Hamzi - Eurandom

Example (A Fokker-Planck Integrable System)
• The stationary Fokker-Planck equation for this example reads
f(x) = σρ ′ ∞(x)/(2ρ∞(x)), so that we find
αx + βx3 = σ ρ
2
′ ∞(x)
ρ∞(x) , ∀ x ∈ supp(ρ∞) .
• To obtain the parameters as a least squares fit to the observations, let
{xi} m i=1 ⊂ R, with xi ∈ supp(ˆρ∞), denote a (finite) sequence of samples.
Since the preceding equation holds for all x ∈ supp(ρ∞) ⊃ supp(ˆρ∞) we
have that
αxi + βx 3 i = ˆσ ˆρ
2
′ ∞(xi)
ˆρ∞(xi) ,
for 1 ≤ i ≤ m. Consequently, the estimators ˆα and ˆ β obtained by a least
squares fit solve the system of linear equations
(∑m i=1 x2 ∑ i
m
i=1 x4i ∑m i=1 x4 ∑ i
m
i=1 x6i ) ( )
ˆα
ˆβ
= ˆσ
2
m∑
i=1
ˆρ ′ ∞(xi)
ˆρ∞(xi)
( xi
which can be solved explicitly provided the system matrix is invertible.
x 3 i
)
. . . . . .
Boumediene Hamzi (Imperial College) On Control and RDS in RKHS June 4th, 2012 53 / 55