T ; V) . We apply this decomposition to the study of the structure of additive functionals in the Revuz correspondence with smooth measures. As a by-product, we also give some existence and uniqueness results for solutions of semilinear equations involving the operator ∂ ∂ t + L t and a functional from the dual W ′ of the space W = { u ∈ L 2 (0 , T ; V) : ∂ t u ∈ L 2 (0 , T ; V ′) } on the right-hand side of the equation. [ABSTRACT FROM AUTHOR]

We prove a theorem on the minimal specific energy for a ± 1 charged particles system, interacting through a class of pair potential v, that may be stated as follows: suppose v may be represented by a scale mixtures of d-dimensional Euclid's hat. If the number of particles n is even, then their interacting energy U n divided by n is minimized by a constant B at the configurations with total charge zero and all particles collapsed to a point; if n is odd, then the ratio U n / (n - 1) is minimized by a constant B ¯ = B at the configurations with total charge ± 1 and all particles collapsed to a point. The theorem is then used to investigate the convergence of the Mayer series for a gas of ± 1 charged particles interacting through the two-dimensional Yukawa pair potential v for inverse temperatures in the collapse interval [ 4 π , 8 π) . The convergence is proved in the present paper up to the second threshold 6 π using the decomposition of the Yukawa potential into scales of modified Bessel functions (standard) and into scale mixtures of Euclid's hat. Moreover, assuming that (i) neutral subclusters of size smaller than an odd number k > 1 do not collapse inside a cluster of size larger than k for β in the threshold interval [ 8 π (k - 2) / (k - 1) , 8 π k / (k + 1)) and (ii) they satisfy a technical condition, then the Mayer series, discarding the first even coefficients of order smaller than k, converges. [ABSTRACT FROM AUTHOR]

For any graph having a suitable uniform Poincaré inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks evolving via time varying, uniformly elliptic conductances, provided the vertex conductances (i.e. reversing measures), increase in time. Such transition density upper bounds apply for discrete time uniformly lazy walks, with the matching lower bounds holding once the parabolic Harnack inequality is proved. [ABSTRACT FROM AUTHOR]

In this paper we investigate an integration by parts formula for Lévy processes by using lower bound conditions of the corresponding Lévy measure. As applications, derivative formula and coupling property are derived for transition semigroups of linear SDEs driven by Lévy processes. [ABSTRACT FROM AUTHOR]

In the present paper, we are going to show that outside a slim set in the sense of Malliavin (or quasi-surely), the signature path (which consists of iterated path integrals in every degree) of Brownian motion is non-self-intersecting. This property relates closely to a non-degeneracy property for the Brownian rough path arising naturally from the uniqueness of signature problem in rough path theory. As an important consequence we conclude that quasi-surely, the Brownian rough path does not have any tree-like pieces and every sample path of Brownian motion is uniquely determined by its signature up to reparametrization. [ABSTRACT FROM AUTHOR]

Given a "Green function" G on a locally compact space X with countable base, a Borel set A in X is called G-semipolar, if there is no measure ν ≠ 0 supported by A such that G ν : = ∫ G (⋅ , y) d ν (y) is a continuous real function on X. Introducing an intrinsic Hausdorff measuremG using G-balls B(x, ρ) := {y ∈ X : G(x, y) > 1/ρ}, it is shown that every set A in X with m G (A) < ∞ is contained in a G-semipolar Borel set. This is of interest, since G-semipolar sets are semipolar in the potential-theoretic sense (countable unions of totally thin sets, hit by a corresponding process at most countably many times), if G is a genuine Green function. The result has immediate consequences for classical potential theory, Riesz potentials and the heat equation (where it solves an open problem). More generally, it is applied to metric measure spaces (X, d, μ), where a continuous heat kernel with upper and lower bounds of the form t−α/βΦj(d(x,y)t− 1/β), j = 1, 2, is given. Then the intrinsic Hausdorff measure on X is equivalent to an ordinary Hausdorff measure mα−β. For the corresponding space-time structure on X × ℝ, the intrinsic Hausdorff measure turns out to be equivalent to an anisotropic Hausdorff measure mα,β. [ABSTRACT FROM AUTHOR]

The aim of this note is to provide an original proof and derive fine properties of the excessive function that characterizes the Laplace transform of the downward first hitting time to a fixed level of a non-degenerate continuous-time branching process. It hinges on a recent result by Choi and Patie (2016) on the potential theory of skip-free Markov chains and reveals, in particular, that the fundamental excessive function that characterizes the first hitting time is a purely excessive function. [ABSTRACT FROM AUTHOR]

