# Questions tagged [brownian-motion]

The brownian-motion tag has no usage guidance.

326
questions

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### Second Skorokhod embedding in high dimensions

The first Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E X^2<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau$ is ...

2
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1
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68
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### Chung's law of the iterated logarithm for Brownian motion

I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup_{r \in [0,u]}|X_r|=\frac{...

0
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1
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### For some $\alpha>0$, $ e^L=P\left(\exp(\alpha\sup_{|s-t|\le\delta}\frac{|B_s-B_t|^2}{|s-t|})<\infty\right) $?

I am reading one lecture note Dynamics for Spherical Models of Spin-Glass and Aging by Alice Guionnet. On page 124, it says that
for some $\alpha>0$,
$$
e^L=P\left(\exp(\alpha\sup_{|s-t|\le\delta}\...

15
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0
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420
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### Quantitative Skorokhod embedding

The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...

6
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### Running maximum/supremum of Brownian motion: add information to make it a Markov process?

Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it ...

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66
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### Feynman-Kac formula with non-zero boundary condition

Let $D \subseteq \mathbb{R}^m$ be a bounded domain. The Feynman-Kac formula for the heat equation with initial condition $u(t, x) = f(x)$ and boundary condition $u(t, x)|_{\partial D} = 0$ is given by
...

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1
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57
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### What new fractional brownian motion (fBm) simulation methods have emerged since 2010? [closed]

I want to describe new methods for simulating fBm, as in the work of Coerjolly and Dieker, but new methods are not very easy to find.

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### Known examples of Laplace transform/distribution of the occupation time of Brownian motion under a moving barrier

Let $(B_t)_{t\geq 0}$ be a Brownian motion with $B_0 = x$. For a function $b:(0,\infty) \to \Bbb R$ let $\Gamma_t^b := \int_0^t 1_{(-\infty , b(s))}(B_s)d s$ the "occupation time" under the &...

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### Alternate proof of Levy’s characterisation of Brownian motion

Levy’s characterisation theorem for Brownian motion states that for a local martingale $X$ with $X_0 = 0$, $X$ is a Brownian motion if and only if it has quadratic variation $\langle X, X \rangle_t = ...

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146
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### Intersection of Brownian motion and finite variation process

Let $B$ be a standard Brownian motion, and $A$ a process of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.
Question: Denoting by $\mathcal L$ the ...

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59
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### Why is Branching Brownian Motion log-correlated?

I need some references(or helps) on understanding why BBM is log-correlated. As I understand it, a random field on some metric space $V$ with distance $d$ is log-correlated if $$\mathbb{E}[X_u X_v]\...

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1
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389
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### Is this a Brownian motion?

I am building a 2D stochastic process as follows. I start with a point $P_0=(0,0)$. Then $P_k=(X_k,Y_k)$ is defined as follows, for $k>0$:
\begin{align}
X_k & =X_{k-1}+R_k \cos(2\pi\theta_k) \\
...

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58
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### Ito formula for fractional BM + drift and supremum bound

Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...

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777
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### The Wiener measure of an open set

There is so much written about the Brownian motion and I suspect the answers to the questions below are hidden in somewhere in the literature but I cannot find them
Denote by $E$ the Banach space ...

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1
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181
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### The joint distribution of the min and max of a Brownian [closed]

The joint distributions of the brownian and both the minimum and the maximum respectively are known. What could be said about the joint distribution of the maximum and the minimum of a Brownian ...

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1
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61
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### Existence of a process on $\mathbb{R}^2$ that looks like two 'independent' brownian bridges $B_1(x)$ and $B_2(x)$ conditioned on $B_1(x)+B_2(x) > 0$

Consider any probability density function $f(x)$ that has mean zero variance one and say all finite moments. You may assume standard normal density if you like.
Given $a_1,a_2>0$, I consider two ...

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1
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365
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### First hitting time for a drifted Brownian motion

While the solution for a first hitting time for a drifted Brownian Motion is well known, I want to post a different question.
Take a continuous-time stochastic process $X_t$ and define the the ...

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1
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155
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### On the range of Holder continuity of Brownian motion

It is known that Brownian motion is almost surely locally Holder continuous, on a range that is random, i.e. depends on the particular path. This question explores the maximal range on which Brownian ...

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46
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### A convergence question in $L^2$ construction of Brownian motion

I feel confused with a particular step in the $L^2$ consturction of Brownian motion.
Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...

5
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1
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108
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### What is the distribution of $2M_1-B_1$ where $M_t$ is the maximum process of the the Brownian motion $B_t$

Let $B_t$ be a standard Brownian motion and let $M_t:=\sup _{s\le t}B_s$ be the maximum process. What is the distribution of $2M_1-B_1$? is it elementary?

