Asymptotically one-dimensional diffusions on the Sierpinski gasket and theabc-gaskets

Summary Diffusion processes on the Sierpinski gasket and theabc-gaskets are constructed as limits of random walks. In terms of the associated renormalization group, the present method uses the inverse trajectories which converge to unstable fixed points corresponding to the random walks on one-dimensional chains. In particular, non-degenerate fixed points are unnecessary for the construction. A limit theorem related to the discrete-time multi-type non-stationary branching processes is applied.     

Estimates of transition densities for Brownian motion on nested fractals

Summary We obtain upper and lower bounds for the transition densities of Brownian motion on nested fractals. Compared with the estimate on the Sierpinski gasket, the results require the introduction of a new exponent,d _J, related to the shortest path metric and chemical exponent on nested fractals. Further, Hölder order of the resolvent densities, sample paths and local times are obtained. The results are obtained using the theory of multi-type branching processes.     

Brownian motion on a homogeneous random fractal

Summary We introduce a simple random fractal based on the Sierpinski gasket and construct a Brownian motion upon the fractal. The properties of the process on the Sierpinski gasket are modified by the random environment. A sample path construction of the process via time truncation is used, which is a direct construction of the process on the fractal from the associated Dirichlet forms. We obtain estimates on the resolvent and transition density for the process and hence a value for the spectral dimension which satisfiesd _s=2d _f/d_w. A branching process in a random environment can be used to deduce some of the sample path properties of the process.     

On reflecting diffusion processes and Skorokhod decompositions

Summary LetG be ad-dimensional bounded Euclidean domain, H¹ (G) the set off in L²(G) such that f (defined in the distribution sense) is in L²(G). Reflecting diffusion processes associated with the Dirichlet spaces (H¹(G), ) on L²(G, dx) are considered in this paper, where A=(a^ij is a symmetric, bounded, uniformly ellipticd×d matrix-valued function such thata ^ij H¹(G) for eachi,j, and H¹(G) is a positive bounded function onG which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with (H¹(G), ) having starting points inG under a mild condition which is satisfied when G has finite (d–1)-dimensional lower Minkowski content.     

Properties of solutions of stochastic differential equations with continuous-state-dependent switching

This work is concerned with several properties of solutions of stochastic differential equations arising from hybrid switching diffusions. The word “hybrid” highlights the coexistence of continuous dynamics and discrete events. The underlying process has two components. One component describes the continuous dynamics, whereas the other is a switching process representing discrete events. One of the main features is the switching component depending on the continuous dynamics. In this paper, weak continuity is proved first. Then continuous and smooth dependence on initial data are demonstrated. In addition, it is shown that certain functions of the solutions verify a system of Kolmogorov's backward differential equations. Moreover, rates of convergence of numerical approximation algorithms are dealt with.     

The skorokhod oblique reflection problem in a convex polyhedron

The Skorokhod oblique reflection problem is studied in the case ofn-dimensional convex polyhedral domains. The natural sufficient condition on the reflection directions is found, which together with the Lipschitz condition on the coefficients gives the existence and uniqueness of the solution. The continuity of the corresponding solution mapping is established. This property enables one to construct in a direct way the reflected (in a convex polyhedral domain) diffusion processes possessing the nice properties.     

The Brownian snake and solutions of Δu=u 2 in a domain

Summary We investigate the connections between the path-valued process called the Brownian snake and nonnegative solutions of the partial differential equation u=u ² in a domain of ^d . In particular, we prove two conjectures recently formulated by Dynkin. The first one gives a complete characterization of the boundary polar sets, which correspond to boundary removable singularities for the equation u=u ². The second one establishes a one-to-one correspondence between nonnegative solutions that are bounded above by a harmonic function, and finite measures on the boundary that do not charge polar sets. This correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the Brownian snake. Our proofs combine probabilistic and analytic arguments. An important role is played by a new version of the special Markov property, which is of independent interest.     

On the semimartingale representation of reflecting Brownian motion in a cusp

Summary LetC be the symmetric cusp {(x, y)²:–x yx ,x0} where >1. In this paper we decide whether or not reflecting Brownian motion inC has a semimartingale representation. Here the reflecting Brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary. Our results carry through for a wide class of asymmetric cusps too.     

