On the Generalized Stochastic Dirichlet Problem—Part I: The Stochastic Weak Maximum Principle

We treat the stochastic Dirichlet problem \(L\lozenge u = h+\nabla f\) in the framework of white noise analysis combined with Sobolev space methods. The input data and the boundary condition are generalized stochastic processes regarded as linear continuous mappings from the Sobolev space \(W_0^{1,2}\) into the Kondratiev space (S)_???1. The operator L is assumed to be strictly elliptic in divergence form \(L\lozenge u=\nabla(A\lozenge\nabla u+b\lozenge u)+c\lozenge\nabla u+d\lozenge u\). Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be generalized random processes, and the product of two generalized processes, denoted by \(\lozenge\), is interpreted as the Wick product. In this paper we prove the weak maximum principle for the operator L, which will imply the uniqueness of the solution to \(L\lozenge u = h+\nabla f\).     

Discreteness of the Spectrum of the Laplacian and Stochastic Incompleteness

We propose geometric conditions equivalent to the discreteness of the spectrum of the Laplacian on a class of Riemannian manifolds with ends close to warped products. For this class of manifolds we establish a relationship between discreteness of the spectrum and stochastic incompleteness.      

Pathwise Uniqueness of the Solutions toVolterra Type Stochastic DifferentialEquations in the Plane

In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in stead of Ito formula, which leads to simplicity the process of proof and extends the result to unbounded coefficients case.     

Existence and Regularity of the Pressure for the Stochastic Navier–Stokes Equations

We prove, on one hand, that for a convenient body force with values in the distribution space (H ^-1(D)) ^d , where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier–Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual space V of the divergence free subspace V of (H ¹ ₀(D)) ^d , in general it is not possible to solve the stochastic Navier–Stokes equations. More precisely, although such body forces have been considered, there is no topological space in which Navier–Stokes equations could be meaningful for them.     

A Stochastic Process Generated by the Lévy Laplacian

In this paper we give a stochastic process generated by the Lévy Laplacian in the white noise analysis with a characterization of the Laplacian.     

Dynkin’s Isomorphism Theorem and the Stochastic Heat Equation

Consider the stochastic heat equation \(\partial_t u = \mathcal{L} u + \dot{W}\), where \(\mathcal{L}\) is the generator of a [Borel right] Markov process in duality. We show that the solution is locally mutually absolutely continuous with respect to a smooth perturbation of the Gaussian process that is associated, via Dynkin’s isomorphism theorem, to the local times of the replica-symmetric process that corresponds to \(\mathcal{L}\). In the case that \(\mathcal{L}\) is the generator of a Lévy process on R ^d , our result gives a probabilistic explanation of the recent findings of Foondun et al. (Trans Am Math Soc, 2007).     

Mild Solutions for the Stochastic Generalized Burgers–Huxley Equation

Journal of Theoretical Probability - In this work, we consider the stochastic generalized Burgers–Huxley equation perturbed by space–time white noise and discuss the global solvability...     

Convergence of a Stochastic Method for the Modeling of Polymeric Fluids

We present a convergence analysis of a stochastic method for numerical modeling of complex fluids using Brownian configuration fields (BCF) for shear flows. The analysis takes into account the special structure of the stochastic partial differential equations for shear flows. We establish the optimal rate of convergence. We also analyze the nature of the error by providing its leading order asymptotics.     

Stochastic conservation laws?

We examine conservation laws, typically the conservation of linear momentum, in the light of a recent successful formulation of fermions as Kerr–Newman type Black Holes, which are created fluctuationally from a background Zero Point Field as in Prigogine's cosmology also. We conclude that these conservation laws are to be taken in the spirit of thermodynamic laws.     

Stochastic Processes Associated with a Sum of the Lévy Laplacians

Abstract

In this paper, we give a stochastic expression of a semigroup generated by a sum of the Lévy Laplacians acting on a class of S-transforms of white noise distributions in terms of an infinite sequence of independent Brownian motions.     

