Positive Harmonic Functions and Diffusion

Stochastic processes and filtrations

Let (Y, ℳ, P) be a probability space and denote points in Y by q. A Borel measurable function η: Y → Rd is called an Rd-valued random variable on (Y,ℳ). The expectation Eη of η with respect to the probability measure P is defined by Eη = ∫yηdP. The Rd-valued random variable η on (Y,ℳ, P) induces a probability measure v on Rd defined by v(B) = P(η ∈ B), for B ∈ Bd, where Bd is the Borel σ-algebra on Rd; v is called the distribution of η.

A family ℳt, t ≥ 0, of σ-algebras satisfying ℳt1 ⊆ ℳt2 for t1 ≤ t2 and ℳt ⊆ ℳ for all t ≥ 0 is called a filtration on (Y, ℳ). If for all t ≥ 0, ℳt includes all sets of P-measure zero, the filtration is called complete on (Y, ℳ, P). If ℳ't is the smallest σ-algebra containing ℳt and all sets of P-measure zero, then ℳ't is called the completion of ℳt.

A measurable map ζ: [0, ∞) × Y →,Rd is called an Rd-valued stochastic process on (Y,ℳ, P). We frequently suppress the variable q ∈ Y and write ζ(t) for ζ(t, q). The stochastic process ζ(t) on (Y, ℳ, P) induces a probability measure v on B([0, ∞), Rd), the space of measurable functions from [0, ∞) to Rd with the sup-norm topology, defined by v(B) = P(ζ(·) ∈ B), for Borel sets B in B([0, ∞), Rd). The measure v is called the distribution of ζ(·).