Let f ∈ L2 (m) and define the functional for u in the domain of E. Prove that ψ is minimized by u = Rλf, and that this function is the unique minimizer. Let Pt be the semigroup associated with a...

Let f ∈ L²
(m) and define the functional

for u in the domain of E. Prove that ψ is minimized by u = R_λf, and that this function is the unique minimizer.

Let P_t
be the semigroup associated with a Dirichlet form and define

J (dx, dy) = Pt(x, dy) m(dx).

(1) Prove that if f , g are continuous with compact support, then

(2) With f and g continuous with compact support, prove that

And

(3) Let k(x) = 1 − P_t1(x). Prove that if E(t) is defined as in, then

(4) Is E^(t)
a Dirichlet form? A regular Dirichlet form?