| The solution u(x,t) to a stochastic heat equation driven by space-time white noise doesn't exist as a function in Rd , for d ≥ 2, while interestingly, U( x,t) = 0t u(x,t)dt exists as a function, for d ≤ 3. In this thesis, a Bessel space of distributions is used and the stochastic integral, created by Ito and developed by Walsh, is extended to be a distribution in Bessel space. L2 Isometry properties are used in the extension. Then we apply this extended stochastic integral to study the properties of solution to a class of heat equation driven by space-time white noise. Existence and uniqueness are proved for the solution. The same result can be obtained by applying same method to study the heat equation driven by spatially homogeneous Gaussian noise. |
