Tom Lindstrøm

The burgers equation with a noisy force and the stochastic heat equation

We consider the multidimensional Burgers equation with a viscosity term and a random force modelled by a functional of time-space white noise, Wick products. Then we show that the nonlinear equation (B) can be transformed into a linear, stochastic heat equation with a noisy potential. This heat equation is solved explicitly in the following two cases.

The Pressure Equation for Fluid Flow in a Stochastic Medium

AbstractAn equation modelling the pressurep(x) =p(x, w) atx ∈D ⊂Rd of an incompressible fluid in a heterogeneous, isotropic medium with a stochastic permeabilityk(x, w) ≥ 0 is the stochastic partial differential equation

Discrete wick calculus and stochastic functional equations

\left\{ {\begin{array}{*{20}c} {div(k(x,{\mathbf{ }}\omega )\diamondsuit \nabla p(x,\omega )){\mathbf{ }} = {\mathbf{ }}--f(x);{\mathbf{ }}x \in D} \\ {\begin{array}{*{20}c} {p(x,{\mathbf{ }}\omega ){\mathbf{ }} = {\mathbf{ }}0;} & {x \in \partial D} \\ \end{array} } \\ \end{array} } \right.

Nonlinear stochastic integrals for hyperfinite Lévy processes

wheref is the given source rate of the fluid, ◊ denotes Wick product.We representk as the positive noise given by the Wick exponential of white noise, and we find an explicit formula for the (unique) solutionp(x, w), which is proved to belong to the space (S)−1 of generalized white noise distributions.

Internal Martingales and Stochastic Integration

SummaryWe give a program for solving stochastic boundary value problems involving functionals of (multiparameter) white noise. As an example we solve the stochastic Schrödinger equation {ie391-1} whereV is a positive, noisy potential. We represent the potentialV by a white noise functional and interpret the product of the two distribution valued processesV andu as a Wick productV ◊u. Such an interpretation is in accordance with the usual interpretation of a white noise product in ordinary stochastic differential equations. The solutionu will not be a generalized white noise functional but can be represented as anL1 functional process.

Random relaxed controls and partially observed stochastic systems

We develop Wick calculus over finite probability spaces and prove that there is a one-to-one correspondence between the solutions of Wick stochastic functional equations and the solutions of the deterministic functional equations obtained by ‘turning off’ the noise. We also point out some possible applications to ordinary and partial stochastic differential equations.

Anderson’s Brownian motion and the Infinite Dimensional Ornstein-Uhlenbeck Process

Stochastic partial differential equations (SPDEs) often have solutions that are known to be pure Schwartz distributions i.e. not functions. To make sense of such equations one needs to introduce some kind of smoothing parameters. This paper is concerned with stability properties of the solutions as one lets the smoothing parameters approach some kind of delta function. The first part of the paper concentrates on linear functionals in connection with SPDEs. In the second part we adress similar problems related to functionals of Hida distributions

Stochastic analysis and applications

I develop a notion of nonlinear stochastic integrals for hyperfinite Lévy processes and use it to find exact formulas for expressions which are intuitively of the form