MC 7: The Ornstein-Uhlenbeck Semigroup
1 Three line summary
There is a natural extension of the Laplacian to the Wiener space.
The Laplacian generates the Ornstein-Uhlenbeck semigroup.
The Ornstein-Uhlenbeck semigroup in finite dimensions is generated by the Ornstein-Uhlenbeck process, from which it derives its name.
2 The Laplacian of a random variable
First, we give some finite-dimensional motivation. Suppose that \(f\in C_c^\infty(\R^d\to\R^d)\) and \(g\in C_c^\infty(\R^d)\). Then an integration by parts shows that the adjoint of the gradient in \(L^2(\R^d)\) is minus the divergence. That is,
\[ \int_{\R^d}f(x) \cdot \nabla g(x) dx=-\int_{\R^d}(\nabla\cdot f(x)) g(x) dx. \]
Then, we define the Laplacian as minus the adjoint of the gradient \(\nabla\) composed with the gradient
\[ \Delta := -\nabla^*\circ \nabla . \]
Which gives the familiar \[\Delta_{\R^d} =\nabla\cdot \nabla=\partial_1^2+\ldots+\partial _d^2\].
Of course, this is all well and good when the domain of \(f,g\) is a finite-dimensional space. Otherwise, there is no Lebesgue measure. We now move to what is our base case in our series of blog posts and consider a probability space \((\Omega,\mathbb{P},\mathcal{F}_t)\) where \(\Ff_t\) is generated by a Wiener process \(W_t\). Then, as we have seen previously, Skorokhod integral \(\delta\) is the adjoint of the Malliavin derivative \(D\) so we would like to define
\[ \Delta := -\delta \circ D. \]
On what kind of random variables can we define this? Well let us take \(X=\sum_{n=0}^{\infty} I_n(f_n)\) with a rapidly decaying chaos expansion, then
\[ \Delta X=-\delta (DX)=-\delta \left(\sum_{n=1}^\infty nI_{n-1}(f_n(\cdot ,t))\right)=-\sum_{n=1}^\infty nI_{n}(f_n). \]
All we require for this expression to make sense is that- the right-hand side is in \(L^2(\Omega)\). That is, by Ito’s \(n\)-th isometry, that
\[ \sum_{n=0}^\infty n^2 \norm{f_n}_{L^2(I_n)}< \infty. \]
Is this a space we’ve dealt with before? Well if we recall our old spaces [\(\mathbb{D](</posts/Malliavin Calculus/3.Malliavin_derivative_1/index.qmd#eq-higher-order>)^{k,p}\)}. Then we have that \[\begin{multline*} \int_{I^2}\norm{D_{t,s}X}_{L^2(\Omega)}^2 ds d t=\int_{I^2}\norm{\sum_{n=2}^\infty n(n-1)I_{n-2}(f_n(\cdot ,s,t))}_{L^2(\Omega)}^2\\=\int_{I^2}\sum_{n=2}^\infty n^2(n-1)^2(n-2)!\norm{f_n(\cdot ,s,t)}_{L^2(I_{n-2})}^2=\sum_{n=2}^\infty n(n-1)n!\norm{f_n(\cdot ,s,t)}_{L^2(I_n)}^2. \end{multline*}\] Where analogous calculations go through if we have more derivatives to get the terms \(n(n-1)\cdots (n-(k-1))\). This shows that
\[ \mathbb{D}^{k,p}:=\left\{X\in L^2(\Omega):\quad \norm{X}_{\mathbb{D}^{k,2}}=\sum_{n=0}^\infty n^kn! \norm{f_n}_{L^p(I_n)}< \infty\right\} . \]
Thus, the domain of \(\Delta\) is exactly \(\mathbb{D}^{2,2}\). This is quite pleasing as, as we have observed earlier, the spaces \(\mathbb{D}^{k,p}\) mimic the Sobolev spaces \(W^{k,p}\), when \(p=2\) this resemblance is quite strong as we have that the norm on \(H^k:=W^{k,2}\) is
\[ \norm{f}_{H^k\R^d)}=\int_{\R^d}\br{\xi }^k \hat{f}(\xi )^2d\xi . \]
Which is formally equal to the one just derived for \(\mathbb{D}^{k,2}.\) It is very interesting to observe that, directly from the definition, we obtain a basis of eigenvalues of \(\Delta\). Let us define
\[ H_n:=\{X\in L^2(\Omega): X=I_n(f_n),\quad \text{for some } f_n \in L^2_S(I^n) \} . \]
That is, \(H_n\) are the random variables that only have the \(n\)-th term in their chaos expansion to be non-zero. Then by the chaos expansion theorem, we know that
\[ L^2(\Omega)=\overline{\oplus_{n=0}^\infty H_n}. \]
