The Ornstein-Uhlenbeck operator
Let (B t , t≥0) be a standardR m -valuedBrownian motion Consider thelinear stochastic differential equation dX t (x) =√
2dB t −X t (x)dt, (2.1) with initial conditionx∈R m Using theItˆo formula, it is immediate to check that the solution to (2.1) is given by
The operator semigroup associated with the Markov process solution to (2.1) is defined by P t f(x) = E m f
Notice that the law of
−(t−s) dB s is Gaussian, mean zero and with co- variance given by
I This fact, together with (2.2), yields
We are going to identify the class of functionsf for which the right hand- side of (2.3) makes sense ; and we will also compute the infinitesimal generator of the semigroup.
We have the following facts about the semigroup which is generated by(X t , t≥0):
1) (P t , t≥0) is a contraction semigroup on L p (R m ;à m ), for all p≥1.
2) For any f ∈ C b 2 (R m ) and everyx∈R m , lim t→0
1) LetXandY be independent random variables with lawà m The law of exp(−t)X+
1−exp(−2t)Y is alsoà m Therefore, (à m ìà m )◦T −1 à m , where T(x, y) = exp(−t)x+
1−exp(−2t)y Then, the definition of P t f and this remark yields
2) This follows very easily by applying the Itˆo formula to the process f(X t ).
3) We must prove that for any g∈L 2 (R m ;à m ),
1−exp(−2t)Y f(X) , whereX andY are two independent standard Gaussian variables This follows easily from the fact that the vector (Z, X), where
1−exp(−2t)Y, has a Gaussian distribution and each component has lawà m
The appropriate spaces to perform the integration by parts men- tioned above are defined in terms of theeigenvalues of the operatorL m
In this article, we will calculate eigenvalues using Hermite polynomials, defined for \( n \geq 0 \) and \( x \in \mathbb{R} \) In the subsequent chapter, we will leverage this relationship within a stochastic framework.
Notice that H 0 (x) = 1 and H n (x) is a polynomial of degree n, for any n ≥ 1 Hence, any polynomial can be written as a sum of Hermite polynomials and therefore the set (H n , n≥0) is dense inL 2 (R, à 1 ). Moreover,
(n!m!) 1 2 δ n,m , whereδ n,m denotes theKronecker symbol Indeed, this is a consequence of the identity
= exp(st), which is proved by a direct computation Thus, √ n!H n , n ≥ 0 is a complete orthonormal system of L 2 (R, à 1 ).
One can easily check that
The operator L1 is non-positive, and the sequence of eigenfunctions (Hn, n≥0) corresponds to the eigenvalues (-n, n≥0) Extending this concept to any finite dimension m≥1 is straightforward Let a = (a1, a2, ), where ai ∈ N, represent a multi-index, with the condition that ai = 0 for any i > m We then define the generalized Hermite polynomial.
Set |a| = m i=1 a i and define L m = m i=1 L i 1 , with L i 1 =∂ x 2 i x i −x i ∂ x i Then
Therefore, the eigenvalues of L m are again (−n, n ≥ 0) and the corre- sponding sequence of eigenspaces are those generated by the sets m i=1 a i !H α i (x i ), m i=1 α i =n, α i ≥0
Notice that if |a|=n, then H a (x) is a polynomial of degree n Denote by P m the set of polynomials on R m Fix p ∈ [1,∞) and k ≥ 0 We define a seminorm onP m as follows:
L p (à m ) , (2.7) where for anys∈R, the operator (I−L m ) s is defined using thespectral decomposition ofL m
2) The norms ã k,p , k ≥ 0, p ∈ [1,∞), are compatible in the following sense: If (F n , n ≥ 1) is a sequence in P m such that lim n→0 F n k,p = 0 and it is a Cauchy sequence in the norm ã k ,p , then lim n→0 F n k ,p = 0.
1) Clearly, by H¨older’s inequality the statement holds true fork=k Hence it suffices to check thatF k,p ≤ F k ,p , for anyk≤k To this end we prove that for anyα ≥0,
Fix F ∈ P m Consider its decomposition in L 2 (R m ;à m ) with re- spect to the orthonormal basis given by the Hermite polynomials,
F = ∞ n=0 J n F Since L m is the infinitesimal generator of P t , the formal relationship P t = exp(L m ) yields P t F = ∞ n=0exp(−nt)J n F. The obvious identity
−t(n+ 1) t α−1 dt, valid for any α >0, yields
Hence, the contraction property of the semigroupP t yields
2) SetG n = (I−L) k 2 F n ∈ P m By assumption, (G n , n≥1) is a Cauchy sequence in L p (à m ) Let us denote by G its limit We want to check thatG= 0 Let H∈ P m , then
R m (I−L) k−k 2 G n (I−L) k−k 2 H dà m = 0. Since P m is dense in L q (à m ), for any q ∈ [1,∞) (see for instance ref. [21]), we conclude thatG= 0 This ends the proof of the Lemma
LetD k,p m be the completion of the set P m with respect to the norm ã k,p defined in (2.7) Set
Lemma 2.2 ensures that the setD ∞ m is well defined Moreover, it is easy to check that D ∞ m is an algebra.
