Stochastic flows for SDEs with non-Lipschitz coefficients

We prove the bicontinuity and homeomorphic property of solutions of stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients.     

Stochastic flows of SDEs with irregular coefficients and stochastic transport equations

In this article we study (possibly degenerate) stochastic differential equations (SDEs) with irregular (or discontinuous) coefficients, and prove that under certain conditions on the coefficients, there exists a unique almost everywhere stochastic (invertible) flow associated with the SDE in the sense of Lebesgue measure. In the case of constant diffusions and BV drifts, we obtain such a result by studying the related stochastic transport equation. In the case of non-constant diffusions and Sobolev drifts, we use a direct method. In particular, we extend the recent results on ODEs with non-smooth vector fields to SDEs.     

Special vekua equations with singular coefficients

Special equations of Vekua-type with singular coefficients are considered. As a first step we study the influence of the coefficients of model equations on the choice of the function spaces for its solutions and on the boundary conditions. As an application we sketch the consideration of boundary value problems for Vekua equations with variable coefficients having a strong singularity at z =0     

Distribution dependent SDEs with singular coefficients

Under integrability conditions on distribution dependent coefficients, existence and uniqueness are proved for distribution dependent SDEs with non-degenerate noise. When the coefficients are Dini continuous in the space variable, gradient estimates and Harnack type inequalities are derived. These generalize the corresponding results derived for classical SDEs, and are new in the distribution dependent setting.     

Scale invariant elliptic operators with singular coefficients

We show that a realization of the operator \({L=|x|^\alpha\Delta +c|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -b|x|^{\alpha-2}}\) generates a semigroup in \({L^p(\mathbb{R}^N)}\) if and only if \({D_c=b+(N-2+c)^2/4 > 0}\) and \({s_1+\min\{0,2-\alpha\} < N/p < s_2+\max\{0,2-\alpha\}}\), where \({s_i}\) are the roots of the equation \({b+s(N-2+c-s)=0}\), or \({D_c=0}\) and \({s_0+\min\{0,2-\alpha\} < N/p < s_0+\max\{0,2-\alpha\}}\), where \({s_0}\) is the unique root of the above equation. The domain of the generator is also characterized.     

Fast-slow stochastic dynamical system with singular coefficients

This paper aims to study the asymptotic behavior of a fast-slow stochastic dynamical system with singular coefficients, where the fast motion is given by a continuous diffusion process while the slow component is driven by an α-stable noise with α ∈ [1, 2). Using Zvonkin’s transformation and the technique of the Poisson equation, we have that both the strong and weak convergences in the averaging principle are established, which can be viewed as a functional law of large numbers. Then we study t...     

Stochastic heat equation with random coefficients

We prove the existence of a unique solution for a one-dimensional stochastic parabolic partial differential equation with random and adapted coefficients perturbed by a two-parameter white noise. The proof is based on a maximal inequality for the Skorohod integral deduced from It?'s formula for this anticipating stochastic integral. Received: 21 November 1997 / Revised version: 20 July 1998     

Evolution semigroups and Hamiltonian flows

The operatorial calculus of Feinsilver is extended to a class of Hamiltonians possessing terms depending on the position variables.     

Second order abstract differential equations with singular coefficients

Summary We study a large class of second order linear abstract differential equations, whose coefficients can be singular. In the framework of suitable « weighted » spaces, we prove some existence and uniqueness results for generalized and ordinary solutions of initial value problems for such equations.This work was partially supported by the G.N.A.F.A. and the Istituto di Analisi Numerica of the C.N.R. (Italy).     

Stochastic discrete Hamiltonian variational integrators

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.     

Hyperbolic-like solutions for singular Hamiltonian systems

We study the existence of unbounded solutions of singular Hamiltonian systems: where is a potential with a singularity. For a class of singular potentials with a strong force , we show the existence of at least one hyperbolic-like solutions. More precisely, for given and , we find a solution q(t) of (*) satisfying Received October 1998     

Gradient estimates for diffusion semigroups with singular coefficients

Uniform gradient estimates are derived for diffusion semigroups, possibly with potential, generated by second order elliptic operators having irregular and unbounded coefficients. We first consider the Rd-case, by using the coupling method. Due to the singularity of the coefficients, the coupling process we construct is not strongly Markovian, so that additional difficulties arise in the study. Then, more generally, we treat the case of a possibly unbounded smooth domain of Rd with Dirichlet boundary conditions. We stress that the resulting estimates are new even in the Rd-case and that the coefficients can be Hölder continuous. Our results also imply a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying diffusion semigroup.     

Spectral theory of Hamiltonian systems with almost constant coefficients

We derive the spectral theory for general linear Hamiltonian systems. The coefficients are assumed to be asymptotically constant and satisfy certain smoothness and decay conditions. These latter constraints preclude the appearance of singular continuous spectra. The results are thus far reaching extensions of earlier theorems of the authors. Two-, three- and four-dimensional systems are studied in greater detail. The results also apply to the case of the Dirichlet index and Dirichlet spectrum.