And by construction of the Laplacian, \(\Delta e_n=-n e_n\) for every \(e_n \in H_n\). In fact, by the uniqueness of the chaos expansion, the elements of \(H_n\) for some \(n \in \N\) are the unique eigenvectors of \(\Delta .\)
3 The Ornstein-Uhlenbeck semigroup
As it turns out, \(\Delta\) defines a semigroup
Definition 1 The Ornstein-Uhlenbeck semigroup is the family of operators \(\Phi(t):L^2(\Omega)\to L^2(\Omega)\)
\[ \Phi(t)X:=\sum_{n=0}^{\infty} e^{-nt}I_n(f_n), \quad\forall t\in I. \]
The term \(e^{-nt}\) is quite reminiscent of the semigroup for the heat equation
\[ e^{t\Delta }u_0:=\int_{\R^d}e^{-4 \pi^2 \xi^2t}\widehat{u_0}(\xi ) d\xi, \]
and will cause an analogous smoothing effect by making the terms in the chaos expansion to decrease faster. To see that \(\Phi\) defines a semigroup first note that, by the linearity of the iterated integrals,
\[ \Phi(t)X:=\sum_{n=0}^{\infty} I_n(e^{-nt}f_n). \]
So as a result
\[ \Phi(t+s)X=\sum_{n=0}^{\infty} e^{-nt}I_n(e^{-ns}f_n)=\sum_{n=0}^{\infty} \Phi(t)\Phi(s)X. \]
Which shows that \(\Phi(t+s)=\Phi(t)\circ \Phi(s)\). Finally, note that
\[ \frac{\Phi(t)X-X}{t}=\sum_{n=0}^{\infty} \left(\frac{e^{-nt}-1}{nt} \right)nI_n(f_n)\to -\sum_{n=0}^{\infty} nI_n(f_n)=\Delta X \in L^2(\Omega) . \]
Where the commutation under the integral sign (with the counting measure) is justified as \((e^{-nt}-1)/(nt)\) is uniformly bounded in \(n\). There’s an explicit formula for \(\Phi(t)\).
Proposition 1 (Mehler’s formula) Let \((\Omega,\Ff_t,\gamma )\) be the Wiener space, then
\[ \Phi(t)X(\omega)=\int_{\Omega}X\left(e^{-t}\omega+\sqrt{1-e^{-2t}}\eta\right) \d \gamma (\eta)\in L^2(\Omega). \]
The proof is technical and can be found in Nualart’s book (Nualart and Nualart 2018) on page 74. Let us try to understand the formula and also the reason for the name of the semigroup. We consider as at the beginning of this post the finite-dimensional case but now with some Gaussian measure \(\mu\)
\[ \mu(A) := \frac{1}{(2\pi)^{d/2}} \int_{A} e^{-\frac{\norm{x}^2}{2}} dx \]
Then, integration by parts shows that
\[ \begin{aligned} \int_{\R^d}f(x) \cdot \nabla g(x) d\mu(x) & =-\int_{\R^d}\nabla\cdot \left(e^{-\frac{\norm{x}^2}{2} }f(x)\right) \nabla g(x) dx \\&=\int_{\R^d}(x\cdot f(x)-\nabla\cdot f(x)) d\mu (x). \end{aligned} \]
That is, the adjoint of the gradient in \(L^2(\R^d,\mu )\) is \(x\cdot -\nabla\cdot\). Notice that we get the extra term that corresponds to multiplication by \(x\cdot\).As a result, the Laplacian on \(L^2(\R^d, \mu )\) is given by
\[ \Delta_\mu =\nabla\cdot \nabla-x\cdot \nabla . \]
Furthermore, by Itô’s formula, \(\Delta_\mu\) is the generator of the SDE
\[ dX(t)=-X(t)d t+ \sqrt{2}dW(t) \]
Let us write \(X_x\) for the solution to the above SDE with initial data \(x \in \R^d\) That is, if we define \[P_tX(x):=E[\varphi(X(t))],\] then
\[ \partial _tP_tX(x)=\Delta_\mu P_tX(x) . \]
The process \(X\) that solves the SDE above is known as the Ornstein-Uhlenbeck process and, by the theory of linear SDEs, is given by
\[ X_x(t)=e^{-t}x+\sqrt{2} \int_{0}^te^{s-t} dW(s). \]
Since
\[ \sqrt{2} \int_{0}^te^{s-t} dW(t)\sim \sqrt{2} e^{-t}\Nn\left(0,\norm{e^\cdot }^2_{L^2([0,t])}\right)=\sqrt{1-e^{-2t}}\Nn(0,1) \]
We deduce that for each fixed \(t\) we can find \(\eta \sim \Nn(0,1)\) with
\[ X(t)=e^{-t}x+\sqrt{1-e^{-2t}}\eta . \]
We then get that
\[ P_t \varphi(x)=\E\left[\varphi\left(e^{-t}x+\sqrt{1-e^{-2t}}\eta \right)\right] \]
And by taking \(\varphi=Id\) we recover Mehler’s formula. This correspondence is expanded on in chapter \(7\) of Hairer’s notes (Hairer 2009).
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