Necessary and sufficient conditions for finiteness of lifetime

We give a geometric characterization for the finiteness of conditioned Brownian motion for a general class of simply connected domains, extending previous results and exhibit some new examples of domains with infinite area and finite lifetime.I would like to thank Professor Rodrigo Bañuelos, my academic advisor, for his help and guidance on this paper which is part of my Ph.D. thesis.     

Lower and upper solutions for elliptic problems in nonsmooth domains

In this paper we prove some existence results of semilinear Dirichlet problems in nonsmooth domains in presence of lower and upper solutions well-ordered or not. We first prove existence results in an abstract setting using degree theory. We secondly apply them for domains with conical points.     

An invariance principle for non-symmetric Markov processes and reflecting diffusions in random domains

Summary We study an invariance principle for additive functionals of nonsymmetric Markov processes with singular mean forward velocities. We generalize results of Kipnis and Varadhan [KV] and De Masi et al. [De] in two directions: Markov processes are non-symmetric, and mean forward velocities are distributions. We study continuous time Markov processes. We use our result to homogenize non-symmetric reflecting diffusions in random domains.     

The skorokhod oblique reflection problem in a convex polyhedron

The Skorokhod oblique reflection problem is studied in the case ofn-dimensional convex polyhedral domains. The natural sufficient condition on the reflection directions is found, which together with the Lipschitz condition on the coefficients gives the existence and uniqueness of the solution. The continuity of the corresponding solution mapping is established. This property enables one to construct in a direct way the reflected (in a convex polyhedral domain) diffusion processes possessing the nice properties.     

Inegalités Barlow-Yor pour les temps locaux d'intersection de deux mouvements Browniens plans

Summary We proveL _p> majorations for the suprema of the intersection local time of two planar Brownian motions. We deduce a sufficient condition on the correlation of two Brownian motions for their i.l.t. to be continuous.

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Inegalités Barlow-Yor pour les temps locaux d'intersection de deux mouvements Browniens plans

Résumé On prouve des majorations en normeL _p pour le suprema des temps locaux d'intersection de deux mouvements browniens plans, renormalisés ou non. On en déduit une condition suffisante sur la corrélation entre deux browniens pour que leur t.l.i. soient continus.

     

Boundary Harnack principle for subordinate Brownian motions

We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in κ$\kappa$-fat open sets (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded κ$\kappa$-fat open sets with respect to these processes with their Euclidean boundaries.     

Markov processes with free-Meixner laws

We study a time-non-homogeneous Markov process which arose from free probability, and which also appeared in the study of stochastic processes with linear regressions and quadratic conditional variances. Our main result is the explicit expression for the generator of the (non-homogeneous) transition operator acting on functions that extend analytically to complex domains.     

Nondegeneracy of elliptic problems with periodic nonlinearities in the plane

A class of asymptotically quadratic functionals on Hilbert spaces, called degenerate, is considered and explored. Our results are applied to obtain a extension, in the planar case, of a result published by Solimini in ‘On the solvability of some elliptic partial differential equations with the linear part at resonance’, J. Math. Anal. Appl., 117 (1986), 138-152. Similar extensions had been previously studied in the literature only for domains with particular geometries.     

Explicit bounds for rational points near planar curves and metric Diophantine approximation

The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in Beresnevich et al. (2007) [10] for C³ non-degenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to C¹ (lowest possible). This way we broaden the notion of non-degeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of Beresnevich et al. (2007) [10] and extend the celebrated theorem of Kleinbock and Margulis (1998) [20] in dimension 2 beyond the notion of non-degeneracy.     

Existence results for critical singular elliptic systems

In this paper, we consider semilinear elliptic systems with both singular and critical growth terms in bounded domains. The existence of a nontrivial solution is obtained by variational methods.     

Reflecting Brownian motions: Quasimartingales and strong Caccioppoli sets

A notion ofstrong Caccioppoli set is defined for bounded Euclidean domains. It is shown that stationary (normally) reflecting Brownian motion on the closure of a bounded Euclidean domain is a quasimartingale on each compact time interval if and only if the domain is a strong Caccioppoli set. A similar result is shown to hold for symmetric reflecting diffusion processes.Research supported in part by NSF Grant DMS 91-01675.Research supported in part by NSF Grants DMS 86-57483 and 90-23335.     

Uniform asymptotic formulae for Green's functions in singularly perturbed domains

Asymptotic formulae for Green's functions for the operator -Δ$-\Delta$ in domains with small holes are obtained. A new feature of these formulae is their uniformity with respect to the independent variables. The cases of multi-dimensional and planar domains are considered.     

Traps for reflected Brownian motion

Consider an open set , d ≥ 2, and a closed ball . Let denote the expectation of the hitting time of B for reflected Brownian motion in D starting from x ∈ D. We say that D is a trap domain if . A domain D is not a trap domain if and only if the reflecting Brownian motion in D is uniformly ergodic. We fully characterize the simply connected planar trap domains using a geometric condition and give a number of (less complete) results for d > 2. Research partially supported by NSF grant DMS-0303310. Research partially supported by NSF grant DMS-0303310. Research partially supported by NSF grant DMS-0201435.     

On the lifetime of a conditioned Brownian motion on a fish bowl

It has been conjectured that for 2-dimensional planar domains the lifetime of a conditioned Brownian motion is maximized for two boundary points. The present example shows that such a claim is in general not true on manifolds. Received: 3 April 2007     

Prescribed mean curvature equations under the transformation with non-orthogonal curvilinear coordinates

The aim of this paper is to investigate the Dirichlet problem of prescribed mean curvature equations. We show the existence of a weak solution. The boundary of domains does not always satisfy the H-convexity condition. Our method is not to construct the barrier functions directly, but to use some uniform estimate for solutions of the approximating regularized solutions.     

On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an condition on b, to a Kato condition on . Received: 21 April 1998 / in final form 26 March 1999     

The Neumann problem for the Helmholtz equation in a starlike planar domain

The interior and exterior Neumann problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so called “superformula” introduced by Gielis. A dedicated numerical procedure based on a computer algebra system is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to classical ones, are obtained. Computed results are found to be in good agreement with theoretical findings on Fourier series expansion presented by Carleson.     

Association rates of diffusion-controlled reactions in two dimensions

A detailed analysis of the main results concerning mathematical models of diffusion-controlled reactions in two dimensions is presented. Specific emphasis is placed on methods for evaluating association rates. After a review of planar models, the effects due to the curvature of the ambient space are investigated. Finally, different possible choices of boundary conditions are considered, and suggestions are given on their aptness to model different physicochemical situations.     

Rational points near planar curves and Diophantine approximation

In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich–Dickinson–Velani [6] and Vaughan–Velani [22]. Furthermore, we complete the Lebesgue theory of Diophantine approximation on weakly non-degenerate planar curves that was initially developed by Beresnevich–Zorin [5] in the divergence case.     

Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains

We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions.