Sur certaines mesures produit conditionnées

H

x·y x y Rd
|x|:= √x·x x p > 0 Lp(µ)
f|f|pµ
µ f ∈L1(µ)
f
Eµ(f) := Eµf:= Rf dµ f µ
Varµ(f) := Eµ((f−Eµf)2)µ ν
ν µ
Ent(ν|µ) := (Eµdν
dµ log dν dµ =Eνlog dν dµ  νµ
+∞.
Ent ν
µEnt(ν|µ)<∞log(dν/dµ)∈L1(ν)
u∈R+7→ ulog(u)Ent(ν|µ)>0
µ=ν µ ν
fµfν
Ent(fν|fµ)Ent(ν|µ)
Ent(·|µ)

Σrr
n
(r1, . . . , rn)ri
i
Σr
S(Σr) Σr
Cr1,...,rn
r:= r!
r1!···rn!
Σr
(r1, . . . , rn)
S(Σr) := 1
rlog Cr1,...,rn
r.
=×log( )
S=κlog(W)

r−1(r1, . . . , rn)
r=r1+··· +rnr−1(r1, . . . , rn)
(p1, . . . , pn)r
Σ∞
Smoy(Σ∞) := lim
r→+∞
1
rlog Cr1,...,rn
r=−
n
X
i=1
pilog(pi),
+
(p1, . . . , pn)7→ S(p1, . . . , pn) := −Pn
i=1 pilog(pi)
{1, . . . , n}S
{1, . . . , n}
Rd
f7→ S(f) := −ZRd f(u) log(f(u)) du,
f
RdS(f⊗g) = S(f) + S(g)f g
RdRd0f⊗g
Rd⊗Rd0=Rd+d0
N(m, Σ) m
Σm
ΣSR
H
v

ft(x, v)R6(x, v)
t
∂tft(x, v) + v· ∇xft(x, v) = Q(ft, ft)(x, v)
Q
ft
x
v· ∇xftH
t7→ H(ft) := ZR3 ft(v) log(ft(v)) dv
t7→ S(ft) = −H(ft)
S S
H
∂tH(ft) = ZR3 Q(ft, ft)(1 + log(ft)) dv 60.
S
fS
f
R3S
FRdW:Rd→RRRdfW(u)du =
1fW:= e−WW(u)
u∈RdfW
RdfW
W
W fWfW
W fw+W=fW