On the solution of a one-dimensional stochastic differential equation with singular drift coefficient

We determine generalized diffusion coefficients and describe the structure of local times for a process defined as a solution of a one-dimensional stochastic differential equation with singular drift coefficient.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 642–655, May, 2004.     

Anisotropic diffusions with singular advections and absorptions part 1: existence

An anisotropic filtration problem with singular advections and strong absorptions, which leads to a quasi-linear degenerate parabolic equation in divergent form, is studied in this paper. The existence for the Cauchy problem as well as the first boundary-initial-value problem is proved.     

On the sector condition and homogenization of diffusions with a Gaussian drift

In this paper we present the functional central limit theorem for a class of Markov processes, whose L²-generator satisfies the so-called graded sector condition. We apply the result to obtain homogenization theorems for certain classes of diffusions with a random Gaussian drift. Additionally, we present a result concerning the regularity of the effective diffusivity tensor with respect to the parameters related to the statistics of the drift. The abstract central limit theorem, see Theorem 2.2, is obtained by applying the technique used in Sethuraman et al. (Comm. Pure Appl. Math. 53 (2000) 972) to the case of infinite particle systems.     

On the existence of minimal surfaces with singular boundaries

In 1931, Jesse Douglas showed that in , every set of rectifiable Jordan curves, with , bounds an area-minimizing minimal surface with prescribed topological type if a ``condition of cohesion' is satisfied. In this paper, it is established that under similar conditions, this result can be extended to non-Jordan curves.

     

On existence of singular solutions

In the paper sufficient conditions are given under which the differential equation y⁽ⁿ⁾=f(t,y,…,y⁽ⁿ⁻²⁾)g(y⁽ⁿ⁻¹⁾) has a singular solution y :[T,τ)→R, τ<∞ fulfilling

     

Stationary distributions for diffusions with inert drift

Consider reflecting Brownian motion in a bounded domain in ${\mathbb R^d}$ that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting Brownian motion and the value of the drift vector has a product form. Moreover, the first component is uniformly distributed on the domain, and the second component has a Gaussian distribution. We also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential.     

Uniqueness for isotropic diffusions with a linear drift

This paper proves that-valued solutions to the SDE are unique in distribution, when D^d is convex and open, D, c>0, is positive and locally Lipschitz on D and zero on D, and {xD:g(x)r} is convex for r sufficiently small. The proof (for =0) is based on the transformation X_te^ctX_t, which removes the drift, and a random time change. Although the set-up is rather specialized the result gives uniqueness for some SDEs that cannot be treated by any of the conventional techniques.Mathematics Subject Classification (2000):60J60, 60H10     

On normal solvability of the Riemann problem with singular coefficient

Suppose is a singular matrix function on a simple, closed, rectifiable contour . We present a necessary and sufficient condition for normal solvability of the Riemann problem with coefficient in the case where admits a spectral (or generalized Wiener-Hopf) factorization with essentially bounded. The boundedness of is not required when takes injective values a.e. on .

     

On the existence of solutions of stochastic differential equations with singular drifts

Summary In this paper we prove the existence of solutions for a stochastic differential equation inR ^d, when the drift and the diffusion term are allowed to depend on a specific way on the local time of thedth coordinate of the process to be constructed. The methods of our construction are of purely probabilistic nature.     

Quasistationary distributions for one-dimensional diffusions with singular boundary points

In the present work we characterize the existence of quasistationary distributions for diffusions on $(0,\infty )$ allowing singular behavior at 0 and $\infty$. If absorption at 0 is certain, we show that there exists a quasistationary distribution as soon as the spectrum of the generator is strictly positive. This complements results of Cattiaux et al. (2009) and Kolb and Steinsaltz (2012) for 0 being a regular boundary point and extends results by Cattiaux et al. (2009) on singular diffusions.     

