Exit time moments, boundary value problems, and the geometry of domains in Euclidean space

Let X _t be a diffusion in Euclidean space. We initiate a study of the geometry of smoothly bounded domains in Euclidean space using the moments of the exit time for particles driven by X _t , as functionals on the space of smoothly bounded domains. We provide a characterization of critical points for each functional in terms of an overdetermined boundary value problem. For Brownian motion we prove that, for each functional, the boundary value problem which characterizes critical points admits solutions if and only if the critical point is a ball, and that all critical points are maxima. Received: 23 January 1997 / Revised version: 21 January 1998     

Stochastic differential equations for Dirichlet processes

We consider the stochastic differential equation dX _t = a(X _t )dW _t + b(X _t )dt, where W is a one-dimensional Brownian motion. We formulate the notion of solution and prove strong existence and pathwise uniqueness results when a is in C ^1/2 and b is only a generalized function, for example,the distributional derivative of a H?lder function or of a function of bounded variation. When b = aa′, that is, when the generator of the SDE is the divergence form operator ℒ = , a result on non-existence of a strong solution and non-pathwise uniqueness is given as well as a result which characterizes when a solution is a semimartingale or not. We also consider extensions of the notion of Stratonovich integral. Received: 23 February 2000 / Revised version: 22 January 2001 / Published online: 23 August 2001     

Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions

Let X be a Lévy process in, , obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.     

Potential theory of subordinate killed Brownian motion in a domain

 Subordination of a killed Brownian motion in a bounded domain D⊂ℝ ^d via an α/2-stable subordinator gives a process Z _t whose infinitesimal generator is −(−Δ| _D )^α/2, the fractional power of the negative Dirichlet Laplacian. In this paper we study the properties of the process Z _t in a Lipschitz domain D by comparing the process with the rotationally invariant α-stable process killed upon exiting D. We show that these processes have comparable killing measures, prove the intrinsic ultracontractivity of the generator of Z _t , prove the intrinsic ultracontractivity of the semigroup of Z _t , and, in the case when D is a bounded C ^1,1 domain, obtain bounds on the Green function and the jumping kernel of Z _t . Received: 4 April 2002 / Revised version: 1 July 2002 / Published online: 19 December 2002 This work was completed while the authors were in the Research in Pairs program at the Mathematisches Forschungsinstitut Oberwolfach. We thank the Institute for the hospitality. The research of the first author is supported in part by NSF Grant DMS-9803240. The research of the second author is supported in part by MZT grant 037008 of the Republic of Croatia. Mathematics Subject Classification (2000): Primary 60J45; Secondary 60J75, 31C25 Key words or phrases: Killed Brownian motions – Stable processes – Subordination – Fractional Laplacian     

The most visited sites of symmetric stable processes

Let X be a symmetric stable process of index α∈ (1,2] and let L ^x _t denote the local time at time t and position x. Let V(t) be such that L _t ^V(t) = sup _{x∈ ℝ} L _t ^x . We call V(t) the most visited site of X up to time t. We prove the transience of V, that is, lim _{t →∞} |V(t)| = ∞ almost surely. An estimate is given concerning the rate of escape of V. The result extends a well-known theorem of Bass and Griffin for Brownian motion. Our approach is based upon an extension of the Ray–Knight theorem for symmetric Markov processes, and relates stable local times to fractional Brownian motion and further to the winding problem for planar Brownian motion. Received: 14 October 1998 / Revised version: 8 June 1999 / Published online: 7 February 2000     

Increment sizes of the principal value of Brownian local time

Let W be a standard Brownian motion, and define Y(t)= ∫₀ ^t ds/W(s) as Cauchy's principal value related to local time. We determine: (a) the modulus of continuity of Y in the sense of P. Lévy; (b) the large increments of Y. Received: 1 April 1999 / Revised version: 27 September 1999 / Published online: 14 June 2000     

