1.4 The Ornstein-Uhlenbeck semigroup
1.4.1 The semigroup of Ornstein-Uhlenbeck
We assume that W ={W(h), h ∈ H} is an isonormal Gaussian process associated to the Hilbert spaceH defined in a complete probability space (Ω,F, P), and thatF is generated by W. We recall that Jn denotes the orthogonal projection on thenth Wiener chaos.
Definition 1.4.1 The Ornstein-Uhlenbeck semigroup is the one-parameter semigroup{Tt, t≥0} of contraction operators onL2(Ω)defined by
Tt(F) = ∞ n=0
e−ntJnF, (1.66)
for anyF ∈L2(Ω) .
There is an alternative procedure for introducing this semigroup. Sup- pose that the process W′ = {W′(h), h ∈ H} is an independent copy of W. We will assume thatW andW′ are defined on the product probability space (Ω×Ω′,F ⊗ F′, P ×P′). For any t > 0 we consider the process Z ={Z(h), h∈H}defined by
Z(h) =e−tW(h) +%
1−e−2tW′(h), h∈H.
This process is Gaussian, with zero mean and with the same covariance function asW. In fact, we have
E(Z(h1)Z(h2)) =e−2th1, h2H+ (1−e−2t)h1, h2H =h1, h2H. LetW : Ω→RH andW′ : Ω′→RH be the canonical mappings associated with the processes{W(h), h∈H}and{W′(h), h∈H}, respectively. Given
a random variableF ∈L2(Ω), we can writeF =ψF◦W, whereψFis a mea- surable mapping fromRH toR, determinedP◦W−1a.s. As a consequence, the random variable ψF(Z(ω, ω′)) =ψF(e−tW(ω) +√
1−e−2tW′(ω′)) is well definedP×P′ a.s. Then, for anyt >0 we put
Tt(F) =E′(ψF(e−tW +%
1−e−2tW′)), (1.67) whereE′denotes mathematical expectation with respect to the probability P′. Equation (1.67) is called Mehler’s formula. We are going to check the equivalence between (1.66) and (1.67). First we will see that both definitions give rise to a linear contraction operator on L2(Ω). This is clear for the definition (1.66). On the other hand, (1.67) defines a linear contraction operator onLp(Ω) for anyp≥1 because we have
E(|Tt(F)|p) = E(|E′(ψF(e−tW+%
1−e−2tW′))|p)
≤ E(E′(|ψF(e−tW+%
1−e−2tW′)|p)) =E(|F|p).
So, to show that (1.66) is equal to (1.67) onL2(Ω), it suffices to check that both definitions coincide whenF= exp
W(h)−12h2H
,h∈H. We have
E′
exp
e−tW(h) +%
1−e−2tW′(h)−1 2h2H
= exp
e−tW(h)−1
2e−2th2H
= ∞ n=0
e−nthnHHn
W(h) hH
= ∞ n=0
e−nt
n! In(h⊗n).
On the other hand,
Tt(F) = Tt
∞
n=0
1
n!In(h⊗n)
= ∞ n=0
e−nt
n! In(h⊗n), which yields the desired equality.
The operatorsTtverify the following properties:
(i) Tt is nonnegative (i.e.,F≥0 impliesTt(F)≥0).
(ii) Tt is symmetric:
E(GTt(F)) =E(F Tt(G)) = ∞ n=0
e−ntE(Jn(F)Jn(G)).
Example 1.4.1 The classical Ornstein-Uhlenbeck (O.U.) process on the real line{Xt, t∈R}is defined as a Gaussian process with zero mean and co- variance function given byK(s, t) =βe−α|s−t|, whereα, β >0ands, t∈R.
This process is Markovian and stationary, and these properties characterize the form of the covariance function, assuming thatK is continuous.
It is easy to check that the transition probabilities of the Ornstein-Uhlen- beck process Xt are the normal distributions
P(Xt∈dy|Xs=x) =N(xe−α(t−s), β(1−e−2α(t−s))).
In fact, for all s < t we have
E(Xt|Xs) = e−α(t−s)Xs, E((Xt−E(Xt|Xs))2) = β(1−e−2α(t−s)).
Also, the standard normal law ν=N(0, β)is an invariant measure for the Markov semigroup associated with the O.U. process.
Consider the semigroup of operators on L2(R,B(R), ν) determined by the stationary transition probabilities of the O.U. process (with α, β = 1).
