Excursion theory for rotation invariant Markov processes

Summary Let (X _t,P ^x) be a rotation invariant (RI) strong Markov process onR ^d{0} having a skew product representation [|X _t |, ], where ( _t ) is a time homogeneous, RI strong Markov process onS ^d–1, |X _t|, and _t are independent underP ^x andA _t is a continuous additive functional of |X _t|. We characterize the rotation invariant extensions of (X _t,P ^x) toR ^d. Two examples are given: the diffusion case, where especially the Walsh's Brownian motion (Brownian hedgehog) is considered, and the case where (X _t,P ^x) is self-similar.     

Ornstein–Uhlenbeck Path Integral and Its Application

Introducing a path integral for the Ornstein–Uhlenbeck process distorted by a potential V(x), we find out the T limit of the probability distributions of X[]:=1/T ₀ ^T V((t))dt for Ornstein–Uhlenbeck process (t), with appropriate values of the exponent that depend on V. The results are compared with those for the Wiener process.     

Large Deviations of Local Times of Lévy Processes

For X(t) a real-valued symmetric Lévy process, its characteristic function is E(e ^iX(t))=exp(–t()). Assume that is regularly varying at infinity with index 1<2. Let L ^x _t denote the local time of X(t) and L* _t =sup _xR L ^x _t . Estimates are obtained for P(L ⁰ _t y) and P(L* _t y) as y and t fixed.     

Some Conditions for a C0-Semigroup to Be Asymptotically Finite-Dimensional

We study the class of bounded C ₀-semigroups T=(T _t ) _t0 on a Banach space X satisfying the asymptotic finite dimensionality condition: codim X ₀(T)<, where X ₀(T):={x X:lim_t T _t x=0}. We prove a theorem which provides some necessary and sufficient conditions for asymptotic finite dimensionality.     

Multivalued Stochastic Differential Equations: Convergence of a Numerical Scheme

In this paper we show the strong mean square convergence of a numerical scheme for a R ^d -multivalued stochastic differential equation: dX _t +A(X _t )dtb(t,X _t )dt+(t,X _t )dW _t and obtain the rate of convergence O(( log(1/)^1/2) when the diffusion coefficient is bounded. By introducing a discrete Skorokhod problem, we establish L ^p -estimates (p2) for the solutions and prove the convergence by using a deterministic result. Numerical experiments for the rate of convergence are presented.     

Donsker's Delta Function of Lévy Process

Let X={X(t):tR} be a Lévy process and a non-decreasing, right continuous, bounded function with (–)=0 (((1+u ²)/u ²)d(u) is the Lévy measure). In this paper we define the Donsker delta function (X(t)–a), t>0 and aR, as a generalized Lévy functional under the condition that (0)–(0–)>0. This leads us to define F(X(t)) for any tempered distribution F, and as an application, we derive an Itô formula for F(X(t)) when has jumps at 0 and 1.     

Interacting Brownian particles and the Wigner law

Summary In this paper, we study interacting diffusing particles governed by the stochastic differential equationsdX _j (t)= _n dB _j (t) –D _jØ_n(X ₁,...,X _n)dt,j=1, 2,...,n. Here theB _jare independent Brownian motions in ^d , and Ø _n (X ₁,...,X _n)= _{n ij} V(X _iX _j) + _ni U(X ₁). The potentialV has a singularity at 0 strong enough to keep the particles apart, and the potentialU serves to keep the particles from escaping to infinity. Our interest is in the behaviour as the number of particles increases without limit, which we study through the empirical measure process. We prove tightness of these processes in the case ofd=1,V(x)=–log|x|,U(x)=x ²/2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure-valued process exists. This process is deterministic, and converges to the Wigner law ast. Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.Supported by SERC grant number GR/H 00444     

A characterization of brownian motion in a lipschitz domain by its killing distributions

Let (Y _t, Q^x) be a strong Markov process in a bounded Lipschitz domainD with continuous paths up to its lifetime , and let (X _t, P^x) be a Brownian motion inD. IfY _– exists in D andQ ^x(Y_– C)=P^x(X_– C) for all Borel subsetsC of D and allx, thenY is a time change ofX.     

Galerkin methods for parabolic equations with nonlinear boundary conditions

A variety of Galerkin methods are studied for the parabolic equationu _t =(a(x) u),x ⁿ ,t (O,T], subject to the nonlinear boundary conditionu _v =g(x,t,u),x,t (O,T] and the usual initial condition. Optimal order error estimates are derived both inL ² () andH ¹ () norms for all methods treated, including several that produce linear computational procedures.The authors were partially supported by The National Science Foundation during the preparation of this paper.     

Semilinear Perturbations of Harmonic Spaces,Liouville Property and a Boundary Value Problem

Let (X,) be a P-harmonic Bauer space and let be a Borel measurable function on X×R satisfying conditions (A) through (D) of Section 2 (e.g., (x,t)=t|t|^–1 where >1). For every Kato family M of potential kernels on X let ^M U(X) denote the set of all real continuous functions on X such that u+K ^M _D (,u)(D) for every open relatively compact subset D of X. We study the existence of a non-trivial function in ^M U(X) which is dominated by a given positive harmonic function on X. If X is a domain of R ^d , is a positive Kato measure on X and L is a second-order differential operator in R ^d , we apply our study to derive a characterization of finite positive measures on the minimal Martin boundary ^M ₁ X for which the boundary value problem Lu=(,u) in X and u= on ^M ₁ X is solvable.     

