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Let $$g_{1},g_{2},\dots $$ be i.i.d. random matrices in $$GL\left( d,\mathbb {R}\right) .$$ For any $$n\ge 1$$ consider the product $$G_{n}=g_{n} \dots g_{1}$$ and the random process $$G_{n}v=g_{n}\dots g_{1}v$$ in $$\mathbb {R}^{d}$$ starting at point $$v\in \mathbb {R}^{d}{\backslash } \left\{ 0\right\} .$$ It is well known that under appropriate assumptions, the sequence $$\left( \log \left\| G_{n}v\right\| \right) _{n\ge 1}$$ behaves like a sum of i.i.d. r.v.'s and satisfies standard classical properties such as the law of large numbers, the law of iterated logarithm and the central limit theorem. For any vector v with $$\left\| v \right\| >1$$ denote by $$\tau _v$$ the first time when the random process $$G_{n}v$$ enters the closed unit ball in $$\mathbb {R}^{d}.$$ We establish the asymptotic as $$n\rightarrow +\infty $$ of the probability of the event $$\left\{ \tau _{v}>n\right\} $$ and find the limit law for the quantity $$\frac{1}{\sqrt{n}} \log \left\| G_{n}v\right\| $$ conditioned that $$\tau _{v}>n$$ . [ABSTRACT FROM AUTHOR]

We establish exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for (1) the fourth order L-Kuramoto–Sivashinsky (L-KS) SPDEs and for (2) the time-fractional stochastic partial integro-differential equations (SPIDEs), driven by the space–time white noise in one-to-three dimensional spaces. Both classes were introduced—with Brownian-time-type kernel formulations—by Allouba in a series of articles starting in 2006, where he presented class (2) in its rigorous stochastic integral equations form. He proved existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above two classes of equations in d = 1 , 2 , 3 . We show that both classes are ( 1 / 2 ) − Hölder continuously differentiable in space when d = 1 , and we give the exact uniform and local moduli of continuity for the gradient in both cases. This is unprecedented for SPDEs driven by the space–time white noise. Our results on exact moduli show that the half-derivative SPIDE is a critical case. It signals the onset of rougher modulus regularity in space than both time-fractional SPIDEs with time-derivatives of order < 1 / 2 and L-KS SPDEs. This is despite the fact that they all have identical spatial Hölder regularity, as shown earlier by Allouba. Moreover, we show that the temporal laws governing (1) and (2) are fundamentally different. We relate L-KS SPDEs to the Houdré-Villa bifractional Brownian motion, yielding a Chung-type law of the iterated logarithm for these SPDEs. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on delicate sample path properties of Gaussian random fields. On the other hand, it builds on and complements Allouba's earlier works on (1) and (2). Similar regularity results hold for the Allen–Cahn nonlinear members of (1) and (2) on compacts via change of measure. [ABSTRACT FROM AUTHOR]

In a setting, where only 'exit measures' are given, as they are associated with a right continuous strong Markov process on a separable metric space, we provide simple criteria for scaling invariant Hölder continuity of bounded harmonic functions with respect to a distance function which, in applications, may be adapted to the special situation. In particular, already a very weak scaling property ensures that Harnack inequalities imply Hölder continuity. Our approach covers recent results by M. Kassmann and A. Mimica as well as cases, where a Green function leads to an intrinsic metric. [ABSTRACT FROM AUTHOR]

We introduce a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle that can be either red or blue. New red and blue particles alternately get emitted from their respective bases and perform random walk. On encountering a particle of the opposite color they kill it and occupy its position. We prove that on the cylinder graph (the product of a path and a cycle) an interface spontaneously forms between red and blue and is maintained in a predictable position with high probability. [ABSTRACT FROM AUTHOR]

In this article we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Our methods rely on local central limit theorems for convergence of random walks on discretizations of smooth domains to Reflected Brownian motion. [ABSTRACT FROM AUTHOR]

