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Due March 22

Problem 1

For any dimension $d\ge 1$,
$\displaystyle \int_{\mathbb R^d} \frac{dx}{(|x|^2 + 1)^{(d+1)/2}} = \frac{\pi^{(d+1)/2}}{\Gamma((d+1)/2)},$
and verify that
$\displaystyle \int_{\mathbb R^d} P_t(x) dx = 1.$
(Hint: Use spherical coordinates and the identity $\displaystyle \int_0^\infty t^\alpha e^{-ts} \frac{dt}{t} = \frac{\Gamma(\alpha)}{s^\alpha}$ for any $\alpha, s>0$.

Problem 2

Let $f$ be integrable on $\mathbb R^d$.
1. For any $h\in\mathbb R^d$, $\displaystyle \int_{\mathbb R^d} f(x) dx = \int_{\mathbb R^d} f(x - h) dx.$
2. For any $r>0$, $\displaystyle \int_{\mathbb R^d} f(x) dx = \frac{1}{r^d} \int_{\mathbb R^d} f\big( \frac{x}{r}\big) dx.$

Problem 3

If $f\in C_0(\mathbb R^d)$, then $u(x,t) \to f(x)$ as $t\to 0$, uniformly in $x\in\mathbb R^d$.

Problem 4

Let $f\in C_c(\mathbb R^d)$. Then $\displaystyle \int_{\mathbb R^d} |f(x - h) - f(x)| dx \to 0$ as $h\to 0$ in $\mathbb R^d$. (Hint: $f$ is uniformly continuous in its compact support.)

Problem 5 (Principle of subordination)

Let $H_t(x)$ be the heat kernel, $\displaystyle H_t(x) = \frac{1}{(4\pi t)^{d/2}} e^{-|x|^2/4t}$. Then
$\displaystyle P_t(x) = \frac{t}{2\sqrt\pi} \int_0^\infty e^{-t^2/4s} H_s(x) \frac{ds}{s^{3/2}}.$
(Hint: use the identifty of the Gamma function as in Problem 1.)