Let $W: \mathbb R^d \to \mathbb R^{d \times d}$ be a matrix-valued function with domain $\mathbb R^d$ and taking values in the set of $d \times d$ real matrices, such that $W(x)$ is positive-definite for every $x \in \mathbb R^d$, and such that $\int_{\mathbb{R}^d} W(x)dx = I_d$. Does there exist a probability density function $p$ over $\mathbb{R}^d$ such that for every appropriately integrable $f: \mathbb{R}^d \to \mathbb{R}^d$ $$\int W(x) f(x) dx = \int p(x) f(x) \ ?$$

Let $W: \mathbb R^d \to \mathbb R^{d \times d}$ be a matrix-valued function with domain $\mathbb R^d$ and taking values in the set of $d \times d$ real matrices, such that $W(x)$ is positive-definite for every $x \in \mathbb R^d$, and such that $\int_{\mathbb{R}^d} W(x) \, dx = I_d$. Does there exist a probability density function $p$ over $\mathbb{R}^d$ such that for every appropriately integrable $f: \mathbb{R}^d \to \mathbb{R}^d$ $$\int W(x) f(x) \, dx = \int p(x) f(x) \,dx \text{ ?}$$