Ergodic BSDEs under weak dissipative assumptions

In this paper we study ergodic backward stochastic differential equations (EBSDEs) dropping the strong dissipativity assumption needed in Fuhrman et al. (2009) [12]. In other words we do not need to require the uniform exponential decay of the difference of two solutions of the underlying forward equation, which, on the contrary, is assumed to be non-degenerate.We show the existence of solutions by the use of coupling estimates for a non-degenerate forward stochastic differential equation with bounded measurable nonlinearity. Moreover we prove the uniqueness of “Markovian” solutions by exploiting the recurrence of the same class of forward equations.Applications are then given for the optimal ergodic control of stochastic partial differential equations and to the associated ergodic Hamilton-Jacobi-Bellman equations.     

Mixed boundary value problems of semilinear elliptic PDEs and BSDEs with singular coefficients

In this paper, we prove that there exists a unique weak (Sobolev) solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.     

Stochastic PDIEs with nonlinear Neumann boundary conditions and generalized backward doubly stochastic differential equations driven by Lévy processes

In this paper, a new class of generalized backward doubly stochastic differential equations (GBDSDEs in short) driven by Teugels martingales associated with Lévy process and the integral with respect to an adapted continuous increasing process is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of stochastic partial differential integral equations (PDIEs in short) with a nonlinear Neumann boundary condition is given.     

Small sets for Neumann boundary conditions

For an open set D ? ?ⁿ and a relatively closed subset E ? D of Lebesgue measure zero, we investigate conditions for the property that Brownian motion with reflexion at the boundary on D and D E are the same. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Carleman estimates for stochastic parabolic equations with Neumann boundary conditions and applications

By a dual method, two Carleman estimates for forward and backward stochastic parabolic equations with Neumann boundary conditions are established. Then they are used to study a null controllability problem and a state observation problem for some stochastic forward parabolic equations with Neumann boundary conditions.     

Multipoint boundary value problems of Neumann type for functional differential equations

We obtain the expression of the explicit solution to a class of multipoint boundary value problems of Neumann type for linear ordinary differential equations and apply these results to study sufficient conditions for the existence of solution to linear functional differential equations with multipoint boundary conditions, considering the particular cases of equations with delay and integro-differential equations.     

On a degenerate variational inequality with Neumann boundary conditions

A stopping time problem for degenerate reflected diffusions is studied in this paper. We give a characterization of the optimal cost as the maximum solution of a degenerate elliptic variational inequality with Neumann boundary conditions.The author would like to thank Professor L. C. Evans for very helpful suggestions on this topic.     

Small-amplitude solutions of the sine-Gordon equation on an interval under dirichlet or Neumann boundary conditions

Summary We give a complete classification of the small-amplitude finite-gap solutions of the sine-Gordon (SG) equation on an interval under Dirichlet or Neumann boundary conditions. Our classification is based on an analysis of the finite-gap solutions of the boundary problems for the SG equation by means of the Schottky uniformization approach.On leave from IPPI, Moscow, Russia     

Nonlinear partial differential equations of fourth order under mixed boundary conditions

Solutions to nonlinear partial differential equations of fourth order are studied. Boundary regularity is proved for solutions that satisfy mixed boundary conditions. Various geometric situations including so called triple points are considered. Regularity is measured in Sobolev-Slobodeckii spaces and the results are sharp in this scale. The approach is based on the use of a first order difference quotient method.     

A fourth-order method of the convection-diffusion equations with Neumann boundary conditions

In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection-diffusion equations with Neumann boundary conditions.     

Blow-up and global solutions for quasilinear parabolic equations with Neumann boundary conditions

Using the upper and lower solution techniques and Hopf's maximum principle, the sufficient conditions for the existence of blow-up positive solution and global positive solution are obtained for a class of quasilinear parabolic equations subject to Neumann boundary conditions. An upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified.     

Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data

In this article we investigate averaging properties of fully nonlinear PDEs in bounded domains with oscillatory Neumann boundary data. The oscillation is periodic and is present both in the operator and in the Neumann data. Our main result states that, when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant, the solutions uniformly converge to the homogenized solution solving a Neumann boundary problem. Furthermore we show that the homogenized Neumann data is continuous with respect to the normal direction of the boundary. Our result is the nonlinear version of the classical result in [3] for divergence-form operators with co-normal boundary data. The main ingredients in our analysis are the estimate on the oscillation on the solutions in half-spaces (Theorem 3.1), and the estimate on the mode of convergence of the solutions as the normal of the half-space varies over irrational directions (Theorem 4.1).     

Optimal control of stochastic differential equations with dynamical boundary conditions

In this paper we investigate the optimal control problem for a class of stochastic Cauchy evolution problems with nonstandard boundary dynamic and control. The model is composed by an infinite dimensional dynamical system coupled with a finite dimensional dynamics, which describes the boundary conditions of the internal system. In other terms, we are concerned with nonstandard boundary conditions, as the value at the boundary is governed by a different stochastic differential equation.     

CONVERGENCE OF BSDEs AND HOMOGENIZATION OF ELLIPTIC SEMI-LINEAR PDEs

We present a result of weak convergence for Backward Stochastic Differential Equations with a random terminal time. Then we can deduce results of homogenization for elliptic semi-linear PDEs with random or periodic coefficients and whose non-linearity may have a quadratic growth in the gradient.     

A nonlocal nonlinear diffusion equation with blowing up boundary conditions

We deal with boundary value problems (prescribing Dirichlet or Neumann boundary conditions) for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation. First, we prove existence, uniqueness and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

A nonlocal p-Laplacian evolution equation with Neumann boundary conditions

In this paper we study the nonlocal p-Laplacian type diffusion equation,

If p>1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation u_t=div(|u|^p−2u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L^∞(0,T;L^p(Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p=1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behavior of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition.     

BSDEs and SDEs with time-advanced and -delayed coefficients

ABSTRACT

This paper introduces a class of backward stochastic differential equations (BSDEs), whose coefficients not only depend on the value of its solutions of the present but also the past and the future. For a sufficiently small time delay or a sufficiently small Lipschitz constant, the existence and uniqueness of such BSDEs is obtained. As an adjoint process, a class of stochastic differential equations (SDEs) is introduced, whose coefficients also depend on the present, the past and the future of its solutions. The existence and uniqueness of such SDEs is proved for a sufficiently small time advance or a sufficiently small Lipschitz constant. A duality between such BSDEs and SDEs is established.     

Optimal control of backward stochastic heat equation with Neumann boundary control and noise

This paper considers a stochastic control problem in which the dynamic system is a controlled backward stochastic heat equation with Neumann boundary control and boundary noise and the state must coincide with a given random vector at terminal time. Through defining a proper form of the mild solution for the state equation, the existence and uniqueness of the mild solution is given. As a main result, a global maximum principle for our control problem is presented. The main result is also applied to a backward linear-quadratic control problem in which an optimal control is obtained explicitly as a feedback of the solution to a forward–backward stochastic partial differential equation.