Fundamental Solutions to the Fractional Heat Operator

In this thesis, we are interested in showing the existence of a fundamental solution to the fractional heat operator. The fractional heat operator is a nonlocal linear operator used to model the time evolution of anomalous diffusion processes whose applicability arises in a wide variety of fields in the physical sciences, engineering, economics, and finance. Fundamental solutions to a partial differential operator are a class of generalized solutions formulated with rich mathematical analysis grounded in classes of well-behaved smooth functions referred to as test functions and their continuous linear functionals referred to as distributions. A primary tool that we use for our existence problem is the Fourier transform. To be able to use the Fourier transform in the generalized function framework, it is necessary to expand the class of test functions to a larger space that is invariant under the Fourier transform. This new space of test functions, called the Space of Schwartz Test Functions, is a space of rapidly decreasing smooth functions whose invariance under the Fourier transform allows us to extend the Fourier transform to the space of distributions on the Schwartz test function space. This Space of Tempered Distributions is similarly invariant under the Fourier transform and thus desirable for our analysis.