Remark 2.1 Let F ∈ D ∞ m Consider a sequence (F n , n ≥ 1) ⊂ P m converging toF in the topology of D ∞ m , that is n→∞ limF−F n k,p = 0, for any k ≥ 0, p ∈ [1,∞) Then L m F is defined as the limit in the topology of D ∞ m of the sequenceF−(I−L m )F n
The adjoint of the differential
We are seeking an operator δ m that serves as the adjoint of the gradient ∇ in the space L²(R m, à m) This operator should operate on functions ϕ: R m → R m, yielding results in the realm of real-valued functions defined on R m, while adhering to the established duality relation.
E m ∇f, ϕ=E m (f δ m ϕ), (2.10) where ã,ã denotes the inner product inR m Assume first that f, ϕ i ∈
P m ,i= 1, , m Then, an integration by parts yields
The definition (2.11) yields the next useful formula δ m (f∇g) =−∇f,∇g −f L m g, (2.12) for anyf, g∈ P m
We remark that the operatorδ 1 satisfies δ 1 H n (x) =xH n (x)−H n (x) =xH n (x)−H n−1 (x)
Therefore it increases the order of a Hermite polynomial by one.
Remark 2.2 All the above identities make sense for f, g ∈ D ∞ m In- deed, it suffices to justify that one can extend the operator ∇ to D ∞ m Here is one possible argument:
LetS(R m ) be the set of Schwartz test functions Consider the isom- etryJ :L 2 (R m , λ m )→L 2 (R m , à m ) defined by
, where λ m denotes Lebesgue measure on R m Following reference [56] (page 142), k≥0D 2,k m =J
Then, for anyF ∈D ∞ m there exists
F˜∈ S(R m ) such thatF =J( ˜F) and one can define∇F =J(∇F˜).
In the upcoming chapter, we will explore Meyer’s result on the equivalence of norms, which demonstrates that the infinite-dimensional counterparts of the spaces D k,p m are the appropriate settings for the application of the Malliavink-th derivative.
An integration by parts formula
Let F : R m → R n be a random vector, F = (F 1 , , F n ) We assume that F ∈D ∞ m (R n ); that is, F i ∈D ∞ m , for any i= 1, , n The
Malliavin matrix of F — also called covariance matrix — is defined by
Notice that by its very definition, A(x) is a symmetric, non-negative definite matrix, for anyx∈R m ClearlyA(x) =DF(x)DF(x) T , where
DF(x) is the Jacobian matrix at x and the superscript T means the transpose.
We want to give sufficient conditions ensuring existence of a density forP◦F −1 We shall apply the criterium of part 1) of Proposition 1.2.
Let us perform some computations showing that (∂ i ϕ)(F), i = 1,
, n, satisfies a linear system of equations Indeed, by the chain rule,
F(x) l , (2.13) l = 1, , n Assume that the matrix A(x) is inversible à m -almost everywhere Then one gets
Taking expectations formally and using (2.12), (2.14) yields
This is an integration by parts formula as in Definition 1.1.
For higher differential orders, things are a little bit more difficult, but essentialy the same ideas would lead to the analogue of formula (1.1) withα= (1, ,1) andG= 1.
The preceding discussion and Proposition 1.2 yield the following result.
1) The matrix A(x) is invertible for every x ∈ R m , à m -almost ev- erywhere.
Then the law of F is absolutely continuous with respect to the Lebesgue measure on R n
The assumptions in 2) show that
∇A −1 i,l ,∇F l +A −1 i,l L m F l is finite Therefore, one can take expectations on both sides of (2.14).
This finishes the proof of the Proposition
Remark 2.3 The proof of smoothness properties for the density re- quires an iteration of the procedure presented in the proof of the Propo- sition 2.1.
Developing Malliavin Calculus in a finite-dimensional Gaussian space offers valuable insights into this complex subject, serving as an excellent training resource due to the explicit nature of the computations involved Stroock emphasizes the importance of the finite-dimensional framework before delving into the fundamental aspects of the Calculus, a strategy also adopted by Ocone.
[48] We have followed essentially his presentation The proof of Lemma2.2 can be found in reference [68]in the general infinite dimensional framework.
Exercises
Hint: Using the definition of P t and the Laplace transform of the Gaus- sian measure, check that
The Basic Operators of Malliavin Calculus
This chapter presents the three fundamental operators essential for constructing the infinite-dimensional Malliavin calculus within a Gaussian space: the Malliavin derivative, the divergence operator (its adjoint), and the Ornstein-Uhlenbeck operator.
In this article, we outline the foundational probability space defined by a real separable Hilbert space H, where we denote the norm and inner product as ã H and ã,ã H, respectively We establish a probability space (Ω,G, à) along with a family of random variables M W(h), for h in H, ensuring that the mapping h→ W(h) is linear Each random variable W(h) is Gaussian, with an expected value of EW(h) = 0, highlighting the key properties of the defined probability space.
In the context of a complete orthonormal system in a Hilbert space H, we construct a family using a canonical probability space defined by a sequence of standard independent Gaussian random variables This space is represented as (Ω, G, P), where Ω is the product of real numbers raised to the power of N, and G is the corresponding Borel σ-algebra For any element h in H, the series involving the inner product of h with the orthonormal basis elements, weighted by the Gaussian random variables, converges in L² to a random variable denoted as W(h) It is important to note that the set M remains closed in this construction.
Gaussian subspace of L 2 (Ω) that is isometric to H In the sequel, we shall assume thatG is the σ-field generated byM.
Here is an example of such aGaussian family LetH=L 2 (A,A, m), where (A,A, m) is a separableσ-finite, atomless measure space For any
F ∈ A with m(F)