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.     

The asymptotic behaviour of local times and occupation integrals of theN-parameter Wiener process in R d

Summary LetL(x, T),xR ^d ,TR ₊ ^N , be the local time of theN-parameter Wiener processW taking values inR ^d . Even in the distribution valued casedd2N,L can be described in a series representation by means of multiple Wiener-Ito integrals. This setting proves to be a good starting point for the investigation of the asymptotic behaviour ofL(x, T) as |x|0 and/orT and of related occupation integrals asT. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws forL(x, T) resp.X _T (f), and of the second order, i.e. normalized convergence laws forL(x, T)–E(L(x, T)) resp.X _T (f)–E(X _T (f)).This research was made during a stay at the LMU in München supported by DAAD     

Geography of the level sets of the Brownian sheet

Summary We describe geometric properties of {W>}, whereW is a standard real-valued Brownian sheet, in the neighborhood of the first hitP of the level set {W>} along a straight line or smooth monotone curveL. In such a neighborhood we use a decomposition of the formW(s, t)=–b(s)+B(t)+x(s, t), whereb(s) andB(t) are particular diffusion processes andx(s, t) is comparatively small, to show thatP is not on the boundary of any connected component of {W>}. Rather, components of this set form clusters nearP. An integral test for thorn-shaped neighborhoods ofL with tip atP that do not meet {W>} is given. We then analyse the position and size of clusters and individual connected components of {W>} near such a thorn, giving upper bounds on their height, width and the space between clusters. This provides a local picture of the level set. Our calculations are based on estimates of the length of excursions ofB andb and an accounting of the error termx.The research of this author was partially supported by NSF grant DMS-9103962, and, during the period of revision, by grant DAAL03-92-6-0323 from the Army Research Office     

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     

Skorokhod embeddings, minimality and non-centred target distributions

In this paper we consider the Skorokhod embedding problem for target distributions with non-zero mean. In the zero-mean case, uniform integrability provides a natural restriction on the class of embeddings, but this is no longer suitable when the target distribution is not centred. Instead we restrict our class of stopping times to those which are minimal, and we find conditions on the stopping times which are equivalent to minimality. We then apply these results, firstly to the problem of embedding non-centred target distributions in Brownian motion, and secondly to embedding general target laws in a diffusion. We construct an embedding (which reduces to the Azema-Yor embedding in the zero-target mean case) which maximises the law of sup_s≤TB_s among the class of minimal embeddings of a general target distribution μ in Brownian motion. We then construct a minimal embedding of μ in a diffusion X which maximises the law of sup_s≤Th(X_s) for a general function h.     

Multilevel large deviations and interacting diffusions

Summary Let ( ^N ) be a sequence of random variables with values in a topological space which satisfy the large deviation principle. For eachM and eachN, let ^{M, N} denote the empirical measure associated withM independent copies of ^N . As a main result, we show that ( ^{M, N} ) also satisfies the large deviation principle asM,N. We derive several representations of the associated rate function. These results are then applied to empirical measure processes ^{M, N} (t) =M ^–1 _i=1 ^N _i ^N (t) 0tT, where ( ₁ ^N ,..., _M ^N (t)) is a system of weakly interacting diffusions with noise intensity 1/N. This is a continuation of our previous work on the McKean-Vlasov limit and related hierarchical models ([4], [5]).Research partially supported by a Natural Science and Engineering Research Council of Canada operating grant     

The law of the Euler scheme for stochastic differential equations

Summary We study the approximation problem ofE f(X _T ) byE f(X _T ⁿ ), where (X _t ) is the solution of a stochastic differential equation, (X _T ⁿ ) is defined by the Euler discretization scheme with stepT/n, andf is a given function. For smoothf's, Talay and Tubaro have shown that the errorE f(X _T ) –f(X _T ⁿ ) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation precedures to accelerate the convergence rate. Here, we prove that the expansion exists also whenf is only supposed measurable and bounded, under an additional nondegeneracy condition of Hörmander type for the infinitesimal generator of (X _t ): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law ofX _T ⁿ and compare it to the density of the law ofX _T .     

The structure of a Brownian bubble

Summary We examine local geometric properties of level sets of the Brownian sheet, and in particular, we identify the asymptotic distribution of the area of sets which correspond to excursions of the sheet high above a given level in the neighborhood of a particular random point. It is equal to the area of certain individual connected components of the random set {(s, t):B(t)>b(s)}, whereB is a standard Brownian motion andb is (essentially) a Bessel process of dimension 3. This limit distribution is studied and, in particular, explicit formulas are given for the probability that a point belongs to a specific connected component, and for the expected area of a component given the height of the excursion ofB(t)-b(s) in this component. These formulas are evaluated numerically and compared with the results from direct simulations ofB andb.The research of this author was partially supported by grants DMS-9103962 from the National Science Foundation and DAAL03-92-6-0323 from the Army Research Office     

Strengthening classical results on convergence rates in strong limit theorems

Let X₁, X₂, . . . be i.i.d. random variables, and set S_n=X₁+ . . . +X_n. Several authors proved convergence of series of the type f(ɛ)=∑_nc_nP(|S_n|>ɛa_n),ɛ>α, under necessary and sufficient conditions. We show that under the same conditions, in fact i.e. the finiteness of ∑_nc_nP(|S_n|>ɛa_n),ɛ>α, is equivalent to the convergence of the double sum ∑_k∑_nc_nP(|S_n|>ka_n). Two exceptional series required deriving necessary and sufficient conditions for E[sup_n|S_n|(logn)^η/n]<∞,0≤η≤1.     

