## 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 $p(t,y) = \frac{1}{(4\pi t)^{n/2}} \sum_{k \in \mathbb{Z}^n} e^{-\frac{\|y+2k\pi\|^2}{4t} }=\frac{1}{(2\pi)^n} \sum_{l\in \mathbb{Z}^n} e^{i l \cdot y -\| l \|^2 t}.$

## 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 $\frac{1}{\pi (1+ x^2)}$.

## 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”.

## MA5311. Take home exam due 03/20

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

## 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. 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 , 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 $[0,1]$.

## MA5311. Non orientable manifolds

Here are some videos to visualize non orientability.

## 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 $D_t =\mathbb{E}(D\mid \mathcal{F}_t), \quad t \ge 0.$