I need help with proving:
1.If $d>1$ then d-dimensional Brownian motion starting at $x$ has 0 probability to actually hit $y$. Note that this is different from the usual notion of recurrence, where the brownian motion only has to get arbitrarily close.
2.If $d>2$ then d-dimensional Brownian motion $B_t$ satisfies $|B_t|\rightarrow \infty$ as $t \rightarrow \infty$ a.s. For this one, I want a specific proof that my professor alluded to but did not give details on. I already know that if you consider a sequence $S_n$ concentric spheres of radius $r^n$ with $r>1$, then if $x$ is of norm $r^N$ then the probability of a brownian motion starting at $x$ to hit $S_{N-1}$ before $S_{N+1}$ is $1/(r^{(d-2)}+1)<1/2$. Now, he suggests that this be related to a random walk with drift. I have trouble making this relationship precise. Please assume that the Brownian motion is defined on some probability space. It is okay to assume that it is everywhere continuous, instead of a.s. continuous, though the greater generality the better. I think the strong Markov property has to be used here to elevate this proof idea from the status of being purely motivational. I don't really need other proofs of 2. There's a proof in Durrett that is legitimate as far as I can see. I just want to know how this proof works.
3.Also, these questions have motivated me to ask questions about what paths of Brownian motions look like. For some reason, people seem terribly interested in Hausdorff dimension. If anybody can say a few words about why anybody should care about Hausdorff dimension, or how one usually goes about computing the Hausdorf dimension of a Brownian path, please let me know.
This question is vastly too broad so let me answer part 1. Consider a Brownian motion starting at $x\ne0$, then one is interested in whether it hits $0$ or not. For every positive $r$, let $T_r$ denote the first hitting time of the sphere centered at $0$ with radius $r$. An idea is to compute $P_x(T_R\leqslant T_r)$ for every $r\lt|x|\lt R$ by solving explicitely a harmonic problem.
Then one can check that $\lim\limits_{r\to0}P_x(T_R\leqslant T_r)=1$, that is, $P_x(T_R\lt T_0)=1$, for every $R\gt|x|$. Now, $T_R\to+\infty$ almost surely when $R\to+\infty$ hence $P_x(T_0=+\infty)=1$. (Note that this is exactly the procedure suggested by your professor to solve part 2.)