In the classical framework, a random walk on a group is a Markov chain with independent and identically distributed increments. In some sense, random walks are time and space homogeneous. This paper is devoted to a class of inhomogeneous random walks on $\mathbb{Z}^d$ termed ‘Markov additive processes’ (also known as Markov random walks, random walks with internal degrees of freedom, or semi-Markov processes). In this model, the increments of the walk are still independent but their distributions are dictated by a Markov chain, termed the internal Markov chain. While this model is largely studied in the literature, most of the results involve internal Markov chains whose operator is quasi-compact. This paper extends two results for more general internal operators: a local limit theorem and a sufficient criterion for their transience. These results are thereafter applied to a new family of models of drifted random walks on the lattice $\mathbb{Z}^d$ .

We obtain series expansions of the q-scale functions of arbitrary spectrally negative Lévy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit q-scale functions. Moreover, we study smoothness properties of the q-scale functions of spectrally negative Lévy processes with infinite jump activity. This complements previous results of Chan et al. (Prob. Theory Relat. Fields 150, 2011) for spectrally negative Lévy processes with Gaussian component or bounded variation.

We revisit the forward algorithm, developed by Irle, to characterize both the value function and the stopping set for a large class of optimal stopping problems on continuous-time Markov chains. Our objective is to renew interest in this constructive method by showing its usefulness in solving some constrained optimal stopping problems that have emerged recently.

Consider a Lamperti–Kiu Markov additive process $(J, \xi)$ on $\{+, -\}\times\mathbb R\cup \{-\infty\}$, where J is the modulating Markov chain component. First we study the finiteness of the exponential functional and then consider its moments and tail asymptotics under Cramér’s condition. In the strong subexponential case we determine the subexponential tails of the exponential functional under some further assumptions.

We derive two-sided bounds for the Newton and Poisson kernels of the W-invariant Dunkl Laplacian in the geometric complex case when the multiplicity $k(\alpha )=1$ i.e., for flat complex symmetric spaces. For the invariant Dunkl–Poisson kernel $P^{W}(x,y)$ , the estimates are

$$ \begin{align*} P^{W}(x,y)\asymp \frac{P^{\mathbf{R}^{d}}(x,y)}{\prod_{\alpha> 0 \ }|x-\sigma_{\alpha} y|^{2k(\alpha)}}, \end{align*} $$

$\alpha $

$\mathbf {R}^{d}$

$\sigma _{\alpha }$

$P^{\mathbf {R}^{d}}$

${\mathbf {R}^{d}}$

$d\ge 3$

The same estimates are derived in the rank one direct product case $\mathbb {Z}_{2}^{N}$ and conjectured for general W-invariant Dunkl processes.

As an application, we get a two-sided bound for the Poisson and Newton kernels of the classical Dyson Brownian motion and of the Brownian motions in any Weyl chamber.

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.

$\mathbb{Z}^{d}$-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.

Let X be the constrained random walk on ℤ+2 having increments (1,0), (-1,1), and (0,-1) with respective probabilities λ, µ1, and µ2 representing the lengths of two tandem queues. We assume that X is stable and µ1≠µ2. Let τn be the first time when the sum of the components of X equals n. Let Y be the constrained random walk on ℤ×ℤ+ having increments (-1,0), (1,1), and (0,-1) with probabilities λ, µ1, and µ2. Let τ be the first time that the components of Y are equal to each other. We prove that Pn-xn(1),xn(2)(τ<∞) approximates pn(xn) with relative error exponentially decaying in n for xn=⌊nx⌋, x ∈ℝ+2, 0<x(1)+x(2)<1, x(1)>0. An affine transformation moving the origin to the point (n,0) and letting n→∞ connect the X and Y processes. We use a linear combination of basis functions constructed from single and conjugate points on a characteristic surface associated with X to derive a simple expression for ℙy(τ<∞) in terms of the utilization rates of the nodes. The proof that the relative error decays exponentially in n uses a sequence of subsolutions of a related Hamilton‒Jacobi‒Bellman equation on a manifold consisting of three copies of ℝ+2 glued to each other along the constraining boundaries. We indicate how the ideas of the paper can be generalized to more general processes and other exit boundaries.