Right inverses of Lévy processes and stationary stopped local times

If X is a symmetric Lévy process on the line, then there exists a non-decreasing, càdlàg process H such that X(H(x)) = x for all x≥ 0 if and only if X is recurrent and has a non-trivial Gaussian component. The minimal such H is a subordinator K. The law of K is identified and shown to be the same as that of a linear time change of the inverse local time at 0 of X. When X is Brownian motion, K is just the usual ladder times process and this result extends the classical result of Lévy that the maximum process has the same law as the local time at 0. Write G _t for last point in the range of K prior to t. In a parallel with classical fluctuation theory, the process Z := (X _t −X _Gt ) _{t ≥0} is Markov with local time at 0 given by (X _Gt ) _{t ≥0}. The transition kernel and excursion measure of Z are identified. A similar programme is outlined for Lévy processes on the circle. This leads to the construction of a stopping time such that the stopped local times constitute a stationary process indexed by the circle. Received: 7 September 1999 / Revised version: 9 November 1999 / Published online: 8 August 2000     

Some properties of the range of super-Brownian motion

We consider a super-Brownian motion X. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting behavior of the volume of the ɛ-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove the support of X _t is capacity-equivalent to [0, 1]² in ℝ^d, d≥ 3, and the range of X, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to [0, 1]⁴ in ℝ^d, d≥ 5. Received: 7 April 1998 / Revised version: 2 October 1998     

A large-deviation result for the range of random walk and for the Wiener sausage

Let {S _n } be a random walk on ℤ ^d and let R _n be the number of different points among 0, S ₁,…, S _{n −1}. We prove here that if d≥ 2, then ψ(x) := lim _{n →∞}(−:1/n) logP{R _n ≥nx} exists for x≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper. We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A⊂ℝ ^d let Λ _t = Λ _t (A) be the d-dimensional Lebesgue measure of the `sausage' ∪_{0≤ s ≤ t} (B(s) + A). Then φ(x) := lim _t→∞: (−1/t) log P{Λ _t ≥tx exists for x≥ 0 and has similar properties as ψ. Received: 20 April 2000 / Revised version: 1 September 2000 / Published online: 26 April 2001     

On the First Hitting Time of a One-dimensional Diffusion and a Compound Poisson Process

It is studied the first-passage time (FPT) of a time homogeneous one-dimensional diffusion, driven by the stochastic differential equation dX(t) = μ(X(t))dt + σ(X(t)) dB _t , X(0) = x ₀, through b + Y(t), where b > x ₀ and Y(t) is a compound Poisson process with rate λ > 0 starting at 0, which is independent of the Brownian motion B _t . In particular, the FPT density is investigated, generalizing a previous result, already known in the case when X(t) = μt + B _t , for which the FPT density is the solution of a certain integral equation. A numerical method is shown to calculate approximately the FPT density; some examples and numerical results are also reported.     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

The optimal stopping problem concerned with ultimate maximum of a Lévy process

For a Lévy process X = (X _t )₀≤t<∞ we consider the time θ = inf{t ≥ 0: sup _s≤t X _s = sup _s≥0 X _s }. We study an optimal approximation of the time θ using the information available at the current instant. A Lévy process being a combination of a Brownian motion with a drift and a Poisson process is considered as an example.     

On Itô s formula for multidimensional Brownian motion

Consider a d-dimensional Brownian motion X = (X ¹,…,X ^d ) and a function F which belongs locally to the Sobolev space W ^1,2. We prove an extension of It? s formula where the usual second order terms are replaced by the quadratic covariations [f _k (X), X ^k ] involving the weak first partial derivatives f _k of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X ^k ] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and backward stochastic integrals. Received: 16 March 1998 / Revised version: 4 April 1999     

Stochastic Equations with Time-Dependent Drift Driven by Levy Processes

The stochastic equation dX _t =dS _t +a(t,X _t )dt, t≥0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ),ξ∈ℝ. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X ₀=x ₀∈ℝ when (ℛe  ψ(ξ))⁻¹=o(|ξ|⁻¹) as |ξ|→∞. These conditions coincide with those found by Tanaka, Tsuchiya and Watanabe (J. Math. Kyoto Univ. 14(1), 73–92, 1974) in the case of a(t,x)=a(x). Our approach is based on Krylov’s estimates for Levy processes with time-dependent drift. Some variants of those estimates are derived in this note.     

