To better understand Markov process, I am trying to summarize basic concepts that appear in Golden triangle which introduces the equivalence between

  • non positive self adjoint operator, L
  • closed symmetric non negative bilinear forms
  • strong continuous self-adjoint contraction semigroup

1. self adjoint operator

adjoint: \left\langle A^{*} f, g\right\rangle_{\mathcal{H}}=\langle f, A g\rangle_{\mathcal{H}} with its domain

\mathcal{D}(A^*)=\{ f \in \mathcal{H}, \exists \text{ } c(f) \ge 0, \forall \text{ } g \in \mathcal{D}(A), | \langle f, Ag \rangle_{\mathcal{H}} | \le c(f) \| g \|_{\mathcal{H}} \}

self-adjoint: symmetric and same domain for operator and its adjoint

2. semigroup

Image
magma with associative = semigroup

3. bilinear form on hilbert space

I will try to update how the above concepts could be related to Markov process and its transition kernels.

ref: https://fabricebaudoin.wordpress.com/2019/01/16/lecture-2-semigroups-on-hilbert-spaces-the-golden-triangle/