Walsh's Brownian Motion-Type of Extensions

Let (X _t ) be a rotation invariant Feller process on the state space F R²{} consisting of finite number of rays, meeting at 0. We study a certain class of possible strong Markov extensions of (X _t ) to F ,{} given the corresponding radial extension to [0, ). A well-known example is the class of Walsh's Brownian motions, in the case where (X _t ) is the Brownian motion on F. It turns out that while the symmetric extension of Walsh's Brownian motion-type always exists, the non-symmetric extension exists iff (X _t ), roughly speaking, does not jump from one ray to another before hitting 0.     

Subsolutions for abstract evolution equations

We study different notions of subsolutions for an abstract evolution equation du/dt+Auf where A is an m-accretive nonlinear operation in an ordered Banach space X with order-preserving resolvents. A first notion is related to the operator d/dt+A in the ordered Banach space L ¹(0, T; X); a second one uses the evolution equation du/dt+A uf where A :x{y;zy for some zAx}; other notions are also considered.     

p-Frames in Separable Banach Spaces

Let X be a separable Banach space with dual X ^*. A countable family of elements {g _i }X ^* is a p-frame (1 p ) if the norm _X is equivalent to the ^p -norm of the sequence {g _i ()}. Without further assumptions, we prove that a p-frame allows every gX ^* to be represented as an unconditionally convergent series g=d _i g _i for coefficients {d _i } ^q , where 1/p+1/q=1. A p-frame {g _i } is not necessarily linear independent, so {g _i } is some kind of overcomplete basis for X ^*. We prove that a q-Riesz basis for X ^* is a p-frame for X and that the associated coefficient functionals {f _i } constitutes a p-Riesz basis allowing us to expand every fX (respectively gX ^*) as f=g _i (f)f _i (respectively g=g(f _i )g _i ). In the general case of a p-frame such expansions are only possible under extra assumptions.     

A continuity property for the inverse of mañé's projection

Let X be a compact subset of a separable Hilbert space H with finite fractal dimension d _F(X), and P ₀an orthogonal projection in H of rank greater than or equal to 2 d _F (X) + 1. For every > 0, there exists an orthogonal projection P in H of the same rank as P ₀, which is injective when restricted to X and such that P – P ₀ < This result follows from Mañé's paper. Thus the inverse (P|_X)^–1 of the restricted mapping P|_X: X PX is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse of Mañé's projection (P| _X )^–1. It is known that when H is finite dimensional then (P| _X )^–1 is Hölder continuous. In this paper we shall prove that if X is a global attractor of an infinite dimensional dissipative evolutionary equation then in some cases (e.g. two-dimensional Navier-Stokes equations with homogeneous Dirichlet boundary conditions) for every x, y X such that , where is a positive constant.     

Cohomologies and analytic differential forms

A proof that homology groups H^k(x; _X) of a complex analytic space X, countable at infinity and locally smoothly contractible, with coefficients in the lattice bundle _X, are canonically isomorphic to the corresponding homology groups of the finite complex of analytic differential forms with the exterior differential d as a coboundary operator.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 569–573, May, 1971.     

Asymptotic properties of solutions of multidimensional stochastic differential equations

Summary LetX _t R ^d be the solution of the stochastic equationdX _t =b(X _t )dt+(X _t )dW _t , whereW _t denotes a standard Wiener process. The aim of the paper is to clarify under which conditions the drift term or the diffusion term is of negligible significance for the long term behaviour ofX _t .     

Perturbations of Generalized Mehler Semigroups and Applications to Stochastic Heat Equations with Levy Noise and Singular Drift

In this paper we solve the Kolmogorov equation and, as a consequence, the martingale problem corresponding to a stochastic differential equation of type dX _t =AX _t dt+b(X _t )dt+dY _t , on a Hilbert space E, where (Y _t ) _t0 is a Levy process on E,A generates a C ₀-semigroup on E and b:EE. Our main point is to allow unbounded A and also singular (in particular, non-continuous) b. Our approach is based on perturbation theory of C ₀-semigroups, which we apply to generalized Mehler semigroups considered on L ²(), where is their respective invariant measure. We apply our results, in particular, to stochastic heat equations with Levy noise and singular drift.     

Brownian charges around loops

Summary Let (X _t ⁿ ) be a Poisson sequence of independent Brownian motions in ^d ,d3; Let be a compact oriented submanifold of d, of dimensiond–2 and volume ; let t be the sum of the windings of (X _s ⁿ , 0st) around ; then t/t converges in law towards a Cauchy variable of parameter /2. A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in ^d .     

Linearisable Third-Order Ordinary Differential Equations and Generalised Sundman Transformations: The Case X ′′′=0

We calculate in detail the conditions which allow the most general third-order ordinary differential equation to be linearised in X (T)=0 under the transformation X(T)=F(x,t), dT=G(x,t)dt.     

Strong approximation of additive functionals

LetX ₁,X ₂,...be a sequence of i.i.d. random variables and putS ₀=0,S _n =X ₁+...+X _n . A strong approximation type result is given forA _N = _i=1 ^N f(S _i ) whereF(x),xR is a real valued function. A similar result is given for ₀ ^t g(B(s))ds. Some weak convergence type implications are also discussed.1991 Mathematics Subject Classification: Primary: 60F15, Secondary: 60J15.Supported by the Hungarian National Foundation for Scientific Research, Grant No. 1905.Supported by an NSERC Canada Grant at Carleton University.Supported by a PSC CUNY Grant No.662349.