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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

For a given quasi-regular positivity preserving coercive form, we construct a family of ($\sigma$-finite) distribution flows associated with the semigroup of the form. The canonical cadlag process equipped with the distribution flows behaves like a strong Markov process. Moreover, employing distribution flows we can construct optional measures and establish Revuz correspondence between additive functionals and smooth measures. The results obtained in this paper will enable us to perform a kind of stochastic analysis related to positivity preserving coercive forms.

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.

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.

We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space–time white noise, or colored noise in spatial dimensions $k\geq1$. Our approach builds on a delicate covering argument developed by M. Talagrand [Ann. Probab. 23 (1995) 767–775; Probab. Theory Related Fields 112 (1998) 545–563] for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic PDEs.

We investigate the metastable behavior of reversible Markov chains on possibly countable infinite state spaces. Based on a new definition of metastable Markov processes, we compute precisely the mean transition time between metastable sets. Under additional size and regularity properties of metastable sets, we establish asymptotic sharp estimates on the Poincar\'e and logarithmic Sobolev constant. The main ingredient in the proof is a capacitary inequality along the lines of V. Maz'ya that relates regularity properties of harmonic functions and capacities. We exemplify the usefulness of this new definition in the context of the random field Curie-Weiss model, where metastability and the additional regularity assumptions are verifiable., Comment: 47 pages. Revised and restructured version with extended remarks. This version differs in style and language to published version in Annals of Applied Probability

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

By using the local dimension-free Harnack inequality established on incomplete Riemannian manifolds, integrability conditions on the coefficients are presented for SDEs to imply the nonexplosion of solutions as well as the existence, uniqueness and regularity estimates of invariant probability measures. These conditions include a class of drifts unbounded on compact domains such that the usual Lyapunov conditions cannot be verified. The main results are extended to second-order differential operators on Hilbert spaces and semilinear SPDEs.

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 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.

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 consider the Feynman-Kac penalization problem for critical measures of symmetric $\alpha $-stable processes via large time asymptotics of Feynman-Kac functionals.

We consider time-dependent random walks among time-dependent conductances. For discrete time random walks, we show that, unlike the time-independent case, two-sided Gaussian heat kernel estimates are not stable under perturbations. This is proved by giving an example of a ballistic and transient time-dependent random walk on Z among uniformly elliptic time-dependent conductances. For continuous time random walks, we show the instability when the holding times are i.i.d. exp(1), and in contrast, we prove the stability when the holding times change by sites in such a way that the base measure is a uniform measure., Comment: 12 pages

We give a stochastic model for the fragmentation phase of a snow avalanche. We construct a fragmentation-branching process related to the avalanches, on the set of all fragmentation sizes introduced by J. Bertoin. A fractal property of this process is emphasized. We also establish a specific stochastic 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., 17 pages

We give an analytic version of the well known Shih's theorem concerning the Markov processes whose hitting distributions are dominated by those of a given process. The treatment is purely analytic, completely different from Shih's arguments and improves essentially his result (in the case when the given processes are transient

Let $\{X_{t}\}_{t \geq 0}$ be the $\alpha$-stable-like or relativistic $\alpha$-stable-like process on $\boldsymbol{R}^{d}$ generated by a certain symmetric jump-type regular Dirichlet form $(\mathcal{E, F})$. It is known in [5-7] that the transition probability density $p(t, x, y)$ of $\{X_{t}\}_{t \geq 0}$ admits the two-sided estimates. Let $\mu$ be a positive smooth Radon measure in a certain class and consider the perturbed form $\mathcal{E}^{\mu}(u, u) = \mathcal{E}(u, u) - (u, u)_\mu$. Denote by $p^{\mu}(t, x, y)$ the fundamental solution associated with $\mathcal{E}^{\mu}$. In this paper, we establish a necessary and sufficient condition on $\mu$ for $p^{\mu}(t, x, y)$ having the same two-sided estimates as $p(t, x, y)$ up to positive constants.

