In this PhD thesis, we first study the (planar) complex valued Ornstein-Uhlenbeck processes (Zt = Xt + iYt, t ≥ 0), where (Xt, t ≥ 0) and (Yt, t ≥ 0) denote its cartesian coordinates. Taking the Ornstein-Uhlenbeck parameter equal to 0 allows to discuss in particular the planar Brownian motion case. More precisely, we study the distribution of several first hitting times related to the winding process around a fixed point. To obtain analytical results, we use and extend Bougerol's identity. We develop some identities in law in terms of (planar) complex valued Ornstein-Uhlenbeck processes, which are equivalent to Bougerol's identity. This allows us to characterize the laws of the hitting times Tc ≡ inf{t : θt = c}, (c > 0) of the continuous winding processes θt, t ≥ 0 associated with our complex Ornstein-Uhlenbeck process. Moreover, we investigate the distribution of the random time T−d,c ≡ inf{t : θt = −d or c}, (c, d > 0) and, more specifically of T−c,c ≡ inf{t : θt = −c or c}, (c > 0). We investigate further Bougerol's identity, and we show that 1/Au(β), where Au(β) denotes the new clock in Bougerol's identity, considered after a suitable measure change from Wiener measure, is infinitely divisible. Using the previous results, we estimate the mean rotation time (MRT) which is the mean of the first time for a planar polymer, modeled as a collection of n rods parameterized by a Brownian angle, to wind around a point. We are led to study the sum of i.i.d. exponentials with a one dimensional Brownian motion in the argument. We find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function. Finally, we obtain an asymptotic formula for the MRT, and the leading order term depends on √n and, interestingly, it also depends weakly upon the mean initial configuration. Our analytical results are confirmed by Brownian simulations.
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