Ph.D. Recipients and their Thesis Abstracts

Probability

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

Random Walks on Solvable Groups

Abstract: We study a number of questions about random walks on solvable groups. For random walks on nilpotent groups, we determine which subgroups are recurrent, and for a random walk on the Heisenberg group, we study the number of distinct visited cosets at time n.

The bulk of the examples considered are about the behavior of random walks away from their starting point in groups of exponential growth. In particular, we examine the rate of escape of some inward biased random walks, as well as some unbiased walks that have an intermediate escape rate. We also compute asymptotics for transition probabilities on some semi-direct products, both at the origin and at more general points.

Incipient Infinite Clusters in 2D Percolation

Abstract: We study several kinds of large clusters in critical two-dimensional percolation, and show that the microscopic (lattice scale) view of these clusters when they are observed from the perspective of one of their sites is described by Kesten's incipient infinite cluster (IIC), as it was conjectured in [Ai97]. This way we relate the IIC to other objects in critical percolation that have been proposed as alternatives [ChChDu, ChCh, Ai97].

The mentioned relationship is established for spanning clusters, large clusters in a finite box, the inhomogeneous model of J. Chayes, L. Chayes and R. Durrett, and the invaded region in invasion percolation without trapping.

Other related theorems are proved. It is shown that for any k greater than or equal to 1 the difference in size between the kth and (k+1)st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity. The distribution of the Chayes-Chayes-Durrett cluster is shown to be singular with respect to the IIC. A new upper bound for the growth rate of the invasion percolation cluster, matching the lower bound up to a constant factor, is obtained.

Asymptotic Density in an n-Threshold Randomly Coalescing and Annihilating Random Walk on the d-Dimensional Integer Lattice

Abstract: We consider a system of particles, each of which performs a continuous time random walk on Z^d. The particles interact only at times when a particle jumps to a site at which there are a least n–1 other particles present. If there are i >= n–1 particles present, then the particle coalesces (is removed from the system) with probability c_i and annihilates (is removed along with another particle) with probability a_i. We call this process the n-threshold randomly coalescing and annihilating random walk. We show that, for n >= 3, if both a_i and a_i + c_i are increasing in i and if the dimension d is at least 2n + 4, then P(t):=P{the origin is occupied at time t} ~ C(d,n) t^{–1/(n–1)}, and E(t):=E{number of particles at the origin at time t} ~ C(d,n) t^{–1/(n–1)}.

The constants C(d,n) are explicitly identified. The proof is an extension of a result obtained by H. Kesten and J. Van Den Berg for the 2-threshold coalescing random walk and is based on an approximation for dE(t)/dt.

Hedging Options with Small Transaction Costs

Abstract: Nonzero proportional transaction costs make perfect replication impossible in the Black-Scholes model. We find the distribution of the replication error caused by readjustments of the hedging portfolio at discrete time intervals. Taking a Leland type limit, we find the limiting behavior of transaction costs being paid. Using these two results, we suggest a method for finding the optimal rebalancing interval.

Asymptotic Behavior of Solutions of One Dimensional Parabolic SPDE

Abstract: We investigate the weak convergence as time t\to\infty of the solutions of SPDE in one dimension with Dirichlet boundary conditions. We first prove a result for the linear case, which is easy since the solution is Gaussian. Then we get a result for nonlinear drift case by proving a new comparison theorem for SPDE. To investigate an elliptic SPDE with Dirichlet boundary condition, we set up an equivalence between an elliptic SPDE and a system of SODE. From there, we construct a counter example that shows non-uniqueness of the solution for SPDE, and look at how the Lipschitz constant of the drift term affects the uniqueness.

Coexistence in a Two Species Reaction Diffusion Process Using a Hydrodynamic Limit

Abstract: A two-species reaction diffusion process on the scaled integer lattice with scaling \epsilon, linear and quadratic birth and death rates and migration according to a random walk with rate 1/\epsilon^2 is considered. As \epsilon\to0 the density fields are shown to converge in probability to the weak solutions of a system of partial differential equations. Properties of these solutions are then shown to imply the existence of a stationary distribution for small \epsilon where both species coexist.