Poisson Kernel and Green Function of the Ball in Real Hyperbolic Spaces

Let (X _t ) _t⩾0 be the n-dimensional hyperbolic Brownian motion, that is the diffusion on the real hyperbolic space having the Laplace–Beltrami operator as its generator. The aim of the paper is to derive the formulas for the Gegenbauer transform of the Poisson kernel and the Green function of the ball for the process (X _t ) _t⩾0. Under additional hypotheses we prove integral representations for the Poisson kernel. This yields explicit formulas in and spaces for the Poisson kernel and the Green function as well.      

Spectral Representation of Gaussian Semimartingales

The aim of the present paper is to characterize the spectral representation of Gaussian semimartingales. That is, we provide necessary and sufficient conditions on the kernel K for X _t =∫ K _t (s) dN _s to be a semimartingale. Here, N denotes an independently scattered Gaussian random measure on a general space S. We study the semimartingale property of X in three different filtrations. First, the ℱ ^X -semimartingale property is considered, and afterwards the ℱ ^X,∞-semimartingale property is treated in the case where X is a moving average process and ℱ _t ^X,∞=σ(X _s :s∈(−∞,t]). Finally, we study a generalization of Gaussian Volterra processes. In particular, we provide necessary and sufficient conditions on K for the Gaussian Volterra process ∫ _−∞ ^t K _t (s) dW _s to be an ℱ ^W,∞-semimartingale (W denotes a Wiener process). Hereby we generalize a result of Knight (Foundations of the Prediction Process, 1992) to the nonstationary case.     

Uniform Shrinking and Expansion under Isotropic Brownian Flows

We study some finite time transport properties of isotropic Brownian flows. Under a certain nondegeneracy condition on the potential spectral measure, we prove that uniform shrinking or expansion of balls under the flow over some bounded time interval can happen with positive probability. We also provide a control theorem for isotropic Brownian flows with drift. Finally, we apply the above results to show that, under the nondegeneracy condition, the length of a rectifiable curve evolving in an isotropic Brownian flow with strictly negative top Lyapunov exponent converges to zero as t→∞ with positive probability. P. Baxendale’s research was supported in part by NSF Grant DMS-05-04853.     

Extremal Behavior of Diffusion Models in Finance

We investigate the extremal behavior of a diffusion X _t given by the SDE , where W is standard Brownian motion, μ is the drift term and σ is the diffusion coefficient. Under some appropriate conditions on X _t we prove that the point process of ε -upcrossings converges in distribution to a homogeneous Poisson process. As examples we study the extremal behavior of term structure models or asset price processes such as the Vasicek model, the Cox–Ingersoll–Ross model and the generalized hyperbolic diffusion. We also show how to construct a diffusion with pre-determined stationary density which captures any extremal behavior. As an example we introduce a new model, the generalized inverse Gaussian diffusion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Queueing approximation of suprema of spectrally positive Lévy process

Let W=sup _0≤t<∞(X(t)−β t), where X is a spectrally positive Lévy process with expectation zero and 0<β<∞. One of the main results of the paper says that for such a process X, there exists a sequence of M/GI/1 queues for which stationary waiting times converge in distribution to W. The second result shows that condition (III) of Proposition 2 in the paper is not implied by all other conditions.     

Inequalities for Exit times and Eigenvalues of Balls

This first result of this paper is about the Laplace transform of u(X _T ) where u is harmonic on some bounded domain Ω, X _t is Brownian motion and T is the exit time from Ω. The following results focus on exit times from balls and Faber–Krahn and reverse Faber–Krahn type inequalities for balls. We also study the behaviour of the first Dirichlet eigenvalue for complex balls under complex interpolation. The method of proof heavily relays on the log-concavity of gaussian measures.