Abstract
We propose an extension of Walsh's classical martingale measure stochastic integral that makes it possible to integrate a general class of Schwartz distributions, which contains the fundamental solution of the wave equation, even in dimensions greater than 3. This leads to a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension, in the case of a driving noise that is white in time and correlated in space. In the particular case of an affine multiplicative noise, we obtain estimates on $p$-th moments of the solution ($p\geq 1$), and we show that the solution is Hölder continuous. The Hölder exponent that we obtain is optimal.
Highlights
The main topic of this thesis is the study of the non-linear stochastic wave equation
After that, inspired from the series expansion obtained in the case of an affine multiplicative noise in Chapter 5 and using similar techniques to those developed by Kloeden and Platen in [14] for stochastic differential equations driven by a Brownian motion, we present an Ito-Taylor type expansion for the solution u(t, x) of (1.5)
Walsh’s stochastic integration with respect to martingale measures is the tool that we use in order to build and study the solution of the non-linear stochastic wave equation (1.5)
Summary
The main topic of this thesis is the study of the non-linear stochastic wave equation. The wave equation arises from Newton’s classical equation of mechanics when we model the displacement of a physical entity (a solid or an incompressible fluid) subject to internal and external forces. This entity can be a piece of wire held at its endpoints or a DNA molecule moving in a fluid, for example. As a motivation for the study of problems with spatial dimension greater than three, we mention that some problems of quantum mechanics, unrelated with the wave equation, for example, need higher dimensional spaces. If we only consider internal forces, Newton’s equation states that u must satisfy the homogeneous wave
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