We give a unified method to obtain the conservativeness of a class of Markov processes associated with lower bounded semi-Dirichlet forms on L ( X; m), including symmetric diffusion processes, some non-symmetric diffusion processes and jump type Markov processes on X, where X is a locally compact separable metric space and m is a positive Radon measure on X with full topological support. Using the method, we give an example in each section, providing the conservativeness of the processes, that are given by the 'increasingness of the volume of some sets(balls)' and 'that of the coefficients on the sets' of the Markov processes. [ABSTRACT FROM AUTHOR]

In this paper, we establish some functional central limit theorems for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. In the particular case when the state $E$ is a finite set and the underlying motion is an irreducible Markov chain on $E$ , our results are superprocess analogs of the functional central limit theorems of Janson (Stoch. Process. Appl. 110:177-245, 2004) for supercritical multitype branching processes. The results of this paper are refinements of the central limit theorems in Ren et al. (Stoch. Process. Appl. 125:428-457, 2015). [ABSTRACT FROM AUTHOR]

In the paper we consider elliptic equations of the form − A u = u − γ ⋅ μ , where A is the operator associated with a regular symmetric Dirichlet form, μ is a positive nontrivial measure and γ > 0 . We prove the existence and uniqueness of solutions of such equations as well as some regularity results. We also study stability of solutions with respect to the convergence of measures on the right-hand side of the equation. For this purpose, we introduce some type of functional convergence of smooth measures, which in fact is equivalent to the quasi-uniform convergence of associated potentials. [ABSTRACT FROM AUTHOR]

We obtain general lower estimates of transition densities of jump Lévy processes. We use them for processes with Lévy measures having bounded support, processes with exponentially decaying Lévy measures for large times and for processes with high intensity of small jumps for small times. [ABSTRACT FROM AUTHOR]

We study darning of compact sets (darning and gluing of finite unions of compact sets), which are not thin at any of their points, in a potential-theoretic framework which may be described, analytically, in terms of harmonic kernels/harmonic functions or, probabilistically, in terms of a diffusion. This is accomplished without leaving our kind of setting so that the procedure can be iterated without any problem. It applies to darning and gluing of compacts in Euclidean spaces (manifolds) of different dimensions, which is of interest pertaining to recent studies on heat kernels. [ABSTRACT FROM AUTHOR]

This paper studies the small time behavior of the heat content for rotationally invariant $$\alpha $$ -stable processes, $$0<\alpha \le 2$$ , in domains of $$\mathbb {R}^d$$ . Unlike the asymptotics for the heat trace, the behavior of the heat content differs depending on the range of $$\alpha $$ according to $$0<\alpha <1$$ , $$\alpha =1$$ and $$1<\alpha \le 2$$ . [ABSTRACT FROM AUTHOR]

The Liouville Brownian motion (LBM), recently introduced by Garban, Rhodes and Vargas and in a weaker form also by Berestycki, is a diffusion process evolving in a planar random geometry induced by the Liouville measure $$M_\gamma $$ , formally written as $$M_\gamma (dz)=e^{\gamma X(z)-{\gamma ^2} \mathbb E[X(z)^2]/2}\, dz$$ , $$\gamma \in (0,2)$$ , for a (massive) Gaussian free field X. It is an $$M_\gamma $$ -symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure $$M_\gamma $$ . In this paper we provide a detailed analysis of the heat kernel $$p_t(x,y)$$ of the LBM. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian upper bound of the form $$p_t(x,y)\le C_{1} t^{-1} \log (t^{-1}) \exp \bigl (-C_{2}((|x-y|^{\beta }\wedge 1)/t)^{\frac{1}{\beta -1}}\bigr )$$ for $$t\in \bigl (0,\frac{1}{2}\bigr ]$$ for each $$\beta >\frac{1}{2}(\gamma +2)^2$$ , and an on-diagonal lower bound of the form $$p_{t}(x,x)\ge C_{3}t^{-1}\bigl (\log (t^{-1})\bigr )^{-\eta }$$ for $$t\in (0,t_{\eta }(x)]$$ , with $$t_{\eta }(x)\in \bigl (0,\frac{1}{2}\bigr ]$$ heavily dependent on x, for each $$\eta >18$$ for $$M_{\gamma }$$ -almost every x. As applications, we deduce that the pointwise spectral dimension equals 2 $$M_\gamma $$ -a.e. and that the global spectral dimension is also 2. [ABSTRACT FROM AUTHOR]