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1
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### $\lim_{r \to +\infty}\frac{1}{\sqrt{2r \ln(\ln(r))}}(B_r-B_{\left \lfloor{\sqrt{2r \ln(\ln(r))}}\right \rfloor})= 0$ a.s.?

Consider a Brownian motion $B$ and let $f(r)=\sqrt{2r \ln(\ln(r))}.$
Is it true that $\lim_{r \to +\infty}\frac{1}{f(r)}(B_r-B_{\left \lfloor{f(r)}\right \rfloor})= 0$ a.s. ?
If so, how to prove it? ...

2
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0
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85
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### Conditional probability of maximum and minimum of Brownian motion

I want to ask for the following problem. Let $(W_t)_{t\geq 0}$ be the standard Brownian motion. For each $t>0$, we call $$m_t =\inf_{0 \leq s \leq t} W_s, \qquad M_t = \sup_{0 \leq s \leq t} W_s.$...

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2
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66
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### Lower-bound on zero-crossing probability of the nonstationary gaussian process $X(t) = tU+(1-t^2)^{1/2}V$, with $(U,V) \sim N(0,I_2)$

Let $(X(t))_{t \in [-1,1]}$ be a centered non-stationary smooth gaussian process with covariation function $\rho(t,s) = \mathbb E[X(t)X(s)]$. For $t_0 \in (-1,1)$ and $\epsilon \in (-1-t_0,1-t_0)$, ...

2
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1
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118
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### Comparison of probabilities that drifted Brownian motion never hits barriers

Let $k , h: \mathbb R_+\to [0,1]$ be non-decreasing and right continuous s.t. $k(t)\le h(t)$ for all $t\ge 0$. Define $\tau_{k}$ (resp. $\tau_h$) by
$$\tau_k : = \inf\{t\ge 0:2+\beta t+ W_t \le k(t)\}\...

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### Problem with fictitious variable in Finite difference method for brownian motion simulation

I'm here asking for help in how to treat the fictitious starting variable x_{i-2} so I can correctly simulate the Brownian Motion with the inertial term (please take a look at the snip below). We can ...

2
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### Continuity of translation operator in fractional white noise analysis

Fix $H\in(\frac{1}{2},1)$, and let $\Omega:=C_0([0,T],\mathbb R^d)$ be the space of $\mathbb R^d$-valued continuous functions. There is a probability measure $P^H$ on $(\Omega,\mathcal B(\Omega))$, ...

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149
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### How to prove excursion process is a Poisson point process?

This question comes from book Ju-Yi Yen and Marc Yor P59 and P60,
On page 59, "Define $\mathcal{Z}_\omega=\{t:B_t(\omega)=0\},$ and $\tau_l$ is the inverse local time. The complement of $\mathcal{...

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### Conditional distribution of Ito integral with deterministic integrand

Let $W$ be a Brownian motion and $\sigma$ a square integrable positive deterministic function. What is the distribution of
$M_s =\sup_{u\leq s}~~I_u$ where $I_s =\int_0^s \sigma(u)\, dW_u$
given $W_t=...

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0
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53
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### Estimate of cumulative probability of geometric Brownian motion

Let $B_\tau$ be the standard BM, $t$ be the initial time, $s$ be the time variable, $r$ and $\theta$ are positive constants. We also assume that $x$ is the initial position of the below geometric ...

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### Random time change from Oksendal's SDE textbook

I have two questions related to the random time change introduced in Oksendal's textbook on SDEs (page 155-156). Specifically, for Lemma 8.5.6., I have no clue as to why we should define $t_j$ in ...

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1
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117
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### How to prove the coupling version of the Donsker's Invariance Principle?

Donsker's invariance principle:
Let $X_1,X_2,...$ be i.i.d. real-valued random variables with mean 0 and variance 1. We define $S_0=0$ and $S_n= X_1+ ... + X_n$ for $n \geq 1$. To get a process in ...

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### Harmonic measure of a punctured disc

Let $D$ be a disc in $\mathbb{C}\cong\mathbb{R}^2 $ and $z_0$ a fixed point of $D$. Is the harmonic measure for $V=D\setminus\{z_0\}$ known? Any reference would also be welcome.

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### Reference request (Brownian local time): for fixed $t$, $a\mapsto L_a(t)$ is a.s. continuous and with compact support

So the title is quite self explanatory.
In the book "Continuous Martingales and Brownian Motion" by Rebuz and Yor, in the proof of Proposition $(2.1)$ of chapter XIII it's stated that:
For ...

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0
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166
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### The quadratic variation of $\int_0^t\int_T^Sg(s,x) \, dW_s^x \, dx$

Consider the process $W^x_t$ which is a Brownian motion for every $x\geq 0$ such that
$$d\langle W_t^x,W_t^y\rangle=Q(x,y)\,dt$$
where $Q$ is some non-negative definite function. Now consider the ...