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., Comment: 31 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

Let $X$ be a regular linear diffusion whose state space is an open interval $E\subseteq\mathbb{R}$. We consider a diffusion $X^*$ which probability law is obtained as a Doob $h$-transform of the law of $X$, where $h$ is a positive harmonic function for the infinitesimal generator of $X$ on $E$. This is the dual of $X$ with respect to $h(x)m(dx)$ where $m(dx)$ is the speed measure of $X$. Examples include the case where $X^*$ is $X$ conditioned to stay above some fixed level. We provide a construction of $X^*$ as a deterministic inversion of $X$, time changed with some random clock. The study involves the construction of some inversions which generalize the Euclidean inversions. Brownian motion with drift and Bessel processes are considered in details., 19 pages

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)

Any negative moment of an increasing Lamperti process(Yt,t≥0) is a completely monotone function of t. This property enticed us to study systematically, for a given Markov process (Yt,t≥0), the functions f such that the expectation of f(Yt) is a completely monotone function of t. We call these functions temporally completely monotone (for Y). Our description of these functions is deduced from the analysis made by Ben Saad and Janßen, in a general framework, of a dual notion, that of completely excessive measures. Finally, we illustrate our general description in the cases when Y is a Lévy process, a Bessel process, or an increasing Lamperti process.

The purpose of this paper is to find optimal estimates for the Green function and the Poisson kernel for a half-line and intervals of the geometric stable process with parameter $\alpha\in(0,2]$. This process has an infinitesimal generator of the form $-\log(1+(-\Delta)^{\alpha/2})$. As an application we prove the scale invariant Harnack inequality as well as the boundary Harnack principle., Comment: 28 pages

This paper concerns the first hitting time $T_0$ of the origin for random walks on $d$-dimensional integer lattice with zero mean and a finite $2+\delta$ absolute moment ($\delta\geq0$). We derive detailed asymptotic estimates of the probabilities $\mathbb{P}_x(T_0=n)$ as $n\to\infty$ that are valid uniformly in $x$, the position at which the random walks start.

We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakao's divergence-like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain class of functions locally in the domain of Dirichlet form and a generalized It\^{o} formula. (With Errata.), Comment: Published in at http://dx.doi.org/10.1214/09-AOP516 and Errata at http://dx.doi.org/10.1214/11-AOP700 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In K. Yano, Y. Yano and M. Yor (2009), limit theorems for the one-dimensional symmetric $\alpha$-stable process normalized by negative (killing) Feynman-Kac functionals were studied. We consider the same problem and extend their results to positive Feynman-Kac functionals of multi-dimensional symmetric $\alpha$-stable processes.

We introduce a class of interesting stochastic processes based on Brownian-time processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of Brownian motion. They generalize the iterated Brownian motion (IBM) of Burdzy and the Markov snake of Le Gall, and they introduce new interesting examples. After defining Brownian-time processes, we relate them to fourth order parabolic PDEs. We then study their exit problem as they exit nice domains in $\Rd$, and connect it to elliptic PDEs. We show that these processes have the peculiar property that they solve fourth order parabolic PDEs, but their exit distribution - at least in the standard Brownian-time process case - solves the usual second order Dirichlet problem. We recover fourth order PDEs in the elliptic setting by encoding the iterative nature of the Brownian-time process, through its exit time, in a standard Brownian motion. We also show that it is possible to assign a formal generator to these non-Markovian processes by giving such a generator in the half-derivative sense., Comment: 15 pages, 3/9 papers from my 2000-2006 collection (preprint version)

We prove diffusive lower bounds on the rate of escape of the random walk on infinite transitive graphs. Similar estimates hold for finite graphs, up to the relaxation time of the walk. Our approach uses nonconstant equivariant harmonic mappings taking values in a Hilbert space. For the special case of discrete, amenable groups, we present a more explicit proof of the Mok-Korevaar-Schoen theorem on the existence of such harmonic maps by constructing them from the heat flow on a F{\o}lner set., Comment: Published in at http://dx.doi.org/10.1214/12-AOP779 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We describe the set of bounded harmonic functions for the Heckman--Opdam Laplacian, when the multiplicity function is larger than 1/2. We prove that this set is a vector space of dimension the cardinality of the Weyl group. We give some consequences in terms of the associated hypergeometric functions., Comment: [V2]: terminology "radial function" changed in "W-invariant function". One reference added

