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Yes, I think you're missing something in the definition. Quoting from Wikipedia's definition, An equation $ H(x,u,Du,D^2 u) = 0 $ in a domain $ \Omega $ is defined to be ''degenerate elliptic'' if …

Assume $t\ge T$. $$ [B_t, B_t^T]_{[T,t]} = \int_T^t dB_t dB_t^T = \int_T^t dB_t (dB_t-dB_{t-T}) = \int_T^t (dB_t)^2 - \int_T^tdB_t dB_{t-T} $$ $$ =\int_T^t dt - \int_T^t dB_t dB_{t-T} = (t-T) - \int_T …

Heuristically, a point $(t,B_t)$ being a extremum is antithetical to fast oscillation, since there is no oscillation on one side of (above/below) $B_t$. However, This is only a heuristic, and One …

In general, if $Y$ is an exponentially distributed "vanishing time" and $X$ is any random time independent of $Y$, then the probability of $X$ occurring before vanishing is actually the Laplace transf …

Yes. See Exercise 1.13(a) of Mörters and Peres, Brownian motion. http://www.stat.berkeley.edu/~peres/bmbook.pdf

I suppose the larger surface area plane will have a greater hitting probability. But what would be a rigorous way of proving that? Depends on whether for each plane $A$, the center of $A$ is the …

If we have a point of increase at $t$, witnessed by $\epsilon$, the question is threefold: why can we assume $t-\epsilon=0$, $B_t(\omega)\le 1$, and $B_{t+\epsilon}(w)-B_t(\omega)\ge 2$. These conditi …

A necessary condition is that $r> \sqrt 2$. Indeed, if $M_n$ has this property then so does $B_n-B_{n-1}$. Let $Y_n = B_n - B_{n-1}$. The variance of $B_n-B_{n-1}$ is 1. The probability that $Y_n>r\ …

Regarding question 1, the limiting probability of not crossing 0 during the time interval $[1,2^n]$ is $$ \lim_{n \to \infty} \mathbb{P}\{K_n = 0\} = 0 $$ since Brownian motion is recurrent (in dimens …

Yes -- Dvoretsky, Erdos and Kakutani 1954 https://books.google.com/books?id=onG8BAAAQBAJ&pg=PA18&lpg=PA18&dq=multiple+points+of+the+Brownian+motion+dense&source=bl&ots=vxQ1n_EC4t&sig=wnDVnqRcN8F1WdCr …

Does there exist $r$ such that with probability one,$$\limsup_{t \to \infty} {{M_t - m_t}\over{\sqrt{t \log \log t}}} = r?$$ Yes, such an $r$ does exist. First note that by the original LIL, with …

Yes, according to Proposition 7 on page 112 of Markov Processes, Brownian Motion, and Time Symmetry by Kai Lai Chung and John B. Walsh, A polar set is very thin; a very thin set is thin; a thin se …

Let $V_t$ and $W_t$ be independent standard Wiener processes ($t\ge 0$, $W_t,V_t\in\mathbb R$). Let $C$ be the event that there is a continuous function $f$ such that for all $s$, $t$, $$ W_t=W_s\iff …

The process $X$ you mention is uniformly continuous on the rationals* in the compact interval $[0,n]$, with probability 1. So you define Brownian motion $B$ to be the unique continuous extension of: $ …

Yes, let $W$ be Brownian motion and let $V$ be the following modification: $V_t=W_t$ except that we pick a number $s\in [0,1]$ according to the uniform distribution, independently of $W$, and let $V_s …