The second term on the right hand side of 41 embodies smoothing of a different type. Suppose that T satisfies the estimate
kDT ϕk
∞
≤ Ckϕk
∞
+ γkDϕk
∞
43 for some positive C and some
γ ∈ 0, 1. Note that this is a variation of what is usually referred to as the Lasota-Yorke inequality [LY73, Liv03] or the Ionescu-Tulcea-Marinescu inequality [ITM50].
Though 43 does not imply that T ϕ belongs to a smoother function space then ϕ, it does imply that
the gradients of T ϕ are smaller then those of ϕ, at least as long as the gradients of ϕ are sufficiently
steep. This is in line with a more colloquial idea of smoothing, though not in line with the traditional mathematical definition used.
5.1.1 Strongly dissipative setting
Where does the assumption C
Π
C
J
+ 2κC
L
come from? This is easy to understand if we consider the “trivial” case Π = 0. In this case, Assumption B.1 is empty and the projection Π
⊥
is the identity. Therefore, the left hand sides from Assumptions B.3 and B.4 coincide, so that one has C
J
= −C
Π
and our restriction becomes C
J
+ κC
L
0. This turns out to be precisely the right condition to impose if one wishes to show that
E kJ
0,n
k → 0 at an exponential rate:
Proposition 5.8 Let Assumptions B.2 and B.3 hold. Then, for any p
∈ [0, ¯p2], one has the bound
E kJ
0,n
k
p
≤ exp pκV u + pC
T
n ,
with κ = η1 − η
′
and C
T
= C
J
+ κC
L
. Proof. Using the fact that
kJ
0,n
k ≤ kJ
n −1,n
kkJ
0,n −1
k, we have the following recursion relation:
E exp p κV u
n
kJ
0,n
k
p
≤ E E exp p
κV u
n
kJ
n −1,n
k
p
| F
n −1
kJ
0,n −1
k
p
≤ E E
kJ
n −1,n
k
¯ p
| F
n −1
p ¯
p
E exp
p¯p ¯
p − p
κV u
n
F
n −1
¯ p
−p ¯
p
kJ
0,n −1
k
p
≤ e
pC
T
E exp p
κV u
n −1
kJ
0,n −1
k
p
, where we made use of Assumptions B.2 and B.3 in the second inequality. It now suffices to apply
this n times and to use the fact that kJ
0,0
k = 1. The assumptions ¯pκ 1 and p ≤ ¯p2 ensure that p¯
p ≤ ¯p − p so that the bound 39 can be used.
We now use this estimate to prove a version of Theorem 5.5 when the system is strongly dissipative:
Proposition 5.9 Let Assumptions B.2 and B.3 hold and set C
T
= C
J
+ κC
L
with κ = η1 − η
′
as before. Then, for any
ϕ : H → R and n ∈ N one has
kDP
n
ϕuk ≤ γ
n
e
κV u
p P
n
kDϕk
2
u . with
γ = e
C
T
. In particular, the semigroup P
t
has the asymptotic strong Feller property whenever C
T
0.
687
Proof. Fixing any ξ ∈ H with kξk = 1, observe that
D P
t
ϕuξ = E
u
Dϕu
t
J
0,t
ξ ≤ Æ
E kJ
0,t
k
2
p
E kDϕk
2
u
t
. Applying Proposition 5.8 completes the proof.
Comparing this result to the bound 41 stated in Theorem 5.5 shows that, the combination of the smoothing Assumption B.4 with Assumption B.1 on the Malliavin matrix allows us to consider the
system as if its Jacobian was contracting at an average rate C
Π
− C
J
2 instead of expanding at a rate C
J
. This is precisely the rate that one would obtain by projecting the Jacobian with Π
⊥
at every second step. The additional term containing
P
2n
ϕ
2
appearing in the right hand side of 41 should then be interpreted as the probabilistic “cost” of performing that projection. Since this “projection”
will be performed by using an approximate inverse to the Malliavin matrix, it makes sense that the larger the lower bound on
M
t
is, the lower the corresponding probabilistic cost.
Remark 5.10 It is worth mentioning, that nothing in this section required that the number of Wiener process be finite. Hence one is free to take d =
∞, as long as all of the solutions and linearization are well defined which places conditions on the g
k
.
5.2 Transfer of variation
