Invariant measures for continuous state Markov processes

The first part of this thesis develops the theoretical foundations of stochastic calculus. Building on these tools, it then examines SDEs under standard regularity assumptions—specifically Lipschitz continuity and sub‑linear growth of drift and diffusion coefficients. Under these conditions, we establish the existence of strong solutions and their pathwise uniqueness. These properties, in turn, yield important structural consequences: solutions form Markov processes, generate Feller semigroups, and can be characterized as diffusions whose infinitesimal generators arise naturally from Itô’s formula. The final part of the thesis focuses on invariant measures associated with semigroups. For SDEs with monotone decreasing drift, the long‑term behavior of strong solutions allows us to prove existence and uniqueness of an invariant measure. More generally, the Krylov–Bogoliubov theorem is used to establish a general criteria to ensure existence of invariant measures.