Analysis on Gaussian Spaces

This project is a seminar I held as the final exam for the course Analysis on Gaussian Spaces. It studies the extension of classical Malliavin calculus to infinite-dimensional spaces, in the case of Gaussian measures $\mu$ on a separable Hilbert space $H$.

1. Gaussian Measures on Hilbert Spaces

A probability measure $\mu$ on a Hilbert space $(H,\mathcal{B}(H))$ is Gaussian if its Fourier transform (characteristic function) is given by:

$$\hat{\mu }(h)={\int }_H{e}^{i⟨h,x⟩}\mu (dx)=e^{i⟨m,h⟩}{e}^{-\frac{1}{2}⟨Qh,h⟩},\forall h\in H.$$

If $\text{Ker}(Q)=\{0\}$, the measure is non-degenerate. A crucial result in this setting is that $Q$ is a compact, symmetric, and positive operator, allowing for an orthonormal basis $\{e_k\}$ of eigenvectors such that:

$$Q{e}_k={\lambda }_k{e}_k.$$

2. The White Noise Function

We define the white noise mapping $W:H\to L^2(H,\mu )$. For $f\in H$, $W_f$ is a Gaussian random variable with mean $0$ and variance $\Vert f{\Vert }^2$.

In the specific case where $H=L^2([0,1])$, the process defined by $B_t=W_{1_{[0,t]}}$ is a Standard Brownian Motion. This links abstract Hilbert space theory back to classical stochastic calculus, via the formula:

$$W_f={\int }_0^1{f}_{s}d{B}_s.$$

3. The Malliavin Derivative

The core of the seminar was the definition of the Malliavin Derivative operator $M$. We start with the space of “exponential” functions $\mathcal{E}(H)$ and define:

$$M:\mathcal{E}(H)\subseteq L^2(H,\mu )\to L^2(H,\mu ;H)$$ $$\phi \mapsto Q^{1/2}D\phi .$$

Resources

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References:

• Giuseppe Da Prato, Introduction to Stochastic Analysis and Malliavin Calculus, Scuola Normale Superiore, 2009.