Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
8,632
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Does this filtration have a name?
In the context of Ethier&Kurtz Markov Processes: Characterization and Convergence (Chapter 4, equation (3.2)) as well as the two papers Martingale problems for conditional distributions of Markov ...
2
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0
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157
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Hunting an invisible target
An invisible target on the integer line starts at $0$. On each round it either stays put, moves to the left or moves to the right by $1$ with probability $\frac{1}{3}$ each. You are then asked to ...
2
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1
answer
85
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Deriving the distribution of standardized variables with empirical mean and standard deviation
I'm working with a set of independent and identically distributed random variables $\{ x_i \}_{i=1}^N$, where each $x_i$ follows a Gaussian distribution $P_X(x) = \mathcal{N}(x; \mu, \sigma^2)$. This ...
2
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101
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Asymptotic Independence of random walks from increments?
Suppose we have two random walks $(S_n:n\geq 1)$ and $(T_n:n\geq 1)$ building from independent identically distributed increment vectors $\{(X_k,Y_k):k\geq 1\}$, i.e. $S_n=\sum_{k=1}^n X_k, T_n=\sum_{...
2
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75
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Multi-scale 3- and 5-arm exponents for critical planar percolation
Consider critical site percolation on the planar triangular lattice. Denote by $A_j(m,n)$ the event that there are $j$ arms (paths from the inner boundary to the outer boundary) of alternating colour ...
2
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99
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The fluctuations of a random path
Suppose I have a $n \times n$ square grid and for each square, I assign 1 with probability $\frac{1}{2}$ and 0 with probability $\frac{1}{2}$. On the boundary, I put 1s on the lower half and 0s on the ...
2
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86
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Embedding a Markov chain in a Markov process
Let $X_{t\ge 0}$ be a Markov process with values in a metric space $(\mathcal{X},d)$ defined on a probabiltiy space $(\Omega,\mathcal{F},\mathbb{P})$ and let $(\tau_n)_{n=1}^{\infty}$ be a sequence of ...
2
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67
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Faster Convergence in CLT for sums and convolutions of Gaussians?
Let $n\in\mathbb{N}$ and $\sigma>0$ be fixed.
I have a certain class $\mathcal{C}$ of random variables I am interested in analyzing.
This contains
$\vec X\sim \mathcal{N}(0, \sigma^2I_n)$
Sums of (...
2
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134
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Sum of arrival times of Chinese Restaurant Process (CRP)
Suppose that a random sample $X_1, X_2, \ldots$ is drawn from a continuous spectrum of colors, or species, following a Chinese Restaurant Process distribution with parameter $|\alpha|$ (or ...
2
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2
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306
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Existence of the limit of periodic measures
Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
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54
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Distance distribution for Cayley graphs of the fintie Heisenberg groups H3(Z/nZ) approaches Gaussian for large "n"?
I wonder several questions about Cayley graphs of finite Heisenberg groups H3(Z/nZ).
Question 1: do we know the diameter dependence on "n", at least for the standard choice of generators ? ...
2
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1
answer
181
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Can we construct close martingales if their terminal marginal laws are close?
Let $M=(M_t)_{0\le t\le 1}$ be a real-valued continuous martingale. Let $\mu := {\rm Law}(M_1)$ and $\varepsilon \in (0,1)$. For any $\nu$ satisfying $W_2(\mu,\nu)\le \varepsilon$, can we construct ...
2
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1
answer
104
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Thinning of (mixed) binomial point process
Let $N= \sum_{i=1}^M \delta_{X_i}$ be a mixed Binomial process over $(\mathbb X, \mathcal X)$. I.e., $M$ is a $\mathbb Z_+$ valued random variable with probability mass function $q_M(m)$, $m=0, 1, \...
2
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89
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Random walk with same directions and different step sizes
Let $X\sim e^{iU}$, where $U$ is uniformly distributed on $(0, 2\pi]$. Define $\chi_1, \cdots, \chi_t$ as i.i.d. random variables with the same distribution as $X$.
Consider the following two random ...
2
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66
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Construct a Bregman divergence from Wasserstein distance
I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance.
More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...