Completeness of Securities Market Models — An Operator Point of View

Abstract: We propose a notion of market completeness which is invariant under change to an equivalent probability measure. Completeness means that an operator T acting on stopping time simple trading strategies has dense range in the weak* topology on bounded random variables. In our setup the claims which can be approximated by attainable ones has codimension equal to the dimension of the kernel of the adjoint operator T* acting on signed measures, which in most cases is equal to the "dimension of the space of martingale measures." From this viewpoint the example of Arzner and Heath (1995) is no longer paradoxical since all these dimensions are 1. We also illustrate how one can check for injectivity of T* and hence for completeness in the case of price processes on a Brownian filtration (e.g. Black-Scholes, Heath-Jarrow-Morton) and price processes driven by a multivariate point process.

Predator Mediated Coexistence

Abstract: Not available.

Brownian Motion and Admissible Limits

Abstract: It is known (see e.g. Rudin (1980) or Korányi (1969)) that every {\Cal M}-harmonic function f in the unit ball of C^n satisfying a certain growth condition has admissible limits at a.e. boundary point and is representable by a boundary function. In Chapter 2, we give a probabilistic proof of this result for bounded {\Cal M}-harmonic functions by investigating the behavior of such a function along the paths of an appropriate diffusion process. In Chapter 3, we consider a related question in the unit disk of R^2.

Optimal Drift on the Unit Interval

Abstract: Consider one dimensional diffusions on the interval [0,1] of the form dX_t = dB_t + b(X_t)DT, with 0 a reflecting boundary, b(x)>= 0, and \int_0^1 b(x) dx= 1. We show that there is a unique drift which minimizes the expected time for X_t to hit 1, starting from X_0=0. In the deterministic case dX_t=b(X_t)\,DT, the optimal drift is the function which is identically equal to 1. By contrast, if dX_t= dB_t+ b(X_t)\,DT, then the optimal drift is the step function which is 2 on the interval [1/4,3/4], and is 0 otherwise. In addition, we solve this problem for arbitrary starting point X_0 = x_0 and find that the unique optimal drift depends on the starting point x_0 in a curious manner.

The original motivation for this problem was a conjecture by Benjamini and Peres. Let N be a fixed positive integer. Among all rooted trees with 2^N leaves at level N such that the root has at least two children, which tree minimizes the expected time for a simple random walk, started at the root, to reach level N? Benjamini and Peres conjecture that the full binary tree minimizes this expected value. We discuss in some detail their conjecture for the case of spherically symmetric trees.

A Note On Greedy Lattice Animals

Abstract: Let {X_v: v\in Z^d} be i.i.d. positive random variables with the common distribution F, for which \int x^d(\log^+ x)^{d + a} dF(x)< \infty for some a> 0. Define M_n and N_n by M_n= max{\sum_{v\in\pi} X_v: \pi a selfavoiding path of length n starting at the origin}, N_n= max{\sum_{v\in\XI} X_v: \XI a lattice animal of size n containing the origin}. Then it has been shown that there exist positive finite constants M and N such that \lim_{n\to\infty} M_n / n= M and \lim_{n\to\infty} N_n / n= N a.s. and in L^1.

In the first part of the thesis we show that M = N if X_0 has bounded support and P{X_0 = R}>= p_c where R is the right end point of the support of X_0 and p_c is the critical probability for site percolation on Z^d, and that M< N in all other cases.

In the second part of the thesis we assume that {X_v(p) : v\in Z^d} are i.i.d. Bernoulli random variables with a success parameter p. When considering this special case, we decorate our various quantities by adding an argument p to indicate the dependence on p, e.g., M_n[p], N_n[p], M[p], and N[p]. Then p\to M[p] and p\to N[p] are increasing continuous functions of p such that M[0] = N[0] = 0 and M[p]= N[p]= 1 for p_c<= p<=1. We show that M[p]\asymp N[p]\asymp p^{1/d} for p\downarrow 0 and we also get some partial results on the power laws of M[p] and N[p] for p\uparrow p_c.

In the last part of the thesis we introduce a continuous version of the theory of greedy lattice animals and we show that this continuous model behaves in the same way as the lattice model does.

Poisson Approximations to Continuous Security Market Models

Abstract: We consider standard continuous models of security markets and construct approximating sequences of models. The return process of the approximating models are one- and two-dimensional linear combinations of compensated Poisson processes. We investigate the convergence of the price process as well as the convergence of value processes and trading strategies for contingent claims of complete and incomplete models.