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.

Let 𝓈 be a finite or countable set. Given a matrix F = (Fij)i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (qij)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≥0Qn ⊗ F*n associated with Q ⊗ F := (qijFij)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 {(Mn, Sn)}n≥0 with discrete recurrent driving chain {Mn}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 recurrence and transience for Lévy processes induced by topological transformation groups acting on complete Riemannian manifolds. In particular the transience–recurrence dichotomy in terms of potential measures is established and transience is shown to be equivalent to the potential measure having finite mass on compact sets when the group acts transitively. It is known that all bi-invariant Lévy processes acting in irreducible Riemannian symmetric pairs of noncompact type are transient. We show that we also have ‘harmonic transience’, that is, local integrability of the inverse of the real part of the characteristic exponent which is associated to the process by means of Gangolli’s Lévy–Khinchine formula.

In this paper we study the balayage of semi-Dirichlet forms. We present new results on balayaged functions and balayaged measures of semi-Dirichlet forms. Some of the results are new even in the Dirichlet forms setting.

Let $X$ be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form $(\varepsilon ,\mathcal{D}(\varepsilon ))$ on ${{L}^{2}}(E;m)$. For $u\,\in \,\mathcal{D}{{(\varepsilon )}_{e}}$, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process $Y$ of $X$. First, let $\hat{X}$ be the dual process of $X$ and $\hat{Y}$ the Girsanov transformed process of $\hat{X} $. We give a necessary and sufficient condition for $(Y,\hat{Y})$ to be in duality with respect to the measure ${{e}^{2u}}m$. We also construct a counterexample, which shows that this condition may not be satisfied and hence $(Y,\hat{Y})$ may not be dual processes. Then we present a sufficient condition under which $Y$ is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup ${{\left( {{\mu }_{t}} \right)}_{t>0}}$. Using appropriate notions of distribution and smooth function spaces, we prove that $L$ is hypoelliptic if and only if ${{\left( {{\mu }_{t}} \right)}_{t>0}}$ is absolutely continuous with respect to Haar measure and admits a continuous density $x\mapsto {{\mu }_{t}}\left( x \right),t>0$, such that ${{\lim }_{t\to 0}}t\log {{\mu }_{t}}\left( e \right)=0$. In particular, this condition holds if and only if any Borel measure $u$ which is solution of $Lu=0$ in an open set $\Omega $ can be represented by a continuous function in $\Omega $. Examples are discussed.

We derive formulas for the first- and higher-order derivatives of the steady state performance measures for changes in transition matrices of irreducible and aperiodic Markov chains. Using these formulas, we obtain a Maclaurin series for the performance measures of such Markov chains. The convergence range of the Maclaurin series can be determined. We show that the derivatives and the coefficients of the Maclaurin series can be easily estimated by analysing a single sample path of the Markov chain. Algorithms for estimating these quantities are provided. Markov chains consisting of transient states and multiple chains are also studied. The results can be easily extended to Markov processes. The derivation of the results is closely related to some fundamental concepts, such as group inverse, potentials, and realization factors in perturbation analysis. Simulation results are provided to illustrate the accuracy of the single sample path based estimation. Possible applications to engineering problems are discussed.

Coercive closed forms on L2-spaces are studied whose associated L2-semigroups are positivity preserving. Earlier work by other authors is extended by further developing the potential theory of such forms and completed by proving an analytic characterization of those of these forms which have a probabilistic counterpart, i.e., are associated with (special standard) Markov processes. Examples with finite and infinite dimensional state spaces are discussed.

Direct and elementary derivation of the classical Poisson kernel for the ball in n = 2 or n = 3 dimensions starting with the usual expression involving the brownian motion bω, the stopping time T on the boundary, and Ex the conditional expectation on paths starting at x.