2
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60
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Iterated chaos expansion
Using the notations from Normal Approximations with Malliavin Calculus, Chapter 2
random variables $F$ in the probability space generated by an iso-normal Gaussian family $X(h)$,
$$E[X(h)X(g)] = \...
2
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66
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Distribution of waiting time conditioned on a fixed time length
FYI, this question is a duplicate from math stack exchange
I ask here again because I got no response.
Suppose, I work in a factory production line. The time for me to finish wrapping product $A$ (or $...
2
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138
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Time reversal of infinite-dimensional SDE
Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
2
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90
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Lower & upper bound on the maximal component given the system of power sums
Given a non-negative vector $x=(x_1,x_2,\dots,x_n)\in\mathbb{R_{>0}^n}$ and $m\in\mathbb{N}$, construct a system of power sum symmetric polynomials (or norms, if you like)
$$
\begin{cases}
x_1+x_2+\...
2
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76
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A question on the convex hull of independent random walks
Consider $m$ independent random walks $X^1_n, \dots, X^m_n$ driven by a probability measure $\mu$ in $ \mathbb{Z}^d$. Assume that the $\mu$ has no drift, that is, the expected value of a $\mu$-...
2
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175
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A few questions on Feller processes
Update. Most of my questions have been answered in the comments. I am adding these answers to the post.
There are at least three definitions of Feller semigroup and the corresponding processes: $C_0 \...
2
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83
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Defining a metric on $\mathbb Z^n$ using Green's function for the simple random walk
Let $G$ be Green's function for the simple random walk on $\mathbb Z^n$ for $n\ge 3$, i.e., $G(x)$ is the expected number of visits to $x$ when the walk starts at the origin.
Define $d(x,y)=G(x-y)^{1/(...
2
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141
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Fractional Brownian motion covariance with a twist
Let $H \in (0, 1)$, $D \in \mathbb{R}$ and assume that the following function
$$
r ( t, s ) = \frac{1}{2} \, \Big[ t^{2H} + s^{2H} - | t - s |^{2H} \Big] + D \, t^H s^H,
\quad t, \, s \geq 0
$$
is ...
2
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0
answers
68
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Reference request : A SPDE model
Let $\Omega_0\subset\mathbb R^d$ be open and bounded with sufficiently smooth boundary $\partial\Omega_0$. Let $O\subset \Omega_0$ be a random open subset. Set $\Omega:=\Omega_0\setminus O$. Consider ...
2
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62
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Joint tail for Brownian motion $P[B_{t_1}>g_1,...,B_{t_n}>g_n]$
Maybe not surprisingly there seems to be a lack of in-depth study of sharp estimates for the joint tail of Brownian motion over different times
$$P[B_{t_1}>g_1,...,B_{t_n}>g_n]$$
for strictly ...
2
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0
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240
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Why is it impossible to create a numerically balanced die with more than 120 sides?
I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...
2
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222
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Ball games: How to allocate $N$ balls into $M$ boxes so as to maximize the expected number of taken balls
Consider the following ball games, which looks like very intuitive and simple but I have tried for a long time.
Assuming we have $M$ identical boxes and $N$ identical balls, we distribute these $N$ ...
2
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0
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91
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Random matrices may be asymptotically free but never free themselves?
It is well known that independent $N\times N$ unitarily-invariant random matrices (or independent families of random matrices) may be asymptotically free as $N\to \infty$ with respect to the ...
2
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53
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Dirichlet series solution to Poisson Point Process question (repost from math.SE)
Reposting here because the bounty on the original math.SE post expired, with no solutions or comments received.
For any discrete subset $S$ of $\mathbb{R}^d$, consider a digraph formed by placing an ...
2
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95
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Equivalence of score function expressions in SDE-based generative modeling
I am studying the paper "Score-Based Generative Modeling through Stochastic Differential Equations" (arXiv:2011.13456) by Yang et al. The authors use the following loss function (Equation 7 ...
2
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222
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Singular values of Kronecker product of random matrices
I'm looking for a way to evaluate $\mathbb{E} \| (\mathbf{X} \mathbf{Q})^+ \|$ for a random matrix $\mathbf{X} \in \mathbb{R}^{r \times n}$ and a (fixed) matrix $\mathbf{Q} \in \mathbb{R}^{n \times \...