Let μ be a positive Kato measure on [ABSTRACT FROM AUTHOR]

The author investigates the relationships of some potential objects for a right Markov process and the same objects for the Girsanov transformed process induced by α-excessive function including Revuz measures, energy functionals, capacities and Lévy systems in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the Martin kernels of general open sets associated with inaccessible points for a large class of purely discontinuous Feller processes in metric measure spaces. Let D be an unbounded open set. Infinity is accessible from D if the expected exit time from D is infinite, and inaccessible otherwise. We prove that under suitable assumptions there is only one Martin boundary point associated with infinity, and that this point is minimal if and only if infinity is accessible from D. Similar results are also proved for finite boundary points of D. [ABSTRACT FROM AUTHOR]

Getoor's conjecture that essentially all Lévy processes satisfy Hunt's hypothesis (H) is a longstanding open problem in potential theory. In the beginning of the paper, we summarize the main results obtained so far for the problem. Then, we present two new necessary and sufficient conditions for the validity of (H). Furthermore, we give applications of these new criteria. First, we give explicit constructions of Lévy processes satisfying (H) in a context where previously known results could not be applied. Second, we show that a large class of pure jump subordinators can be decomposed into the summation of two independent subordinators such that both of them satisfy (H). [ABSTRACT FROM AUTHOR]

In this article we characterize the closed cones respectively generated by the symmetric inverse M -matrices and by the inverses of symmetric row diagonally dominant M -matrices. We show the latter has a finite number of extremal rays, while the former has infinitely many extremal rays. As a consequence we prove that every potential is the sum of ultrametric matrices. [ABSTRACT FROM AUTHOR]

In this article we characterize inverse M -matrices and potentials whose inverses are supported on trees. In the symmetric case we show they are a Hadamard product of tree ultrametric matrices, generalizing a result by Gantmacher and Krein [12] done for inverse tridiagonal matrices. We also provide an algorithm that recognizes when a positive matrix W has an inverse M -matrix supported on a tree. This algorithm has quadratic complexity. We also provide a formula to compute W − 1 , which can be implemented with a linear complexity. Finally, we also study some stability properties for Hadamard products and powers. [ABSTRACT FROM AUTHOR]

To obtain groups with bounded harmonic functions (which are amenable), one of the most frequent way is to look at some semi-direct products ( e.g. lamplighter groups). The aim here is to show that many of these semi-direct products do not admit harmonic functions with gradient in ℓ , for $p\in [1,\infty [$ . [ABSTRACT FROM AUTHOR]

We establish metastability in the sense of Lebowitz and Penrose under practical and simple hypotheses for Markov chains on a finite configuration space in some asymptotic regime. By comparing restricted ensembles and quasi-stationary measures, and introducing soft measures as an interpolation between the two, we prove an asymptotic exponential exit law and, on a generally different time scale, an asymptotic exponential transition law. By using potential-theoretic tools, and introducing “ ( κ , λ ) -capacities”, we give sharp estimates on relaxation time, as well as mean exit time and transition time. We also establish local thermalization on shorter time scales. [ABSTRACT FROM AUTHOR]

Let $(X,\mathcal {W})$ be a balayage space, $1\in \mathcal {W}$ , or - equivalently - let $\mathcal W$ be the set of excessive functions of a Hunt process on a locally compact space X with countable base such that $\mathcal {W}$ separates points, every function in $\mathcal {W}$ is the supremum of its continuous minorants and there exist strictly positive continuous $u,v\in \mathcal {W}$ such that u/ v → 0 at infinity. We suppose that there is a Green function G > 0 for X, a metric ρ for X and a decreasing function $g\colon [0,\infty )\to (0,\infty ]$ having the doubling property such that G ≈ g ∘ ρ. Assuming that the constant function 1 is harmonic and balls of ( X, ρ) are relatively compact, it is shown that every positive harmonic function on X is constant (Liouville property) and that Wiener's test at infinity shows, if a given set A in X is unavoidable, that is, if the process hits A with probability one, wherever it starts. An application yields that locally finite unions of pairwise disjoint balls B( z, r ), z ∈ Z, which have a certain separation property with respect to a suitable measure λ on X are unavoidable if and only if, for some/any point x ∈ X, the series ${\sum }_{z\in Z} g(\rho (x_{0},z))/g(r_{z}) $ diverges. The results generalize and, exploiting a zero-one law for hitting probabilities, simplify recent work by S. Gardiner and M. Ghergu, A. Mimica and Z. Vondraček, and the author. [ABSTRACT FROM AUTHOR]

