A Skew Stochastic Heat Equation

Stochastic heat equation with white-noise drift

On the Exact Controllability of a Nonlinear Stochastic Heat Equation

The bounded open subsetG of Rn is assumed to have a smooth boundary and g is continuous from L2(0,T ;L2(G)) into L2(0,T ;L2(G)). The existence of a solution of the linear version of (1.1) is established in Section 3 of the paper. The nonlinear case is considered in Section 4. The main assumptions of the paper and some preliminary results are given in Section 2. Exact controllability of the wave...

Variations of the Solution to a Stochastic Heat Equation

We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, F (t), has a nontrivial quartic variation. This process, therefore, has infinite quadratic variation and is not a semimartingale. It follows that the classical Itô calculus does not apply. Motivated by heuristic ideas ab...

Variations of the solution to a stochastic heat equation II

We consider the solution u(x, t) to a stochastic heat equation. For fixed x, the process F (t) = u(x, t) has a nontrivial quartic variation. It follows that F is not a semimartingale, so a stochastic integral with respect to F cannot be defined in the classical Itô sense. We show that for sufficiently differentiable functions g, a stochastic integral ∫ g(F ) dF exists as a limit in distribution...

On a Linear Stochastic Wave Equation Modeling Heat Flow

We consider a stochastic Klein-Gordon wave equation modeling heat flow in a linear field that is coupled to thermal reservoirs at different temperatures. We discuss, in a perturbative context, the approach to a stationary, non-equilibrium state, and show that the state is supported on field configurations which are Hölder continuous, with any exponent less than 1/2. We determine the heat flux t...