On singular stochastic control and optimal stopping of spectrally negative jump diffusions

We consider a broad class of singular stochastic control problems of spectrally negative jump diffusions in the presence of potentially nonlinear state-dependent exercise payoffs. We analyse these problems by relying on associated variational inequalities and state a set of sufficient conditions under which the value of the considered problems can be explicitly derived in terms of the increasing minimal r-harmonic map. We also present a set of inequalities bounding the value of the optimal policy and prove that increased policy flexibility increases both the value of the optimal strategy as well as the rate at which this value grows.     

Integral inequalities for the fundamental solutions of diffusions on manifolds with divergence-free drift

The main purpose of this article is to present uniform integral inequalities for the fundamental solutions of diffusions on compact manifolds with divergence free drift vector fields. The method relies on the fact that the heat flow depends on the isoperimetric function. The isoperimetric function is used to construct a suitable comparison manifold. The heat kernel of this comparison manifold gives uniform bounds for the fundamental solutions of the original diffusion problem. The results presented here can be used to solve some open problem of Bhattacharya and Götze about diffusions with periodic, divergence free drift vector fields (see [Bhagöt1] and [Bhagöt2]). Mathematics Subject Classification (2000): 58J35, 47D07This research was supported by the German Exchange Service DAAD     

Anisotropic diffusions with singular advections and absorptions part 2: uniqueness

An anisotropic filtration problem with singular advections and strong absorptions, which leads to a quasilinear degenerate parabolic equation in divergent form, is studied in this paper. The uniqueness and the comparison principle are proved for the bounded and continuous solutions to the Cauchy problem and the first boundary-initial-value problems.     

Diffusions with singular drift related to wave functions

Summary Schrödinger equations are equivalent to pairs of mutually time-reversed non-linear diffusion equations. Here the associated diffusion processes with singular drift are constructed under assumptions adopted from the theory of Schrödinger operators, expressed in terms of a local space-time Sobolev space.By means of Nagasawa's multiplicative functionalN _s ^t , a Radon-Nikodym derivative on the space of continuous paths, a transformed process is obtained from Wiener measure. Its singular drift is identified by Maruyama's drift transformation. For this a version of Itô's formula for continuous space-time functions with first and second order derivatives in the sense of distributions satisfying local integrability conditions has to be derived.The equivalence is shown between weak solutions of a diffusion equation with singular creation and killing term and the solutions of a Feynman-Kac integral equation with a locally integrable potential function.     

The normal approximation rate for the drift estimator of multidimensional diffusions

We consider the problem of the density and drift estimation by the observation of a trajectory of an \mathbbR^d{\mathbb{R}^{d}}-dimensional homogeneous diffusion process with a unique invariant density. We construct estimators of the kernel type based on discretely sampled observations and study their asymptotic distribution. An estimate of the rate of normal approximation is given.     

On the existence of positive solutions of singular nonlinear elliptic equations with Dirichlet boundary conditions

This paper is concerned with the existence of positive solutions of the singular nonlinear elliptic equation with a Dirichlet boundary condition

     

Exact adaptive pointwise drift estimation for multidimensional ergodic diffusions

The problem of pointwise adaptive estimation of the drift coefficient of a multivariate diffusion process is investigated. We propose an estimator which is sharp adaptive on scales of Sobolev smoothness classes. The analysis of the exact risk asymptotics allows to identify the impact of the dimension and other influencing values—such as the geometry of the diffusion coefficient—of the prototypical drift estimation problem for a large class of multidimensional diffusion processes. We further sketch generalizations of our results to arbitrary diffusions satisfying suitable Bernstein-type inequalities.     

Homogenization with respect to Gibbs measures for periodic drift diffusions on lattices

A homogenization problem for infinite dimensional diffusion processes indexed by ${\mathbf{Z}}^d$ having periodic drift coefficients is considered. By an application of the uniform ergodic theorem for the infinite dimensional diffusion processes based on logarithmic Sobolev inequalities, an $L^1$ type homogenization property of the processes with respect to an invariant measure is proved. This is the, so far, best possible analogue in infinite dimensions to a known result in the finite dimensional case (cf. [G. Papanicolaou, S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Seria Coll. Math. Soc. Janos Bolyai, vol. 27, North-Holland, 1979. [4]]). To cite this article: S. Albeverio et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).