Hastings studied Carleson measures for non-negative subharmonic functions on the polydisk and characterized them by a certain geometric condition relative to Lebesgue measure σ. Cima & Wogen and Luecking proved analogous results for weighted Bergman spaces on the unit ball and other open subsets of $\mathbb {C}^{n}$ . We consider a similar problem on a homogeneous tree, and study how the characterization and properties of Carleson measures for various function spaces depend on the choice of reference measure σ. [ABSTRACT FROM AUTHOR]

We give a stochastic model for the fragmentation phase of an avalanche. We construct a fragmentation-branching process related to the avalanches, on the set of all fragmentation sizes introduced by Bertoin. A fractal property of this process is emphasized. We also establish a specific stochastic differential equation of fragmentation. It turns out that specific branching Markov processes on finite configurations of particles with sizes bigger than a strictly positive threshold are convenient for describing the continuous time evolution of the number of the resulting fragments. The results are obtained by combining analytic and probabilistic potential theoretical tools. [ABSTRACT FROM AUTHOR]

A randomized subspace action algorithm is investigated for fusion frame signal recovery problems and for the problem of recovering a signal from projections onto random subspaces. It is noted that Kaczmarz bounds provide upper bounds on the algorithm's error moments. The main question of which probability distributions on a random subspace lead to provably fast convergence is addressed. In particular, it is proved which distributions give minimal Kaczmarz bounds, and hence give best control on error moment upper bounds arising from Kaczmarz bounds. Uniqueness of the optimal distributions is also addressed. [ABSTRACT FROM AUTHOR]

In the paper we prove the existence of probabilistic solutions to systems of the form − A u = F( x, u) + μ, where F satisfies a generalized sign condition and μ is a smooth measure. As for A we assume that it is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on L. This class includes local and nonlocal operators corresponding to Dirichlet forms as well as some operators which are not in the variational form. To study the problem we introduce new concept of compactness property relating the underlying Markov process to almost everywhere convergence. We prove some useful properties of the compactness property and provide its characterization in terms of Meyer's property (L) of Markov processes and in terms of order compactness of the associated resolvent. [ABSTRACT FROM AUTHOR]

As a generalization to the heat semigroup on the Heisenberg group, the diffusion semigroup generated by the subelliptic operator $$\L: = \frac{1}{2}\sum\nolimits_{i = 1}^m {X_i^2on{\kern 1pt} {\mathbb{R}^m} \times {\mathbb{R}^d}} $$ is investigated, where $${X_i}\left( {x,y} \right) = \sum\limits_{k = 1}^m {{\sigma _{ki}}{\partial _{xk}} + } \sum\limits_{l = 1}^d {{{\left( {{A_l}x} \right)}_i}{\partial _{yl}}} ,\left( {x,y} \right) \in {\mathbb{R}^{m + d}},1 \leqslant i \leqslant m$$ for σ an invertible m×m-matrix and {Al}1≤ l≤ d some m×m-matrices such that the Hörmander condition holds. We first establish Bismut-type and Driver-type derivative formulas with applications on gradient estimates and the coupling/Liouville properties, which are new even for the heat semigroup on the Heisenberg group; then extend some recent results derived for the heat semigroup on the Heisenberg group. [ABSTRACT FROM AUTHOR]