Recently, in [Preprint (2006)], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups. In this paper, we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and measure-valued potentials in bounded domains. Our process $Y$ is a diffusion process whose generator can be formally written as $L+\mu\cdot\nabla-\nu$ with Dirichlet boundary conditions, where $L$ is a uniformly elliptic second-order differential operator and $\mu=(\mu^1,...,\mu^d)$ is such that each component $\mu^i$, $i=1,...,d$, is a signed measure belonging to the Kato class $\mathbf{K}_{d,1}$ and $\nu$ is a (nonnegative) measure belonging to the Kato class $\mathbf{K}_{d,2}$. We show that scale-invariant parabolic and elliptic Harnack inequalities are valid for $Y$. In this paper, we prove the parabolic boundary Harnack principle and the intrinsic ultracontractivity for the killed diffusion $Y^D$ with measure-valued drift and potential when $D$ is one of the following types of bounded domains: twisted H\"{o}lder domains of order $\alpha\in(1/3,1]$, uniformly H\"{o}lder domains of order $\alpha\in(0,2)$ and domains which can be locally represented as the region above the graph of a function. This extends the results in [J. Funct. Anal. 100 (1991) 181--206] and [Probab. Theory Related Fields 91 (1992) 405--443]. As a consequence of the intrinsic ultracontractivity, we get that the supremum of the expected conditional lifetimes of $Y^D$ is finite., Comment: Published in at http://dx.doi.org/10.1214/07-AOP381 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study the precise relationship between the subordinate killed and killed subordinate processes in the case of an underlying Hunt process, and show that, under minimal conditions, the former is a subprocess of the latter obtained by killing at a terminal time. Moreover, we also show that the killed subordinate process can be obtained by resurrecting the subordinate killed one at most countably many times.

We extend the concept of intrinsic ultracontractivity to non-symmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of non- symmetric second order elliptic operators in bounded Lipschitz domains.

We give potential theoretic estimates for the probability that a set $A$ contains a double point of planar Brownian motion run for unit time. Unlike the probability for $A$ to intersect the range of a Markov process, this cannot be estimated by a capacity of the set $A$. Instead, we introduce the notion of a capacity with respect to two gauge functions simultaneously. We also give a polar decomposition of $A$ into a set that never intersects the set of Brownian double points and a set for which intersection with the set of Brownian double points is the same as intersection with the Brownian path., Published in at http://dx.doi.org/10.1214/009117907000000169 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper we shall derive asymptotic expansions of the Green function and the transition probabilities of Markov additive (MA) processes $(\xi_n, S_n)$ whose first component satisfies Doeblin's condition and the second one takes valued in $Z^d$. The derivation is based on a certain perturbation argument that has been used in previous works in the same context. In our asymptotic expansions, however, not only the principal term but also the second order term are expressed explicitly in terms of a few basic functions that are characteristics of the expansion. The second order term will be important for instance in computation of the harmonic measures of a half space for certain models. We introduce a certain aperiodicity condition, named Condition (AP), that seems a minimal one under which the Fourier analysis can be applied straightforwardly. In the case when Condition (AP) is violated the structure of MA processes will be clarified and it will be shown that in a simple manner the process, if not degenerate, are transformed to another one that satisfies Condition (AP) so that from it we derive either directly or indirectly (depending on purpose) the asymptotic expansions for the original process. It in particular is shown that if the MA processes is irreducible as a Markov process, then the Green function is expanded quite similarly to that of a classical random walk on $Z^d$.