This semigroup is a particular case of the Ornstein-Uhlenbeck semigroup introduced in Definition 1.4.1, if we take(Ω,F, P) = (R,B(R), ν),H =R, and W(t)(x) =tx for any t ∈R. In fact, if {Xs, s∈ R} is a real-valued O.U. process, for any bounded measurable function f on R we have for t≥0ands∈R
R
f(y)P(Xs+t∈dy|Xs=x) =
R
f(y)N(e−tx,1−e−2t)(dy)
=
R
f(e−tx+%
1−e−2ty)ν(dy)
= (Ttf)(x).
LetW be a Brownian measure on the real line. That is,{W(B), B∈ B(R)} is a centered Gaussian family such that
E(W(B1)W(B2)) =
R
1B1∩B2(x)dx.
Then the process
Xt=% 2αβ
t
−∞
e−α(t−u)dWu
has the law of an Ornstein-Uhlenbeck process of parameters α, β. Further- more, the process Xt satisfies the stochastic differential equation
dXt=%
2αβdWt−αXtdt.
Consider now the case whereH =L2(T,B, à) andàis aσ-finite atomless measure. Using the above ideas we are going to introduce an Ornstein- Uhlenbeck process parametrized byH. To do this we consider a Brownian measureBonT×R, defined on some probability space (Ω,F,P) and with intensity equal to 2à(dt)dx. Then we define
Xt(h) = t
−∞
T
h(τ)e−(t−s)B(dτ , ds). (1.68) It is easy to check thatXt(h) is a Gaussian zero-mean process with covari- ance function given by
E(X t1(h1)Xt2(h2)) =e−|t1−t2|h1, h2H. Consequently, we have the following properties:
(i) For any h∈ H, {Xt(h), t ∈R} is a real-valued Ornstein-Uhlenbeck process with parametersα= 1 andβ =h2H.
(ii) For anyt≥0,{Xt(h), h∈H}has the same law as {W(h), h∈H}. Therefore, for any random variable F ∈L0(Ω) we can define the com- position F(Xt). That is, F(Xt) is short notation for ψF(Xt), where ψF is the mapping from RH to Rdetermined by ψF(W) =F. Let Ftdenote theσ-field generated by the random variablesB(G), whereGis a measur- able and bounded subset of T×(−∞, t]. The following result establishes the relationship between the process Xt(h) and the Ornstein-Uhlenbeck semigroup.
Proposition 1.4.1 For anyt≥0, s∈R, and for any integrable random variableF we have
E(F(X s+t)|Fs) = (TtF)(Xs). (1.69) Proof: Without loss of generality we may assume that F is a smooth random variable of the form
F =f(W(h1), . . . , W(hn)),
where f ∈ Cp∞(Rn), h1, . . . , hn ∈ H, 1 ≤ i ≤ n. In fact, the set S of smooth random variables is dense in L1(Ω), and both members of Eq.
(1.69) are continuous in L1(Ω). We are going to use the decomposition Xs+t=Xs+t−e−tXs+e−tXs. Note that
(i) {e−tXs(h), h∈H}isFs-measurable, and
(ii) the Gaussian family{Xs+t(h)−e−tXs(h), h∈H}has the same law as{√
1−e−2tW(h), h∈H}, and is independent ofFs.
Therefore, we have
E(F (Xs+t)|F˜s) = E(f (Xs+t(h1), . . . , Xs+t(hn))|Fs)
= E
f(Xs+t(h1)−e−tXs(h1) +e−tXs(h1)), . . . , Xs+t(hn)−e−tXs(hn) +e−tXs(hn))|Fs
= E′
f%
1−e−2tW′(h1) +e−tXs(h1),
. . . ,%
1−e−2tW′(hn) +e−tXs(hn)
= (TtF)(Xs),
whereW′ is an independent copy ofW, andE′ denotes the mathematical
expectation with respect toW′.
Consider, in particular, the case of the Brownian motion. That means Ω =C0([0,1]), andP is the Wiener measure. In that case, T = [0,1], and the process defined by (1.68) can be written as
Xt(h) = 1
0
h(τ)Xt(dτ), whereXt(τ) =t
−∞
τ
0 e−(t−s)W(dσ, ds), andW is a two-parameter Wiener process on [0,1]×R with intensity 2dtdx. We remark that the stochastic process{Xt(ã), t∈R}is a stationary Gaussian continuous Markov process with values onC0([0,1]), which has the Wiener measure as invariant mea- sure.