MA5311. Take home exam

Exercise 1. Solve Exercise 44 in Chapter 1 of the book.

Exercise 2.  Solve Exercise 3 in Chapter 1 of the book.

Exercise 3.  Solve Exercise 39 in Chapter 1 of the book.

Exercise 4. The heat kernel on \mathbb{S}^1 is given by p(t,y) =\frac{1}{2\pi}\sum_{m \in \mathbb{Z}} e^{-m^2 t} e^{im y} =\frac{1}{\sqrt{4\pi t}} \sum_{k \in \mathbb{Z}} e^{-\frac{(y -2k\pi)^2}{4t} }.

  • By using the subordination identity e^{-\tau | \alpha | } =\frac{\tau}{2\sqrt{\pi}} \int_0^{+\infty} \frac{e^{-\frac{\tau^2}{4t}-t \alpha^2}}{t^{3/2}} dt, \quad \tau \neq 0, \alpha \in \mathbb{R}, show that for \tau > 0, \frac{1+e^{-2\pi \tau}}{1-e^{-2\pi \tau}} =\frac{1}{2\pi} \sum_{k \in \mathbb{Z}} \frac{2\tau}{\tau^2+n^2}
  • The Bernoulli numbers B_k are defined via the series expansion \frac{x}{e^x -1}=\sum_{k=0}^{+\infty} B_k \frac{x^k}{k!}. By using the previous identity show that for k \in \mathbb{N}, k \neq 0, \sum_{n=1}^{+\infty} \frac{1}{n^{2k}} =(-1)^{k-1} \frac{(2\pi)^{2k} B_{2k} }{2(2k)!}.

 

Exercise 5. Show that the heat kernel on the torus \mathbb{T}^n=\mathbb{R}^n / (2 \pi \mathbb{Z})^n is given by p(t,y) = \frac{1}{(4\pi t)^{n/2}} \sum_{k \in \mathbb{Z}^n} e^{-\frac{\|y+2k\pi\|^2}{4t} }=\frac{1}{(2\pi)^n} \sum_{l\in \mathbb{Z}^n} e^{i l \cdot y -\| l \|^2 t}.

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MA5161. Take home exam

Exercise 1. The Hermite polynomial of order n is defined as
H_n (x)=(-1)^n e^{\frac{x^2}{2}} \frac{d^n}{dx^n} e^{-\frac{x^2}{2}}.

  • Compute H_0, H_1,H_2,H_3.
  • Show that if (B_t)_{t \ge 0} is a Brownian motion, then the process \left(t^{n/2}H_n (\frac{B_t}{\sqrt{t}})\right)_{t \ge 0} is a martingale.
  • Show that
    t^{n/2}H_n (\frac{B_t}{\sqrt{t}})=n! \int_0^t \int_0^{t_1} ... \int_0^{t_{n-1}} dB_{s_1}...dB_{s_n}.

 

Exercise 2. (Probabilistic proof of Liouville theorem) By using martingale methods, prove that if f:\mathbb{R}^n \to \mathbb{R} is a bounded harmonic function, then f is constant.

Exercise 3. Show that if (M_t)_{t \ge 0} is a local martingale of a Brownian filtration (\mathcal{F}_t)_{t \ge 0}, then there is a unique progressively measurable process (u_t)_{t \ge 0} such that for every t \ge 0, \mathbb{P} \left(\int_0^t u_s^2 ds < +\infty \right)=1 and M_t=\mathbb{E} (M_0)+ \int_0^{t} u_s dB_s.
Exercise 4 [Skew-product decomposition]
Let (B_t)_{t \ge 0} be a complex Brownian motion started at z \neq 0.

  1. Show that for t \ge 0,
    B_t=z \exp\left( \int_0^t \frac{dB_s}{B_s} \right).
  2.  Show that there exists a complex Brownian motion (\beta_t)_{t \ge 0} such that
    B_t=z \exp{\left( \beta_{\int_0^t \frac{ds}{\rho_s^2} }\right)},
    where \rho_t =| B_t |.
  3. Show that the process (\rho_t)_{t \ge 0} is independent from the Brownian motion (\gamma_t)_{t \ge 0}=(\mathbf{Im} ( \beta_t))_{t \ge 0}.
  4. We denote \theta_t=\mathbf{Im}\left( \int_0^t \frac{dB_s}{B_s} \right) which can be interpreted as a winding number around 0 of the complex Brownian motion paths. For r>| z|, we consider the stopping time
    T_r =\inf \{ t \ge 0, | B_t | = r \}.
  5. Compute for every r>| z|, the distribution of the random variable
    \frac{1}{\ln (r/|z|)}\theta_{T_r}.
  6. Prove Spitzer theorem: In distribution, we have the following convergence
    \frac{ 2 \theta_t}{\ln t} \to_{+\infty} C,
    where C is a Cauchy random variable with parameter 1 that is a random variable with density \frac{1}{\pi (1+ x^2)}.

 

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MA5311. HW due April 7

Solve Exercises 10,14,15,16,18,19 in Chapter 1 of the book “The Laplacian on  a Riemannian manifold”.

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MA5311. Take home exam due 03/20

Solve Problems 1,2,8,9,10,11 in Milnor’s book. (Extra credit for problem 6)

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MA5161. Take home exam. Due 03/20

Exercise 1.