Let G(x,y) and G_D(x,y) be the Green functions of rotationally invariant symmetric \alpha-stable process in R^d and in an open set D respectively, where 00, then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded \kappa-fat open set, which includes a bounded John domain. The 3G we consider is of the form G_D(x,y)G_D(z,w)/G_D(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form G_D(x,y)G_D(z,w)/G_D(x,w) appears in non-local Schrodinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on \kappa-fat open sets. As an application, we discuss relativistic \alpha-stable processes (relativistic Hamiltonian when \alpha=1) in \kappa-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic \alpha-stable processes in \kappa-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in \kappa-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Levy processes., Comment: 32 pages

Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the Revuz measure used in the time change. In this paper we give a complete characterization of time changed processes of an arbitrary symmetric Markov process, in terms of the Beurling--Deny decomposition of their associated Dirichlet forms and of Feller measures of the process. In particular, we determine the jumping and killing measure (or, equivalently, the L\'{e}vy system) for the time-changed process. We further discuss when the trace Dirichlet form for the time changed process can be characterized as the space of finite Douglas integrals defined by Feller measures. Finally, we give a probabilistic characterization of Feller measures in terms of the excursions of the base process., Comment: Published at http://dx.doi.org/10.1214/009117905000000657 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

It is shown in this paper that the transition kernel corresponding to a spatially inhomogeneous random walk on ${\mathbf{Z}}^d$ admits upper and lower Gaussian estimates.

We will propose a sufficient condition which guarantees the pathwise uniqueness for jump type equations in multi-dimensional case. An example given in Section 3 shows that the condition is nearly best possible. Comparing our results with those known in the case of Brownian equations, we claim that essential difference between these two cases. It seems to be remarkable that we could explain these phenomena in the language of the Potential theory. Our principal method in the paper is based on the Fourier analysis, where effective tools such as Bessel functions, hypergeometric functions play essential roles.

Let $X= (X_t) _{t \geq 0}$ be a self-similar Markov process with values in the non-negative half-line, such that the state $0$ is a trap. We present a necessary and sufficient condition for the existence of a self-similar recurrent extension of $X$ that leaves $0$ continuously. This condition is expressed in terms of the Levy process associated with $X$ by the Lamperti transformation.

We give a sufficient condition to construct non-trivial $\mu$-symmetric diffusion processes on a locally compact separable metric measure space $(M,\rho , \mu )$. These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of $\Gamma$-limits for approximating non-local Dirichlet forms. For various fractals, we can use existing estimates to verify our assumptions. This shows that our general method of constructing diffusions can be applied to these fractals.

We extend the classical Douglas integral, which expresses the Dirichlet integral of a harmonic function on the unit disk in terms of its value on boundary, to the case of conservative symmetric diffusion in terms of Feller measure, by using the approach of time change of Markov processes., Comment: Published at http://dx.doi.org/10.1214/009117904000000649 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In the context of a strongly local Dirichlet space we show that if a function mapping the real line to itself (and fixing the origin) operates by composition on the left to map the Dirichlet space into itself, then the function is necessarily locally Lipschitz continuous. If, in addition, the Dirichlet space contains unbounded elements, then the function must be globally Lipschitz continuous. The proofs rely on a co-area formula for condenser potentials.

We study Girsanov's theorem in the context of symmetric Markov processes, extending earlier work of Fukushima-Takeda and Fitzsimmons on Girsanov transformations of ``gradient type.'' We investigate the most general Girsanov transformation leading to another symmetric Markov process. This investigation requires an extension of the forward-backward martingale method of Lyons-Zheng, to cover the case of processes with jumps., Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000432

Starting from a central limit theorem for geometric random walks we give an elementary construction of couplings between Brownian motions on Riemannian manifolds. This approach shows that cut locus phenomena are indeed inessential for Kendall's and Cranston's stochastic proof of gradient estimates for harmonic functions on Riemannian manifolds with lower curvature bounds. Moreover, since the method is based on an asymptotic quadruple inequality and a central limit theorem only it may be extended to certain non smooth spaces which we illustrate by the example of Riemannian polyhedra. Here we also recover the classical heat kernel gradient estimate which is well known from the smooth setting.

