A stability property of the stochastic heat equation

The Stochastic Heat Equation Driven by a Gaussian Noise: Germ Markov Property

Let u = {u(t, x); t ∈ [0, T ], x ∈ R} be the process solution of the stochastic heat equation ut = ∆u+ Ḟ , u(0, ·) = 0 driven by a Gaussian noise Ḟ , which is white in time and has spatial covariance induced by the kernel f . In this paper we prove that the process u is locally germ Markov, if f is the Bessel kernel of order α = 2k, k ∈ N+, or f is the Riesz kernel of order α = 4k, k ∈ N+.

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 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...

Stochastic heat equation with white-noise drift