Interpretation of Notation of Laplacian and Brownian Motion

Question

Let $f: \mathbb{R}^d \to \mathbb{R}$ be a twice differentiable function. In particular, $\Delta f$ is well defined. Let $W := (W_t)_{t \geq 0}$ be a $d$-dimensional standard Brownian Motion. Sometimes, we define quantity such as $\Delta f(W_t)$ (e.g. in this lecture, where one defines a Martingale) . My question is that: a) What does the notation mean? b) strictly speaking, with the notation as it is, doesn't the notation imply the use of chain rule (i.e. that we also need to take the derivative of a Brownian motion with respect to $t$, which isn't defined)?

Answer 1

Since $\Delta f$ is a real valued function defined on $\mathbb R^d$ and $W_t$ is an $\mathbb R^d$-valued random variable, $\Delta f(W_t)$ is a real valued random variable.