Let d ≥ 1 and Z be a subordinate Brownian motion on R with infinitesimal generator Δ + ψ(Δ), where ψ is the Laplace exponent of a one-dimensional non-decreasing Lévy process (called subordinator). We establish the existence and uniqueness of fundamental solution (also called heat kernel) p( t, x, y) for non-local operator ℒ = Δ + ψ(Δ) + b · ∇, where b is an R-valued function in Kato class K. We show that p( t, x, y) is jointly continuous and derive its sharp two-sided estimates. The kernel p( t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for (L, C (R)) and X is a weak solution of $X_t = X_0 + Z_t + \int_0^t {b(X_s )ds,} t \geqslant 0$. Moreover, we prove that the above stochastic differential equation has a unique weak solution. [ABSTRACT FROM AUTHOR]

Generalizing the L-Kuramoto–Sivashinsky (L-KS) kernel from our earlier work, we give a novel explicit-kernel formulation useful for a large class of fourth order deterministic, stochastic, linear, and nonlinear PDEs in multispatial dimensions. These include pattern formation equations like the Swift–Hohenberg and many other prominent and new PDEs. We first establish existence, uniqueness, and sharp dimension-dependent spatio-temporal Hölder regularity for the canonical (zero drift) L-KS SPDE, driven by white noise on { R + × R d } d = 1 3 . The spatio-temporal Hölder exponents are exactly the same as the striking ones we proved for our recently introduced Brownian-time Brownian motion (BTBM) stochastic integral equation, associated with time-fractional PDEs. The challenge here is that, unlike the positive BTBM density, the L-KS kernel is the Gaussian average of a modified, highly oscillatory, and complex Schrödinger propagator. We use a combination of harmonic and delicate analysis to get the necessary estimates. Second, attaching order parameters ε 1 to the L-KS spatial operator and ε 2 to the noise term, we show that the dimension-dependent critical ratio ε 2 / ε 1 d / 8 controls the limiting behavior of the L-KS SPDE, as ε 1 , ε 2 ↘ 0 ; and we compare this behavior to that of the less regular second order heat SPDEs. Finally, we give a change-of-measure equivalence between the canonical L-KS SPDE and nonlinear L-KS SPDEs. In particular, we prove uniqueness in law for the Swift–Hohenberg and the law equivalence—and hence the same Hölder regularity—of the Swift–Hohenberg SPDE and the canonical L-KS SPDE on compacts in one-to-three dimensions. [ABSTRACT FROM AUTHOR]

We describe singular diffusion in bounded subsets $${\Omega}$$ of $${\mathbb{R}^{n}}$$ by form methods and characterize the associated operator. We also prove positivity and contractivity of the corresponding semigroup. This results in a description of a stochastic process moving according to classical diffusion in one part of $${\Omega}$$ , where jumps are allowed through the rest of $${\Omega}$$ . [ABSTRACT FROM AUTHOR]

Motivated by the study of the directed polymer model with mobile Poissonian traps or catalysts and the stochastic parabolic Anderson model with time-dependent potential, we investigate the asymptotic behavior of where $$\theta >0$$ is a constant, $$\overline{V}$$ is the renormalized Poisson potential of the form and $$\omega _s$$ is the measure-valued process consisting of independent Brownian particles whose initial positions form a Poisson random measure on $$\mathbb {R}^d$$ with Lebesgue measure as its intensity. Different scaling limits are obtained according to the parameter $$p$$ and dimension $$d$$ . For the logarithm of the negative exponential moment, the range of $$\frac{d}{2}2)$$ , the exponential moments become infinite for all $$t>0$$ . [ABSTRACT FROM AUTHOR]

Recurrence criteria and related capacitary inequalities for a class of symmetric pure jump processes are given under suitable upper volume growth conditions for the underlying metric measure space. Recurrence problems for symmetric jump processes on d-sets are also investigated. [ABSTRACT FROM AUTHOR]

We prove a scale-invariant boundary Harnack principle for inner uniform domains in metric measure Dirichlet spaces. We assume that the Dirichlet form is symmetric, strongly local, regular, and that the volume doubling property and two-sided sub-Gaussian heat kernel bounds are satisfied. We make no assumptions on the pseudo-metric induced by the Dirichlet form, hence the underlying space can be a fractal space. [ABSTRACT FROM AUTHOR]

We study massive (reccurent) sets with respect to a certain random walk [ABSTRACT FROM AUTHOR]

