Is the distribution of a Banach space valued Lévy process uniquely determined by its characteristic function?

Question

Let $E$ be a $\mathbb R$-Banach space. Remember that if $\mu$ is a finite measure on $\mathcal B(E)$ then $$\Phi_\mu:E'\to\mathbb C\;,\;\;\;\varphi\mapsto\int\mu({\rm d}x)e^{{\rm i}\varphi(x)}$$ is called the characteristic function of $\mu$.

Let

• $(\Omega,\mathcal A,\operatorname P)$ be a probability space;
• $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$;
• $(L_t)_{t\ge0}$ be a $E$-valued process on $(\Omega,\mathcal A,\operatorname P)$.

Remember that $L$ is called $\mathcal F$-Lévy if

1. $L$ is $\mathcal F$-adapted;
2. $L_0=0$;
3. $L_{s+t}-L_s$ and $\mathcal F_s$ are independent for all $t\ge s\ge0$;
4. $L_{s+t}-L_s\sim L_s$ for all $t\ge s\ge0$.

Assume $L$ is $\mathcal F$-Lévy. As usual, let $\mathcal L(X)$ denote the distribution of a random variable $X$.

Are we able to show that there is a continuous $f:E'\to\mathbb C$ with $f(0)=0$ and $$\Phi_{\mathcal L(L_t)}=e^{-{\rm i}tf(\varphi)}\tag1$$ for all $\phi\in E'$ and $t\ge0$? Moreover, I would like to conclude that the distribution $\mathcal L(L)$ of the process $L$ is uniquely determined by $f$ and $\mathcal L(L_0)$.

I think we first need to ensure that a finite measure $\mu$ on $\mathcal B(E)$ is uniquely determined by $\Phi_\mu$. I know that this is true when $E=\mathbb R^d$, $d\in\mathbb N$. The proof is relying on the fact that $C_c(\mathbb R^d)$ is separating for the space of finite measures on $\mathcal B(\mathbb R^d)$. Maybe this can be generalized.

Answer 1

If the Banach space $E$ is separable, then, by Lemma 2.1, there does exist a unique continuous $f\colon E'\to\mathbb C$ with $f(0)=0$ such that your condition (1) holds. Moreover, then the function $f$ is sequentially $w^*$-continuous.

Also, then the smallest $\sigma$-algebra generated by all continuous linear functionals on $E$ coincides with the Borel $\sigma$-algebra over $E$, and so, the distribution of $L_t$ is uniquely determined by $f$. Therefore and because $L_\cdot$ is a Lévy process, all the finite-dimensional distributions of $L_\cdot$ are uniquely determined by $f$.