On Gaussian Invariant Measure to First Order PDEs

It is vital to consider invariant measures of dynamical systems induced by partial differential equations with irregular coefficients because of their application and importance in applied and pure mathematics. This thesis investigates the existence of invariant measures for the Lasota equation in threefold.
Firstly, we consider a special choice of the coefficients using the interpolation theory. We show that the law of the Liouville Fractional Brownian Motion with the Hurst parameter H is an invariant measure of the Lasota equation with the drift coefficient a(x) = x and the multiplication parameter λ = H − (1/2). Secondly, we study the existence and the uniqueness of mild solutions and prove the existence of invariant measures to the linear Lasota equation assuming only some basic properties of the coefficients a and c. Lastly, we consider the nonlinear Lasota equation and study the existence and the uniqueness of a global mild solution with a new set of assumptions for the coefficient c and prove the existence of invariant measures.