Dynkin's isomorphism theorem and the Ray-Knight theorems

Summary In the case of diffusions, we show that the isomorphism theorem of Dynkin and the Ray-Knight theorems can be derived from each other. Our proof uses additivity properties of squared Bessel processes and an absolute continuity relation between squared Bessel processes of dimensions one and three.Research supported in part by Air Force Office of Scientific Research (USAFOSR 89-0261)This work was carried out while visiting the Technion     

Markov snakes and superprocesses

Summary We suggest the name Markov snakes for a class of path-valued Markov processes introduced recently by J.-F. Le Gall in connection with the theory of branching measure-valued processes. Le Gall applied this class to investigate path properties of superdiffusions and to approach probabilistically partial differential equations involving a nonlinear operator v–v ². We establish an isomorphism theorem which allows to translate results on continuous superprocesses into the language of Markov snakes and vice versa. By using this theorem, we get limit theorems for discrete Markov snakes.Partially supported by National Science Foundation Grant DMS-9301315 and by The US Army Research Office through the Mathematical Sciences Institute at Cornell University     

A class of path-valued Markov processes and its applications to superprocesses

Summary Let ( _s ) be a continuous Markov process satisfying certain regularity assumptions. We introduce a path-valued strong Markov process associated with ( _s ), which is closely related to the so-called superprocess with spatial motion ( _s ). In particular, a subsetH of the state space of ( _s ) intersects the range of the superprocess if and only if the set of paths that hitH is not polar for the path-valued process. The latter property can be investigated using the tools of the potential theory of symmetric Markov processes: A set is not polar if and only if it supports a measure of finite energy. The same approach can be applied to study sets that are polar for the graph of the superprocess. In the special case when ( _s ) is a diffusion process, we recover certain results recently obtained by Dynkin.     

The law of iterated logarithm for additive functionals and martingale additive functionals of Harris recurrent Markov processes

We use Nummelin splitting in continuous time in order to prove laws of iterated logarithm for additive functionals of a Harris recurrent Markov process, with deterministic or random renormalization.     

Differentiability preserving properties of a class of semigroups

It is well known that for a large class of Markov process the associated semi-group T(t)f(x)=f(y)P(t,x;dy) satisfies the Kolmogorov backward differential equation, that is, if u(t,x)=T(t)f(x) then and .In this paper we are considering the opposite problem: given the diffusion and drift coefficients we study the differentiability preserving properties of the semigroup T(t) having as infinitesimal generator .More specifically, for a large class of functions a(x) and b(x), we will prove for k=0, ..., 3 the existence of T(t) such that T(t): C ^k (I) C ^k (I) and the existence of a constant _k such that |T(t)f| _k |f| _k exp ( _k t) for fC ^k (I). Moreover an explicit expression of _k in terms of the coefficients a(x) and b(x) is obtained. As a side result we obtain the necessity of the boundary conditions imposed.This paper is a revised version of the author's Ph. D. dissertation at University of Massachusetts under W. Rosenkrantz     

Harnack inequality on configuration spaces: The coupling approach and a unified treatment

In this paper, we establish the dimension-free Harnack inequality on configuration spaces by using the coupling argument. Furthermore, a unified treatment is also used to prove the equivalence between the Harnack inequality on configuration space and that on the corresponding base space under a very mild condition.     

Harnack inequalities for Ornstein-Uhlenbeck processes driven by Lévy processes

By using the existing sharp estimates of the density function for rotationally invariant symmetric α-stable Lévy processes and rotationally invariant symmetric truncated α-stable Lévy processes, we obtain that the Harnack inequalities hold for rotationally invariant symmetric α-stable Lévy processes with α∈(0,2) and Ornstein-Uhlenbeck processes driven by rotationally invariant symmetric α-stable Lévy process, while the logarithmic Harnack inequalities are satisfied for rotationally invariant symmetric truncated α-stable Lévy processes.     

A duality relation for entrance and exit laws for Markov processes

Markov processes X_t on (X, F_X) and Y_t on (Y, F_Y) are said to be dual with respect to the function f(x, y) if E_xf(X_t, y) = E_yf(x, Y_t for all x ? X, y ? Y, t ? 0. It is shown that this duality reverses the role of entrance and exit laws for the processes, and that two previously published results of the authors are dual in precisely this sense. The duality relation for the function f(x, y) = 1_{x<y} is established for one-dimensional diffusions, and several new results on entrance and exit laws for diffusions, birth-death processes, and discrete time birth-death chains are obtained.     

Some zero-one laws for additive functionals of Markov processes

Summary Let (X _t,P ^x ) be anm-symmetric Markov process with a strictly positive transition density. Consider the additive functionalA _t : = ₀ ^t f (X _s ) wheref:E[0, ] is a universally measurable function on the state spaceE. Among others, we prove thatP ^x (A _t <)=1, for somexE and somet>0, already impliesP ^x (A _t <)=1, for quasi everyxE and allt>0. The latter is also equivalent toP ^x (A _t <)>0, for quasi everyxE and allt>0, and to the analytic condition , for a sequence of finely open Borel setsF _n such thatEF _n is polar. In the special cases of Brownian motion and Bessel process, these results were obtained earlier by H.J. Engelbert, W. Schmidt, X.-X. Xue and the authors.     

Conditioned Brownian motion in planar domains

Summary We give an upper bound for the Green functions of conditioned Brownian motion in planar domains. A corollary is the conditional gauge theorem in bounded planar domains.Supported in part by NSF grant DMS-9100244 and an AMS Centennial Fellowship