On the Well-Posedness for Stochastic Schr¨odinger Equations with Quadratic Potential

The authors investigate the influence of a harmonic potential and random perturbations on the nonlinear Schrödinger equations. The local and global well-posedness are proved with values in the space Σ(? ⁿ ) = {f ∈ H ₁(? ⁿ ), |·|f ∈ L ²(? ⁿ )}. When the nonlinearity is focusing and L ²-supercritical, the authors give sufficient conditions for the solutions to blow up in finite time for both confining and repulsive potential. Especially for the repulsive case, the solution to the deterministic equation with the initial data satisfying the stochastic blow-up condition will also blow up in finite time. Thus, compared with the deterministic equation for the repulsive case, the blow-up condition is stronger on average, and depends on the regularity of the noise. If ? = 0, our results coincide with the ones for the deterministic equation.     

Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.     

A Note on the Rate of Convergence of the Euler–Maruyama Method for Stochastic Differential Equations

Stochastic processes are approximated by wavelet operators. For the stochastic process and the scale function satisfying some conditions, a new method is given to estimate the degree of approximation.     

Convergence for a Splitting-Up Scheme for the 3D Stochastic Navier-Stokes-α Model

We propose and analyze a splitting-up scheme for the numerical approximation of the 3D stochastic Navier-Stokes-α model. We prove the convergence of the scheme to the unique variational solution of the 3D stochastic Navier-Stokes-α model when the time step tends to zero.     

On the Value Function of Weakly Coercive Problems in Nonlinear Stochastic Control

In this paper we investigate via a dynamic programming approach some nonlinear stochastic control problems where the control set is unbounded and a classical coercivity hypothesis is replaced by some weaker assumptions. We prove that these problems can be approximated by finite fuel problems; show the continuity of the relative value functions and characterize them as unique viscosity solutions of a quasi-variational inequality with suitable boundary conditions.     

On the Generalized Stochastic Dirichlet Problem—Part II: Solvability,Stability and the Colombeau Case

In this paper we consider the stochastic Dirichlet problem \(L\lozenge u=h+\nabla f\) in the framework of white noise analysis combined with Sobolev space and Colombeau algebra methods. The operator L is assumed to be strictly elliptic in divergence form \(L\lozenge u=\nabla(A\lozenge\nabla u+b\lozenge u)+c\lozenge\nabla u+d\lozenge u\). Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be generalized random processes, and the product of two generalized processes is interpreted as the Wick product. Generalized random processes are considered as linear bounded mappings from the Sobolev space \(W_0^{1,2}\) into the Kondratiev space (S)_???1. In this paper we prove existence and uniqueness of the problem of this form in the case when the operator L generates a coercive bilinear form, and then extend this result to the general case. We also consider the case when the coefficients of L, the input data and the boundary condition are Colombeau-type generalized stochastic processes.     

A Functional LIL for Stochastic Integrals and the Lévy Area Process

A functional law of the iterated logarithm is obtained for processes given by certain stochastic integrals. This extends earlier results by Shi⁽¹²⁾ and Rémillard⁽¹⁰⁾ who established analogues of the classical limit results of Chung⁽⁴⁾ for a variety of processes, including Lévys stochastic area process. The functional aspects of our results are motivated by a paper of Wichura⁽¹³⁾ on Brownian motion. Proofs depend on small ball probability estimates, and yield the small ball probabilities of the weighted sup-norm for the processes given by these stochastic integrals.     

Stochastic bilinear spectral systems †

A class of bilinear stochastic partial differential equations is investigated using a semigroup approach. Existence of a mild solution is obtained by proving a maximal inequality for stochastic convolution integrals with a stochastic evolution operator U(t,s) as integrand; moreover, we show the existence of a regular version in t. Under an additional assumption we show the existence of a continuous version of U (.,.) in the space of bounded operators on the state space. Finally, we analyse a p.d.e. model of a simply supported beam to illustrate the applicability of our results to modelling uncertainty in large flexible space structures     

Stochastic Nonlinear Complementarity Problems: Stochastic Programming Reformulation and Penalty-Based Approximation Method

We consider a class of stochastic nonlinear complementarity problems. We first reformulate the stochastic complementarity problem as a stochastic programming model. Based on the reformulation, we then propose a penalty-based sample average approximation method and prove its convergence. Finally, we report on some numerical test results to show the efficiency of our method.