Markov process

Markov process ->(defines via regaular conditional probability wrt filtration) transition functions ->(induces) semigroup on some domain ->(defines) generator

Kolmogorov’s theorem: the connection between “Markov process” and “transition functions”
Hille-Yisoda: the connection between “strongly continuous contraction semigroup” and “linear operator on a Banach space”. \

In discrete time, the martingale problem characterizes the Markov process but in continuous time
Martingale problem: the solution may not be unique, and not all solutions all Markov processes
Solution of martingale problem (martingale property wrt $P_x$ of $f(X_t) - \int_0^t \mathcal{L}f(X_s) \mathrm{d} s$ ) -> transition function (via $P_x$) ->(induces) semigroup ->(defines) generator $L$ (extension of $\mathcal{L}$)
Generator (+closed operator wrt graph norm) of a strongly continuous contraction semigroup and is an extension of $(\mathcal{L}, \mathcal{A})$-> uniquenss semigroup with a generaor that extends $\mathcal{L}$

Log Sobolev inequality in EDG

Goal: Want to show a Log-Sobolev inequality for EDG process.

Consequence: Exponential Convergence to equilibrium