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### A polar open set in a topological subspace?

Suppose $U$ is a bounded open set in $\mathbb{R}^m$ with ($m\geq2$). Is it possible to have a non-empty set $E$ in the boundary $ \partial U$ of $U$ that is open in $ \partial U$ and is polar?
A set $...

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394
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### Largeness of the set of zeroes of a Brownian motion

Definitions:
A measurable subset $S$ of $\mathbb R$ is said to be mesoscopic if there exists a continuous function $f: \mathbb R \to \mathbb R$ such that $f(S)$ is Lebesgue measurable and has nonzero ...

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### The long run average amount of time the deviation of Brownian motion spends above its expected value

Let $B_t$ be a standard one dimensional Brownian motion. Is it true that
$$\lim_{s \to \infty} \frac{\int_{[0, s]} \mathbf 1_{ \{|B_t| \geq \sqrt{2t/\pi} \} } \ dt}{s}$$
exists almost surely?

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### Why the distribution of M(t) is the same as X(t)?

Let $ B(t)(t\geq 0) $ be the standard Brownian motion and $ M(t)=\max_{0\leq s\leq t}{B(s)} $. If we define $ X(t)=M(t)-B(t) $ as a new stochastic process, how can I show that $ X(t) $ has the same ...

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### L2-closure of absolutely continuous stochastic processes?

Assume we have a possibly multidimensional Brownian motion on a probability space $(\Omega,\mathcal F,\mathbb P)$ where $(\mathcal F_t)_{t\in[0;T]}$ is the Brownian standard filtration. Let $\Vert X\...

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197
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### Bounding Brownian motion and an Ito process simultaneously

Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...

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### Probability that a $d$-dimensional Brownian bridge is greater than a given parameter

Let $(W_t)_{t\in[0,T]}$ be a Brownian bridge such that $W_0=a$ and $W_T=b$, the probability that $\forall t\in[0,T],W_t\geqslant x$ given the parameter $x\leqslant\min(a,b)$ is well known :
$$ \mathbb{...

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### Decay rate of transition density of a SDE system

Consider the following SDE system
$$dx_t = b(y_t)dt + dw^1_t, \quad dy_t = dw^2_t.$$
Here the drift $b(\cdot)$ is a smooth function that may decay slowly. For example, $|b(x)| \le C/|x|^\sigma$ for ...

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58
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### Extension of the Kelvin transform

Suppose $B=B(y,r)$ is ball in $\mathbb{R}^m$ ($m\geq2$), and $u$ a superharmonic function on a neighborhood of the closure $\overline{B}$ of $B$. We know that the Kelvin transform of $u$ with respect ...

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1
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149
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### Generalized Fokker-Planck equation

Consider the diffusion process
$$
d X = \mu(X, t) dt + \sigma(X, t) dY.
$$
When $Y$ is a Brownian motion, we know that the density follows the Fokker-Planck equation. Here I'm considering the general ...

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0
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75
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### Laplace Equation for Brownian Motion

So, I know that there is this theorem (taken from here):
For Laplace's equation $\Delta u = 0$ on a domain $D$ and $u=f$ on $\partial D$ (and some regularity conditions on $D$), we have
$$
u(x) = \...

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1
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48
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### Bound moments wrt. known initial and final moments

Let $X$ be an $L^p$ random variable, where $p\in (0,1)$ and $W_t$ usual Brownian motion (with $W_t$ independent from $X$). I'd like to bound
$$\mathbb E|X+W_t|^p$$
purely in terms of $\mathbb E|X|^p$ ...

1
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55
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### Superharmonicity of the distance function

Suppose $V$ is a convex open proper subset of $\mathbb{R}^m$ ($m\geq2$). It is known that the function $u(x)=$dist$(x,\partial V)$ is superharmonic on $V$. Is there a similar result without $V$ being ...

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78
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### Probability to cross an envelopp for 1D random walk?

Imagine we have an evolving sequence composed of 1 and -1 (ex: -1-11-111...) where the probability to get -1 or 1 is 1/2. n is the lengh of my sequence.
I can make an analogy with random walk: let ...

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2
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135
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### A question on minimum principle

Suppose $D$ be an unbounded domain of $\mathbb{R}^m$ for $m\geq3$, and $u$ is superharmonic on $D$. We know that if $\liminf_{x\to y}u(x)\geq0$ for all $y$ in $\partial^\infty D$ (the boundary of $D$ ...

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55
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### An open set whose complement is non-thin at infinity

Let $x^*$ designate the inverse of a point $x\in\mathbb{R}^m$ under the Kelvin transformation with respect to the circle of center 0 and radius 1. Recall that
$$x^*=|x|^{-2}x.$$
For a set $E$, we set $...