Conditioned Diffusions which are Brownian Bridges

Let X _t be a one-dimensional diffusion of the form dX _t=dB _t+(X _t)dt. Let Tbe a fixed positive number and let be the diffusion process which is X _t conditioned so that X ₀=X _T=x. If the drift is constant, i.e., , then the conditioned diffusion process is a Brownian bridge. In this paper, we show the converse is false. There is a two parameter family of nonlinear drifts with this property.     

Solutions of Δ<Emphasis Type="Italic">u</Emphasis>=4<Emphasis Type="Italic">u</Emphasis><Superscript><Emphasis Type="Italic">2</Emphasis></Superscript> with Neumann’s conditions using the Brownian snake

We consider a Brownian snake (W_s,s0) with underlying process a reflected Brownian motion in a bounded domain D. We construct a continuous additive functional (L_s,s0) of the Brownian snake which counts the time spent by the end points _s of the Brownian snake paths on D. The random measure Z=_sdLs is supported by D. Then we represent the solution v of u=4u² in D with weak Neumann boundary condition 0 by using exponential moment of (Z,) under the excursion measure of the Brownian snake. We then derive an integral equation for v. For small it is then possible to describe negative solution of u=4u² in D with weak Neumann boundary condition . In contrast to the exit measure of the Brownian snake out of D, the measure Z is more regular. In particular we show it is absolutely continuous with respect to the surface measure on D for dimension 2 and 3.Mathematics Subject Classification (2000):60J55, 60J80, 60H30, 60G57, 35C15, 35J65     

Continuity in Law with Respect to the Hurst Parameter of the Local Time of the Fractional Brownian Motion

We give a result of stability in law of the local time of the fractional Brownian motion with respect to small perturbations of the Hurst parameter. Concretely, we prove that the law (in the space of continuous functions) of the local time of the fractional Brownian motion with Hurst parameter H converges weakly to that of the local time of , when H tends to H ₀.      

Large deviations for Brownian motion in a random scenery

We prove large deviations principles in large time, for the Brownian occupation time in random scenery . The random field is constant on the elements of a partition of ^d into unit cubes. These random constants, say consist of i.i.d. bounded variables, independent of the Brownian motion {B_s,s0}. This model is a time-continuous version of Kesten and Spitzer's random walk in random scenery. We prove large deviations principles in ``quenched' and ``annealed' settings.Mathematics Subject Classification (2000):60F10, 60J55, 60K37     

The advantage of small machines in a stochastic fluid production process

We consider a stochastic fluid production model, where m machines which are subject to breakdown and repair, produce a fluid at ratep > 0 per machine if it is working. This fluid is fed into an infinite buffer with stochastic output rate. Under the assumption that the machine processes are independent and identically distributed, we prove that the buffer content at timet is less or equal in the increasing convex ordering to the buffer content at time t of a model withm m machines and production ratep =m/m p. This formulation includes a conjecture posed by Mitra [6]. More-over, it is shown how to extend this result to Brownian flow systems, systems obtained by diffusion approximation and simple stochastic flow networks like tandem buffer and assembly systems.     

On almost sure noncentral limit theorems

LetX(t) be a fractional Brownian motion or Hermite process of indexH. SetX _m (t)=m ^–H X(mt), which we view as an element ofC[0, 1]. Let {x} denote a point mass at x. Then The corresponding results for certain partial sums in the domain of attraction toX(t) are shown to hold.     

Exponential Convergence in Probability for Empirical Means of Brownian Motion and of Random Walks

Given a Brownian motion (B _t) _t0 in R ^d and a measurable real function f on R ^d belonging to the Kato class, we show that 1/t ₀ ^t f(B _s ) ds converges to a constant z with an exponential rate in probability if and only if f has a uniform mean z. A similar result is also established in the case of random walks.     

Integration by parts for a Lie group valued Brownian motion

LetG be a Lie group ofd×d matrices and be theLLie algebra ofG. We choose some Euclidean norm on , and an orthonormal basis (D ₁,...D _m ) relative to it. Let be the corresponding left invariant vector fields onG. In this paper we derive an integration by parts formula for aG-valued Brownian motion corresponding to the Laplacian .     

Comparison results for the lower tail of Gaussian seminorms

Let =( _n ) be i.i.d.N(0, 1) random variables andq(x), q(x):R [0, ) be seminorms. We investigate necessary and sufficient conditions that the ratio ofP(q()<) andP(q()<) goes to a positive constant as 0⁺. We give satisfactory answers forl ₂-norms and also some results for sup-norms andl _p-norms. Some applications are given to the rate of escape of infinite dimensional Brownian motion, and we give the lower tail of the Ornstein-Uhlenbeck process and a weighted Brownian bridge under theL ₂-norms.     