On Path Properties of Superdiffusions

Abstract: A super-Brownian motion X is a measure-valued Markov process describing an evolution of random cloud. It arises as the limit of a system of indistinguishable particles which move according to the law of a Brownian motion and which die at random time leaving a random number of offspring. The branching mechanism of X depends on a class of functions \psi which includes the family \psi(z) = z^\alpha, 1 < \alpha <= 2. Path properties of the super-Brownian are well known due to the work of Dawson, Dynkin, Perkins, Le Gall and others. Connections between the probabilistic and analytic theories have been established by Dynkin, Le Gall, a.o. The main results of this dissertation are as follows.

(A) Assume that \psi(z) = z^\alpha, 1 < \alpha <= 2. Using Dynkin's analytic characterization of the class of G-polar sets of X and a nonlinear potential theory, we establish relations between G-polar sets and the restricted Hausdorff dimension introduced by Taylor and Watson. We give new proofs of Dynkin's criteria for the R-polarity and H-polarity (established earlier by Dawson, Iscoe, Perkins and Le Gall under more restrictive assumptions).

(B) Assume that \psi(z) = z^\alpha, 1 < \alpha <= 2. We establish a connection between polar sets on the boundary (i.e., R_D-polar sets) for the super-Brownian in a smooth domain D \subset R^d and removability of boundary singularity for solutions of a boundary value problem for the corresponding nonlinear PDE. Then we characterize the class of R_D-polar sets in terms of Hausdorff dimension. Using a new kind of capacity on the boundary suggested by Dynkin, we establish partial relations between polar sets on the boundary and capacity zero sets.

(C) Assume that \psi depends neither on position nor on time. Let q(D) be the total mass which escapes the domain D. We study the asymptotic behavior of q(D_n) as D_n\uparrow R^d. As an application, we evaluate the probability that the range of X is compact.

The Regularity of Solutions to the Heat Equation Over Group-Valued Path Space

Abstract: In the group-valued path space W_G= {\gamma\in C[0,1]; G | \gamma(0) = e}, one has a natural Laplacian operator which is associated to the finite energy subgroup. The main purpose of this thesis is to solve the following heat equation on W_G:

and to prove some regularity properties of the solutions. As in the finite dimensional case, we construct a W_G-valued Brownian motion and use the convolution of the transition measures (heat kernels) of the W_G-valued Browning motion and the initial condition f to solve the equation. As an extension of those techniques in proving the regularity theorems for solutions of the heat equation, we also prove some regularity properties for the Brownian bridge measure on the based loop group.

On the Essential Self-Adjointness of Dirichlet Operators on Nonlinear Path Space

Abstract: There are two points of view in studying loop groups depending on the domain one chooses for the differential operators. If G is a compact Lie group and g is its Lie algebra, one can consider functions defined on

W_g= {b: [0, 1] \rightarrow g; b continuous, b(0) = 0}

or functions defined on

W_G= {g: [0, 1] \rightarrow G; g continuous, g(0) = e.}

Getzler and Malliavin have formulated their work on analysis over Loop Spaces in terms of functions on W_g. We choose to work with spaces of functions on W_G since they reflect topological properties of G. We call W_G nonlinear path space. We make substantial progress towards the solution of the problem of essential self-adjointness of the number operator over the nonlinear path space in the natural presumed core. Consider the Cameron-Martin space H over g and for h in H let \partial _h be the directional derivative on W_G. The adjoint operator \partial*_h with respect to the "Wiener measure" P on W_G is given by –\partial_h+ j_h. For each partition 0 < T_1 < ... < T_m of [0,1] define the subspace {\cal F}^m_h of L^2(W_G, P) by pulling back C^{\infty}_c(R x G^m)with respect to the map \theta_m(\gamma) = (j_h, \gamma(T_1), ..., \gamma(T_m)) on W_G. The completion \bar{{\cal F}}^m_h of {\cal F}^m_h in L^2(W_G, P) is isomorphic to L^2(R x G^m, \rho_m) where \rho_m is the density of the distribution law of \theta_m. We appeal to Malliavin calculus to prove smoothness of \rho_m and then use Wielen's technique to prove that the Dirichlet operator \partial*_h \partial _h|_{{\cal F}^m_h} as an operator on L^2(R x G^m, \rho_m) is essentially self-adjoint in {\cal F}^m_h. The essential self-adjointness of \partial*_h \partial _h in U {\cal F}^m_h is deduced easily due to the invariance of {\cal F}^m_h with respect to \partial*_h \partial _h.

Last modified: February 6, 2004