Under mild conditions on measures used in the perturbation, we establish the L p -independence of spectral radius for generalized Feynman–Kac semigroups without assuming the irreducibility and the boundedness of the function appeared in the continuous additive functionals locally of zero energy in the framework of symmetric Markov processes. These results are obtained by using the gaugeability approach developed by the first named author as well as the recent progress on the irreducible decomposition for Markov processes proved by the third author and on the analytic characterizations of gaugeability for generalized Feynman–Kac functionals developed by the second and third authors. [ABSTRACT FROM AUTHOR]

We suggest a method for constructing a positive harmonic function for a wide class of transition kernels on Z + . We also find natural conditions under which this harmonic function has a positive finite limit at infinity. Further, we apply our results on harmonic functions to asymptotically homogeneous Markov chains on Z + with asymptotically negative drift which arise in various queueing models. More precisely, assuming that the Markov chain satisfies Cramér's condition, we study the tail asymptotics of its stationary distribution. In particular, we clarify the impact of the rate of convergence of chain jumps towards the limiting distribution. [ABSTRACT FROM AUTHOR]

Let {Xt}t≥0 be the symmetric α-stable process with generator H=(-Δ)α/2 for 0<α≤2. For a positive Radon measure μ, we define the Schrödinger operator Hμ=H-μ and consider the fundamental solution of the equation ∂u/∂t=-Hμu. If μ is critical, the behavior of the fundamental solution is different from that of the transition density function of {Xt}t≥0. In this paper, we give a certain ergodic-type limit theorem for fundamental solutions of critical Schrödinger operators. [ABSTRACT FROM AUTHOR]

In our study of electrical networks we develop two themes: finding explicit formulas for special classes of functions defined on the vertices of a transient network, namely monopoles, dipoles, and harmonic functions. Secondly, our interest is focused on the properties of electrical networks supported on Bratteli diagrams. We show that the structure of Bratteli diagrams allows one to describe algorithmically harmonic functions as well as monopoles and dipoles. We also discuss some special classes of Bratteli diagrams (stationary, Pascal, trees), and we give conditions under which the harmonic functions defined on these diagrams have finite energy. [ABSTRACT FROM AUTHOR]

Convergence is proved for solutions un of Dirichlet problems in regions with many small excluded sets (holes), as the holes become smaller and more numerous. The problem is formulated probabilistically in the context of general Dirichlet forms, for random and nonrandom excluded sets. Sufficient conditions are given in Theorem 2.1 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 Rényi. Stable convergence of the entrance times implies convergence of the solutions un of the corresponding Dirichlet problems. Theorem 2.1 applies to Dirichlet forms such that the Markov process associated with the form has continuous paths and satisfies an absolute continuity condition for occupation time measures (Eq. 2.4). Conditions for convergence are formulated in terms of the sum of the expectations of the equilibrium measures for the excluded sets. The proof of convergence uses the fact that any martingale with respect to the natural filtration of the process must be continuous. In the case that the excluded sets are iid random, Theorem 2.1 strengthens previous results for the classical Brownian motion setting. [ABSTRACT FROM AUTHOR]

In this note, the authors survey the existing convergence results for random variables under sublinear expectations, and prove some new results. Concretely, under the assumption that the sublinear expectation has the monotone continuity property, the authors prove that convergence in capacity is stronger than convergence in distribution, and give some equivalent characterizations of convergence in distribution. In addition, they give a dominated convergence theorem under sublinear expectations, which may have its own interest. [ABSTRACT FROM AUTHOR]

In a recent work Levine et al. (Ann Henri Poincaré 17:1677-1711, 2016. 10.1007/s00023-015-0433-x) 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. [ABSTRACT FROM AUTHOR]

In the first part of the paper we introduce a new parametrization for the manifold underlying quadratic analogue of the usual Heisenberg group introduced in Accardi et al. (Infin Dimens Anal Quantum Probab Relat Top 13:551-587, 2010) which makes the composition law much more transparent. In the second part of the paper the new coordinates are used to construct an inductive system of ∗-algebras each of which is isomorphic to a finite tensor product of copies of the one-mode quadratic Weyl algebra. We prove that the inductive limit ∗-algebra is factorizable and has a natural localization given by a family of ∗-sub-algebras each of which is localized on a bounded Borel subset of R. Moreover, we prove that the family of quadratic analogues of the Fock states, defined on the inductive family of ∗-algebras, is projective hence it defines a unique state on the limit ∗-algebra. Finally we complete this ∗-algebra under the (minimal regular) C∗-norm thus obtaining a C∗-algebra. [ABSTRACT FROM AUTHOR]