A local limit theorem for random walks conditioned to stay positive

We consider a real random walk S_n=X₁+...+X_n attracted (without centering) to the normal law: this means that for a suitable norming sequence a_n we have the weak convergence S_n/a_n⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S₁>0,...,S_n>0) and let S_n⁺ denote the random variable S_n conditioned on : it is known that S_n⁺/a_n ↠ ϕ⁺(x) dx, where ϕ⁺(x):=x exp (−x²/2)1_(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X₁ has an absolutely continuous law: in this case the uniform convergence of the density of S_n⁺/a_n towards ϕ⁺(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation Theory for random walks.     

N/V-limit for stochastic dynamics in continuous particle systems

We provide an N/V-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on ℝ ^d ,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝ ^d with finite volume (Lebesgue measure) V=|Λ|<∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above N-particle dynamic in Λ as N→∞ and V→∞ such that N/V→ρ, where ρ is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate relation between N and V. Then tightness is shown by using the Lyons–Zheng decomposition. The equilibrium measures of the accumulation points are identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle, infinite volume stochastic dynamics via the associated martingale problem. Assuming a property closely related to Markov uniqueness and weaker than essential self-adjointness, via Mosco convergence techniques we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials ϕ of Ruelle type and all temperatures, densities, and dimensions d≥1, respectively. ϕ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides as a by-product an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary condition.     

Regularity of Skorohod integral processes based on integrands in a finite Wiener chaos

Summary Letf be a square integrable kernel on them-dimensional unit cube,U the Skorohod integral process in them ^th Wiener chaos associated with it. Isoperimetric inequalities for functions on Wiener space yield the exponential integrability of the increments ofU. To this result we apply the majorizing measure technique to show thatU possesses a continuous version and give an upper bound of its modulus of continuity.     

Sample function behavior of increasing processes of class L

Summary We consider increasing processes {X(t)t0} of classL, that is, increasing self-similar processes with inswpendent increments. Leth(t) be an increasing positive function on (0,) withh(0+)=0 andh()=. By virtue of the zero-one laws, there existsc (resp.C) [0,] such that lim inf (resp. lim sup)X(t)/h(t)=c (resp.C) a.s. both ast tends to 0 and ast tends to . We decide a necessary and sufficient condition for the existence ofh(t) withc orC=1 and explicitly constructh(t) in caseh(t) exists withc orC=1. Moreover, we give a criterion to classify functionsh(t) withc (orC)=0 andh(t) withc (orC)= in caseh(t) does not exist withc (orC)=1.     

Passage times of random walks and Lévy processes across power law boundaries

We establish an integral test involving only the distribution of the increments of a random walk S which determines whether limsup _n→∞(S_n/n^κ) is almost surely zero, finite or infinite when 1/2<κ<1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller [9] concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of κ≥0. The results, and those of [9], are also extended to Lévy processes.This work is partially supported by ARC Grant DP0210572.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

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.     

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.     

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.     

Sur la mesure de sortie du super mouvement brownien

Summary We study some properties of the exit measure of super Brownian motion from a smooth domainD inR ^d . In particular, we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary. As an application, we compute the Hausdorff dimension of the support of the exit measure. In dimension 2, we prove that the exit measure is absolutely continuous with respect to the Lebesgue measure on the boundary. In connection with Dynkin's work, our results give some information on the behavior of solutions of u=u ² inD, and are related to the characterization of removable singularities at the boundary. As a consequence of our estimates, we give a sufficient condition for the uniqueness of the positive solution of u=u ² inD that tends to on an open subsetO of D and to 0 on the complement in D of the closure ofO. Our proofs use the path-valued process studied in [L2, L3].

     

Conditioned super-Brownian motion

Summary We investigate classes of conditioned super-Brownian motions, namely H-transformsP ^H with non-negative finitely-based space-time harmonic functionsH(t, ). We prove thatH ^H is the unique solution of a martingale problem with interaction and is a weak limit of a sequence of rescaled interacting branching Brownian motions. We identify the limit behaviour of H-transforms with functionsH(t, )=h(t, (1)) depending only on the total mass (1). Using the Palm measures of the super-Brownian motion we describe for an additive spacetime harmonic functionH(t, )=h(t, x) (dx) theH-transformP ^H as a conditioned super-Brownian motion in which an immortal particle moves like an h-transform of Brownian motion.