We give a power law for the homogeneity degree of the Martin kernel at infinity of the fractional Laplacian for the right circular cone when the angle of the cone tends to zero. [ABSTRACT FROM AUTHOR]

We show that given a Borel right process there exists a dual process which is also a right Markov process. However, it is necessary to enlarge the initial space to a new Lusin topological space and the dual process is a right process with respect to a second Lusin topology. As a result both processes can be identified as solutions to martingale problems. Another application is the proof that the Riesz decomposition holds (in potential and harmonic components) for the excessive functions and the set of all potentials becomes solid in order. [ABSTRACT FROM AUTHOR]

We derive a large deviation principle for the density profile of occupation times of random interlacements at a fixed level in a large box of $${\mathbb { Z}}^d$$ , $$d \ge 3$$ . As an application, we analyze the asymptotic behavior of the probability that atypically high values of the density profile insulate a macroscopic body in a large box. As a step in this program, we obtain a similar large deviation principle for the occupation-time measure of Brownian interlacements at a fixed level in a large box of $${\mathbb { R}}^d$$ , and we derive a new identity for the Laplace transform of the occupation-time measure, which is based on the analysis of certain Schrödinger semi-groups. [ABSTRACT FROM AUTHOR]

We give a proof that the first eigenfunction of the α-symmetric stable process on a bounded Lipschitz domain in $\mathbb {R}^{d}$, d≥1, is superharmonic for α=2/ m, where m>2 is an integer. This result was first proved by M. Kaßmann and L. Silvestre for the ball (private communication) and for C domains in Kaßmann and Silvestre (). For α=1, the result for Lipschitz domains was proved in Bañuelos and Kulczycki (J. Funct. Anal. 211, 355-423, 2004, Theorem 4.7). [ABSTRACT FROM AUTHOR]

This paper studies the heat content for Schrödinger operators of the fractional Laplacian ${(-{\Delta })^{\frac {\alpha }{2} }}$, 0< α≤2 in ${\mathbb {R}}^{d}$, d≥1. Employing probabilistic and analytic techniques, a small time asymptotic expansion formula is given and the 'heat content invariants' are identified. To the best of our knowledge, these results are new even in the case of the Laplacian, α = 2. [ABSTRACT FROM AUTHOR]

We study a nonlinear semigroup associated with a nonexpansive mapping on an Hadamard space and establish its weak convergence to a fixed point. A discrete-time counterpart of such a semigroup, the proximal point algorithm, turns out to have the same asymptotic behavior. This complements several results in the literature-both classical and more recent ones. As an application, we obtain a new approach to heat flows in singular spaces for discrete as well as continuous times. [ABSTRACT FROM AUTHOR]

In classical PDE theory, Widder's theorem gives a representation for non-negative solutions of the heat equation on $$\mathbb{R }^n$$. We show that an analogous theorem holds for local weak solutions of the canonical 'heat equation' on a symmetric local Dirichlet space satisfying a local parabolic Harnack inequality. [ABSTRACT FROM AUTHOR]

This paper features a comparison inequality for the densities of the moment measures of super-Brownian motion. These densities are defined recursively for each n≥1 in terms of the Poisson and Green's kernels, hence can be analyzed using the techniques of classical potential theory. When n=1, the moment density is equal to the Poisson kernel, and the comparison is simply the classical inequality of Harnack. For n>1 we find that the constant in the comparison inequality grows at most exponentially with n. We apply this to a class of X-harmonic functions H of super-Brownian motion, introduced by Dynkin. We show that for a.e. H in this class, $H^{\nu }(\mu )<\infty $ for every μ. [ABSTRACT FROM AUTHOR]

We study diffusion processes driven by a Brownian motion with regular drift in a finite dimension setting. The drift has two components on different time scales, a fast conservative component and a slow dissipative component. Using the theory of Dirichlet form and Mosco-convergence we obtain simpler proofs, interpretations and new results of the averaging principle for such processes when we speed up the conservative component. As a result, one obtains an effective process with values in the space of connected level sets of the conserved quantities. The use of Dirichlet forms provides a simple and nice way to characterize this process and its properties. [ABSTRACT FROM AUTHOR]