Besov Regularity of Stochastic Integrals with Respect to the Fractional Brownian Motion with Parameter H > 1/2

Let {B _t ,t[0,1]} be a fractional Brownian motion with Hurst parameter H > 1/2. Using the techniques of the Malliavin calculus we show that the trajectories of the indefinite divergence integral ^t ₀ u _s B _s belong to the Besov space _p,q for all , provided the integrand u belongs to the space . Moreover, if u is bounded and belongs to for some even integer p2 and for some large enough, then the trajectories of the indefinite divergence integral ^t ₀ u _s B _s belong to the Besov space _p, ^H .     

Critical Dimensions for the Existence of Self-Intersection Local Times of the N-Parameter Brownian Motion in R d

Fix two rectangles A, B in [0, 1] ^N . Then the size of the random set of double points of the N-parameter Brownian motion in R ^d , i.e, the set of pairs (s, t), where sA, tB, and W _s=W _t, can be measured as usual by a self-intersection local time. If A=B, we show that the critical dimension below which self-intersection local time does not explode, is given by d=2N. If A B is a p-dimensional rectangle, it is 4N–2p (0pN). If A B = , it is infinite. In all cases, we derive the rate of explosion of canonical approximations of self-intersection local time for dimensions above the critical one, and determine its smoothness in terms of the canonical Dirichlet structure on Wiener space.     

Estimating the Diffusion Coefficient for Diffusions Driven by fBm

When the Hurst coefficient of a fBm B _t ^H is greater than 1/2, it is possible to define a stochastic integral with respect to B _t ^H as the pathwise limit of Riemann sums. In this article we consider diffusion equations of the type X_t = x₀ + ₀ ^T (X_s) dB_s ^H. We then construct a simple-to-use estimator of the diffusion coefficient (x), based on the number of crossings of level x of the process X _t. We then study consistency in probability of this estimator and calculate convergence rates in probability.     

Iterated Brownian Motion in Parabola-Shaped Domains

Iterated Brownian motion Z_t serves as a physical model for diffusions in a crack. If τ_D(Z) is the first exit time of this processes from a domain D⊂ℝⁿ, started at z∈D, then P_z[τ_D(Z)>t] is the distribution of the lifetime of the process in D. In this paper we determine the large time asymptotics of which gives exponential integrability of for parabola-shaped domains of the form P_α={(x,Y)∈ℝ×ℝⁿ⁻¹:x>0, |Y|<Ax^α}, for 0<α<1, A>0. We also obtain similar results for twisted domains in ℝ² as defined in DeBlassie and Smits: Brownian motion in twisted domains, Preprint, 2004. In particular, for a planar iterated Brownian motion in a parabola we find that for z∈℘

On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains

We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W _m ¹( _s ) [W _m ¹( _s )]^* in a sequence of perforated domains _s . Under a certain condition imposed on the local capacity of the set \ _s , we prove the following principle of compensated compactness: , where r _s(x) and z _s(x) are sequences weakly convergent in W _m ¹() and such that r _s(x) is an analog of a corrector for a homogenization problem and z _s(x) is an arbitrary sequence from whose weak limit is equal to zero.     

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.     

Zero–One Law for some Brownian Functionals

We prove that for a>0, (B _t) one-dimensional standard Brownian motion and ₀=inf{t>0 : B _t=0} the following zero–one law is valid

The uniform modulus of continuity of iterated Brownian motion

LetX be a Brownian motion defined on the line (withX(0)=0) and letY be an independent Brownian motion defined on the nonnegative real numbers. For allt0, we define theiterated Brownian motion (IBM),Z, by setting . In this paper we determine the exact uniform modulus of continuity of the process Z.Research supported by NSF grant DMS-9122242.     

Small Deviations for Gaussian Markov Processes Under the Sup-Norm

Let {X(t); 0t1} be a real-valued continuous Gaussian Markov process with mean zero and covariance (s, t) = EX(s) X(t) 0 for 0<s, t<1. It is known that we can write (s, t) = G(min(s, t)) H(max(s, t)) with G>0, H>0 and G/H nondecreasing on the interval (0, 1). We show that

Weak convergence of the scaled median of independent Brownian motions

We consider the median of n independent Brownian motions, denoted by M _n (t), and show that $\sqrt{n}\,M_nWe consider the median of n independent Brownian motions, denoted by M _n (t), and show that converges weakly to a centered Gaussian process. The chief difficulty is establishing tightness, which is proved through direct estimates on the increments of the median process. An explicit formula is given for the covariance function of the limit process. The limit process is also shown to be H?lder continuous with exponent γ for all γ < 1/4.      

Existence,uniqueness, stochastic persistence and global stability of positive solutions of the logistic equation with random perturbation

This paper discusses a randomized logistic equation (1) with initial value x(0)=x₀>0, where B(t) is a standard one‐dimension Brownian motion, and θ∈(0, 0.5). We show that the positive solution of the stochastic differential equation does not explode at any finite time under certain conditions. In addition, we study the existence, uniqueness, boundedness, stochastic persistence and global stability of the positive solution. Copyright © 2006 John Wiley & Sons, Ltd.