Let x∈Rd, d ≥ 3, and f:Rd→R be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let aij:Rd→R be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to 퓛f(x)=12∑i=1d∑j=1d∂∂xiaij(·)∂f∂xj(x)+∫Rd\{0}[f(y)-f(x)]J(x,y)dywhere J:Rd×Rd→R is a symmetric measurable function. Let q:Rd→R. We specify assumptions on a, q, and J so that non-negative bounded solutions to 퓛f+qf=0satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to 퓛f=0. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions under which the fractional semilinear elliptic equation Δα2u=ρ(x)φ(u) admits nonnegative nontrivial bounded solutions in the whole space ℝN. [ABSTRACT FROM AUTHOR]

We revisit some problems arising in the context of multiallelic discrete-time and deterministic evolutionary dynamics driven first by fitness differences and then by segregation distortion. In the model with fitness, we describe classes of fitness matrices exhibiting polymorphism. In the segregation case, still in search for conditions of polymorphism, we focus on a class of skew-symmetric matrices with a unique strictly positive kernel vector. Our main results reduce the study of these cases to the analysis of stochastic matrices. [ABSTRACT FROM AUTHOR]

Abstract: The space‐time distribution, Q A ( x , d t d ξ ) say, of Brownian hitting of a bounded Borel set A of R d is studied. We derive the asymptotic form of the leading term of the time‐derivative Q A ( x , d t d ξ ) / d t for each d = 2 , 3 , …, valid uniformly with respect to the starting point x of the Brownian motion, which result significantly extends the classical ones for Q A ( x , d t d ξ ) itself by Hunt ( d = 2 ), Joffe and Spitzer ( d ⩾ 3). The results obtained are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joining the origin to a distant point. [ABSTRACT FROM AUTHOR]

We give a general analytic characterization of the gaugeability for generalized Feynman-Kac functionals involving continous additive functionals locally of zero energy generalizing the earlier work on the analytic characterization of the gaugeability of Feynman-Kac functional for absorbing Brownian motion on a bounded domain given by Aizenman-Simon (Commun Pure Appl Math 35(2), 209-273, 1982). The result also extends the previous known results on the analytic characterization of the gaugeability for generalized Feynman-Kac functionals in the framework of symmetric Markov processes satisfying irreducibility condition and absolute continuity condition. [ABSTRACT FROM AUTHOR]

We study in detail and explicitly solve the version of Kyle's model introduced in a specific case in Back and Baruch (Econometrica 72:433-465, 2004), where the trading horizon is given by an exponentially distributed random time. The first part of the paper is devoted to the analysis of time-homogeneous equilibria using tools from the theory of one-dimensional diffusions. It turns out that such an equilibrium is only possible if the final payoff is Bernoulli distributed as in Back and Baruch (Econometrica 72:433-465, 2004). We show in the second part that the signal the market makers use in the general case is a time-changed version of the one they would have used had the final payoff had a Bernoulli distribution. In both cases, we characterise explicitly the equilibrium price process and the optimal strategy of the informed trader. In contrast to the original Kyle model, it is found that the reciprocal of the market's depth, i.e., Kyle's lambda, is a uniformly integrable supermartingale. While Kyle's lambda is a potential, i.e., converges to 0, for the Bernoulli-distributed final payoff, its limit in general is different from 0. [ABSTRACT FROM AUTHOR]

We give a probabilistic numerical approach for the nonlinear Dirichlet problem associated with a branching process. Main tools are the probabilistic representation of the solution with the measure-valued branching process, as well as specific techniques for the numerical solution of linear partial differential equations, introduced and developed by Milstein and Tretyakov, and Monte Carlo methods. [ABSTRACT FROM AUTHOR]

We give a method to associate a right Markov process to a resistance form, that is different from the one based on the regularity assumption for the associated Dirichlet form. We use tools from potential theory associated to resolvents of kernels and cones of potentials (H-cones). The induced capacity and processes obtained by time change with the inverses of continuous additive functionals also occur. [ABSTRACT FROM AUTHOR]

We prove the existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular (possibly disconnected) domains of harmonicity, in the context of general metric measure spaces. As a corollary, we prove the uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R with positive continuous density of the Lévy measure; stable-like processes in R and in domains; and stable-like subordinate diffusions in metric measure spaces. [ABSTRACT FROM AUTHOR]

In this paper we prove a uniform and scale invariant boundary Harnack principle at infinity for a large class of purely discontinuous Feller processes in metric measurespaces. [ABSTRACT FROM AUTHOR]

We study the wave equation on one-dimensional self-similar fractal structures that can be analyzed by the spectral decimation method. We develop efficient numerical approximation techniques and also provide uniform estimates obtained by analytical methods. [ABSTRACT FROM AUTHOR]