The history of stochastic integration and the modelling of risky asset prices both begin with Brownian motion, so let us begin there too. The earliest attempts to model Brownian motion mathematically can be traced to three sources, each of which knew nothing about the others: the first was that of T. N. Thiele of Copenhagen, who effectively created a model of Brownian motion while studying time series in 1880 [80]. 1 ; the second was that of L. Bachelier of Paris, who created a model of Brownian motion while deriving the dynamic behavior of the Paris stock market, in 1900 (see, [1, 2, 11]); and the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of matter, in 1905 [21](See [63] for a discussion of Einstein’s model and his motivations.) Of these three models, those of Thiele and Bachelier had little impact for a long time, while that of Einstein was immediately influential. We go into a little detail about what happened to Bachelier, since he is now seen by many as the founder of modern Mathematical Finance. Ignorant of the work of Thiele (which was little appreciated in its day) and preceding the work of Einstein, Bachelier attempted to model the market noise of the Paris Bourse. Exploiting the ideas of the Central Limit Theorem, and realizing that market noise should be without memory, he reasoned that increments of stock prices should be independent and normally distributed. He combined his reasoning with the Markov property and semigroups, and connected Brownian motion with the heat equation, using that the Gaussian kernel is the fundamental solution to the heat equation. He was able to define other processes related to Brownian motion, such as the maximum change during a time interval (for one dimensional Brownian motion), by using random walks and letting the time steps go to zero, and by then taking

The potential kernel of a positive left additive functional (of a strong Markov process $X$) maps positive functions to strongly supermedian functions and satisfies a variant of the classical domination principle of potential theory. Such a kernel $V$ is called a regular strongly supermedian kernel in recent work of L. Beznea and N. Boboc. We establish the converse: Every regular strongly supermedian kernel $V$ is the potential kernel of a random measure homogeneous on $[0,\infty[$. Under additional finiteness conditions such random measures give rise to left additive functionals. We investigate such random measures, their potential kernels, and their associated characteristic measures. Given a left additive functional $A$ (not necessarily continuous), we give an explicit construction of a simple Markov process $Z$ whose resolvent has initial kernel equal to the potential kernel $U_{\!A}$. The theory we develop is the probabilistic counterpart of the work of Beznea and Boboc. Our main tool is the Kuznetsov process associated with $X$ and a given excessive measure $m$.

For a Markov process associated with a not necessarily symmetric regular Dirichlet form, if the form satisfies the sector condition, then any semipolar sets are exceptional. On the other hand, in the case of the space-time Markov process associated with a family of time dependent Dirichlet forms, there exist non-exceptional semipolar sets. The main purpose of this paper is to show that any semipolar set $B=J\times \Gamma$ of the direct product type of a subset $J$ of time and a subset $\Gamma$ of space is exceptional if $J$ has positive Lebesgue measure.

We define a sequence of mappings $\Gamma_k:D_0(R_+)^k\to D_0(R_+)^k$ and prove the following result: Let $N_1,\ldots,N_n$ be the counting functions of independent Poisson processes on $R_+$ with respective intensities $\mu_1 \lt \mu_2 \lt \cdots \lt \mu_n$. The conditional law of $N_1,\ldots,N_n$, given that $$N_1(t)\le\cdots\le N_n(t), \mbox{ for all }t\ge 0,$$ is the same as the unconditional law of $\Gamma_n(N)$. From this, we deduce the corresponding results for independent Poisson processes of equal rates and for independent Brownian motions (in both of these cases the conditioning is in the sense of Doob). This extends a recent observation, independently due to Baryshnikov (2001) and Gravner, Tracy and Widom (2001), which relates the law of a certain functional of Brownian motion to that of the largest eigenvalue of a GUE random matrix. Our main result can also be regarded as a generalisation of Pitman's representation for the 3-dimensional Bessel process.

In the frame of Borel right Markov processes, we investigate, following an analytical point of view, the Revuz correspondence between classes of potential kernels and their associated measures, improving upon the results of Revuz, Azema, Getoor and Sharpe, Fitzsimmons, Fitzsimmons and Getoor and Dellacherie, Maisonneuve and Meyer. In the probabilistic approach of the problem, the kernels that occur are the potential operators of different types of homogeneous random measures. We completely characterize the hypothesis (B) of Hunt in terms of Revuz measures.