> <\body> <\hide-preamble> <\padded-normal|0.5fn|0.5fn> >||> > > \; \; <\center> > \; \; <\very-small> Version 1, 2020.05.12. Tutorial classes: Mon May 25th 16\U18 \ (Zoom) Min Liu \| Wed May 27th 16\U18 (Zoom) Daria Frolova. Solutions in groups of 2 (at most). To be handled in or format via eCampus not later than 4pm Thursday May 21th. Use this sheet for your solutions and write them under the corresponding exercise. Fill out your names below. \; \; \; <\exercise> [Pts 2] Let =B/|\|>> where is a Brownian motion in > and 2>. Prove that the time\Uchanged process <\equation*> Z=Y>,T=A,A=|\|>\s, is a diffusion taking values in the unit sphere =\:=1|}>> with generator <\equation*> \f=f-x*xf|\x\x>|)>-xf|\x>,x\S. where > is the Laplacian in > and where diffusion here means continuous time process solving the martingale problem for this generator. <\exercise> [Pts 2+2+2+1+1] 2>)> Let > be a Brownian motion on > starting at >. Let <\equation*> ,N|)>\e>|)>,sin|)>|)>. We will assume without proof that <\equation*> >e>\s=+\,a.s. <\enumerate-alpha> Prove that > is a Brownian motion on > changed of time (starting from where?) ; Compute the Euclidean norm ,N|)>|\|>> of the vector ,N|)>> and deduce that a Brownian motion in > never visit the point >, that is <\equation*> \t\0:B=|)>=0. Conclude that never visit any given point >. Use the Markov property to deduce from (c) that t\0:B=|)>=0.> [Hint: consider t\1/n:B=|)>> as 0>.] Prove that a Brownian motion in > with 2> does not visit any given point \>. \; \; <\exercise> [Pts 2+2+2+1+1] 3>)> Let be a Brownian motion in > starting from \\0>. We say that a process is transient if |\|>\\> as \> almost surely. <\enumerate-alpha> Prove that the process =1/|\|>> is a positive local martingale.\ Prove that >=lim\> M> exists almost surely. Compute |]>> and deduce that >=0>. This implies that is transient. Show that whatever the starting point is, is always transient. Prove that a Brownian motion in > with 3> is transient. <\exercise> [Pts 2] Let \\> be an holomorphic function and be a planar Brownian motion (with the identification of > with >). Prove that the process =f|)>> is a continuous local martingale with values in >. Deduce that it is a complex Brownian motion changed of time. This property is called conformal invariance of Brownian motion.\ \; \; <\initial> <\collection> <\references> <\collection> >