Let \alpha, \varepsilon, c >0. Let (X_t)_{t\in [0,1]} be a continuous Gaussian process such that for s,t \in [0,1],

\mathbb{E} \left( \| X_t - X_s \|^{\alpha} \right) \leq c \mid t-s \mid^{1+\varepsilon},
Show that for every \gamma \in [0, \varepsilon/\alpha), there is a positive random variable \eta such that \mathbb{E}(\eta^p)<\infty, for every p \ge 1 and such that for every s,t \in [0,1], \|X_t-X_s\| \le \eta |t-s|^{\gamma}, \quad a.s. \textbf{Hint:} You may use without proof the Garsia-Rodemich-Rumsey inequality: Let p \ge 1 and \alpha >p^{-1}, then there exists a constant C_{\alpha,p} >0 such that for any continuous function f on [0,T], and for all t,s \in [0,T] one has:
\|f(t)-f(s)\|^p \le C_{\alpha,p} |t-s|^{\alpha p-1} \int_0^T \int_0^T \frac{ \|f(x)-f(y)\|^p}{ |x-y|^{\alpha p+1}} dx dy.

Exercise.(Non-canonical representation of Brownian motion)

  • Show that for t \ge 0, the Riemann integral \int_0^t \frac{B_s}{s} ds almost surely exists.
  • Show that the process \left( B_t-\int_0^t \frac{B_s}{s}ds\right)_{t \ge 0} is a standard Brownian motion.

Exercise. [Non-differentiability of the Brownian paths]
1) Show that if f: (0,1) \to \mathbb{R} is differentiable at t \in (0,1), then there exist an interval (t-\delta,t+\delta) and a constant C>0 such that for s \in (t-\delta,t+\delta),
| f(t)-f(s) | \le C |t-s|.
2)  For n \ge 1, let
M_n = \min_{1 \le k \le n} \{ \max \{| B_{k/n}-B_{(k-1)/n}|, | B_{(k+1)/n}-B_{k/n}|, | B_{(k+2)/n}-B_{k+1/n}| \} \}.
Show that \lim_{n \to +\infty} \mathbb{P}( \exists C>0, nM_n \le  C)=0.

3)  Deduce that
\mathbb{P} \left( \exists t \in (0,1), B \text{ is differentiable at } t\right)=0.

Exercise.[Fractional Brownian motion] Let 0<H<1. 1) Show that for s \in \mathbb{R}, the function f_s(t)=(|t-s|^{H-\frac{1}{2}} - \mathbf{1}_{(-\infty, 0]}(t)|t|^{H-\frac{1}{2}}) \mathbf{1}_{(-\infty, s]}(t) is square integrable on \mathbb{R}. 2)  Deduce that R(s,t)=\frac{1}{2} \left( s^{2H} +t^{2H} -|t-s|^{2H} \right), \quad s,t \ge 0 is a covariance function. 3) A continuous and centered Gaussian process with covariance function R is called a fractional Brownian motion with parameter H. Show that such process exists and study its Holder sample path regularity. 4) Let (B_t)_{t \ge 0} be a fractional Brownian motion with parameter H. Show that for any h \geq 0, the process (B_{t+h} - B_h)_{t \ge 0} is a fractional Brownian motion. 5) Show that for every c >0, the process (B_{ct})_{t \geq 0} has the same law as the process (c^H B_t)_{t \geq 0}

Exercise. (Brownian bridge)
Let T>0 and x \in \mathbb{R}.

  1. Show that the process
    X_t =\frac{t}{T}x +B_t -\frac{t}{T}B_T, \quad, 0 \le t \le T,
    is a Gaussian process. Compute its mean function and its covariance function.
  2.  Show that (X_t)_{0 \le t \le T} is a Brownian motion conditioned to be x at time T, that is for every 0\le t_1 \le \cdots \le t_n <T, and A_1,\cdots,A_n Borel sets of \mathbb{R},
    \mathbb{P}( X_{t_1} \in A_1, \cdots, X_{t_n} \in A_n) = \mathbb{P}( B_{t_1} \in A_1, \cdots, B_{t_n} \in A_n | B_T =x).
  3. Let (\alpha_n)_{n \ge 0}, (\beta_n)_{n \ge 1} be two independent sequences of i.i.d. Gaussian random variables with mean 0 and variance 1. By using the Fourier series decomposition of the process B_t -t B_1, show that the random series
    X_t=t \alpha_0 +\sqrt{2} \sum_{n=1}^{+\infty} \left( \frac{\alpha_n}{2\pi n} (\cos (2\pi nt)-1)+\frac{\beta_n}{2\pi n} \sin (2\pi nt)\right)
    is a Brownian motion on [0,1].
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MA5311. Non orientable manifolds

Here are some videos to visualize non orientability.

 

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HW4 MA5161. Due February 24

Exercise. Let (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P}) be a filtered probability space that satisfies the usual conditions. We denote

\mathcal{F}_{\infty}=\sigma \left( \mathcal{F}_t , t \ge 0 \right)

and for t\ge 0, \mathbb{P}_{/\mathcal{F}_t} is the restriction of \mathbb{P} to \mathcal{F}_t. Let \mathbb{Q} be a probability measure on \mathcal{F}_{\infty} such that for every t \ge 0,

\mathbb{Q}_{/\mathcal{F}_t} \ll \mathbb{P}_{/\mathcal{F}_t}.

  • Show that there exists a right continuous and left limited martingale (D_t)_{t \ge 0} such that for every t \ge 0,D_t=\frac{d\mathbb{Q}_{/\mathcal{F}_t}}{d\mathbb{P}_{/\mathcal{F}_t}},\text{ }\mathbb{P}-a.s.
  • Show that the following properties are equivalent:
    1) \mathbb{Q}_{/\mathcal{F}_\infty} \ll \mathbb{P}_{/\mathcal{F}_\infty};
    2) The martingale (D_t)_{t \ge 0} is uniformly integrable;
    3) (D_t)_{t \ge 0} converges in L^1;
    4) (D_t)_{t \ge 0} almost surely converges to an integrable and \mathcal{F}_\infty measurable random variable D such that D_t =\mathbb{E}(D\mid \mathcal{F}_t), \quad t \ge 0.

 

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