The space C[0, 1]

The support of a measure $\lambda$ is the smallest closed set $F$ such that $\lambda \left(F^{c}\right)=0$. Let $\mathbb {P}$ be a Wiener measure on $C[0,1]$, i.e., the law of a Brownian motion on $[0,1]$. Use Exercise 13.4 to prove that the support of $\mathbb {P}$ is all of $C[0,1]$.

Let $(\mathcal{S}, d)$ be a complete separable metric space and let $\mathcal{R}$ be a subset of $\mathcal{S}$. Then $(\mathcal{R}, d)$ is also a metric space. If $X_{n}$ converges weakly to $X$ with respect to the topology of $(\mathcal{S}, d)$ and each $X_{n}$ and $X$ take values in $\mathcal{R}$, does $X_{n}$ converge weakly to $X$ with respect to the topology of $(\mathcal{R}, d)$ ? Does the answer change if $\mathcal{R}$ is a closed subset of $\mathcal{S}$ ?

If $X_{n}$ and $X$ take values in $\mathcal{R}$ and $X_{n}$ converges weakly to $X$ with respect to the topology of ( $\mathcal{R}, d$ ), does $X_{n}$ converge weakly to $X$ with respect to the topology of $(\mathcal{S}, d)$ ? What if $\mathcal{R}$ is a closed subset of $\mathcal{S}$ ?

Give a proof of Theorem 32.2 using (32.7) in place of Theorem 8.1.

Suppose $(X, W, \mathbb {P})$ is a weak solution to

where $W$ is a one-dimensional Brownian motion and $\sigma$ and $b$ are bounded and continuous, but we do not assume that $\sigma$ is bounded below by a positive constant. Suppose the solution to (32.8) is unique in law.

Suppose $\sigma_{n}$ and $b_{n}$ are Lipschitz functions which are uniformly bounded and which converge uniformly to $\sigma$ and $b$, respectively. Let $X_{t}(n)$ be the unique pathwise solution to

the probability measure here is $\mathbb {P}$. Prove that $X(n)$ converges weakly to $X$ with respect to $C[0,1]$.

Let $W$ be a $d$-dimensional Brownian motion and let $\left\{ X_{t}, t \in [0,1]\right\}$ be the solution to (24.22). If $x \in \mathbb {R}^{d}$, prove that the support of $\mathbb {P}[x]$ is all of $C[0,1]$.