2
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119
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Measure algebra for a family of probability measures
Let $(X,B,P)$ be a probability space, $I_P$ the $\sigma$-ideal of $P$-null sets and
\begin{align}
B_P = B \ltimes I_P &= \{ A \mathbin{\triangle} N \mid A \in B, N \in I_P \}
\end{align}
the ...
2
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0
answers
67
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What is an efficient non-adaptive group testing scheme if the number of defectives, $d$, grows proportionally to the number of items, $n$?
Suppose that for some $p \in \left(0, 1\right)$ and some $n \in \mathbb{N}$, we have $n$ independent Bernoulli random variables, $X_{1}, X_{2}, \dots, X_{n}$, each with mean $p$. We shall call $X_{1}, ...
2
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84
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A question related to the jumps of a Levy process
The Lévy–Khintchine formula says that any Lévy process, $X=(X(t), t \geq 0)$, has a specific form for its characteristic function. More precisely, for all $t \geq 0$, $u \in \mathbb R^d$:
$$
\mathbb{E}...
2
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133
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An attempt to define expected value of a Riemannian manifold valued random variable - what'll go wrong?
Let $X:\Omega\to (M,g)$ be a random variable taking values in a Riemannian manifold $(M,g)$ with the Riemannian volume form denoted by $dvol_g(x).$ We know that there's no standard way to generalize ...
2
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0
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45
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$\sigma$-compactness of probability measures under a refined topology
Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
2
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60
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Linear regression with heavy-tailed gradient noise and generalized CLT
I want to investigate the distribution of the stochastic gradients when applying SGD to linear regression, in very simple but non-trivial situations. For dimension $1$ and when estimating only the ...
2
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0
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124
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Almost sure convergence of a recursively defined sequence
Let $(X_k)_{k\ge 1}$ be a sequence of i.i.d. positive random variables of exponential distribution $\mathcal E(1)$, i.e.
$$\mathbb P[X_k>x]=e^{-x},\quad \forall x\ge 0.$$
Fix some $s\ge 0$. For ...
2
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0
answers
106
views
Log Sobolev inequality for log concave perturbations of uniform measure
Suppose $\Omega$ is a convex bounded open set of $\mathbb{R}^n$ (I would be happy with just $\Omega$ as the $n$-dimensional cube). Let $\mu$ be the uniform measure on $\Omega$ and consider the ...
2
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0
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118
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Can a diffusion process admit an invariant measure with a non-differentiable density?
The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
2
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1
answer
240
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Functional integral formulas for the wave equation and other hyperbolic PDEs
The Feynman–Kac formula provides a functional (Wiener) integral representation of the solution $u$ to the heat equation
\begin{align*}
\partial_t u &= \frac{1}{2}\Delta_x u,\\
u(0,x) &= ...
2
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0
answers
114
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Martingale regularization
Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u \in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$
I was wondering if there ...
2
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94
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Limiting value of $\dfrac{1}{m}\mathrm{tr}(FAF^\top (FBF^\top)^{-1})$, where $F$ has iide $N(0,1)$ entries and $A,B$ are deterministic
Let $F=F_{m,d}$ be a random $m \times d$ matrix with iid entries from $N(0,1)$. Let $A=A_d$ and $B=B_d$ be deterministic $d \times d$ positive-definite matrices. In case it helps, it may be assumed ...
2
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0
answers
62
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Recursive sequence of renewal type : when does one term dominate them all?
Let $(b_n)_{n \geq 0}$ be an increasing sequence of non negative real numbers.
Let $(u_n)_{n \geq 0}$ be recursively defined by $u_0 =1$ and
$$u_{n} = \sum_{k=0}^{n-1} u_{k} b_{n-k}$$
Find a ...
2
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0
answers
120
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An additive combinatoric probability question
I asked the question in cs-theory stack exchange but was advised a pure math forum would be more apt. Link to the question:
https://cstheory.stackexchange.com/questions/52930/an-additive-combinatoric-...
2
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0
answers
233
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Correlation decay rate
Let $T$ be a continuous transformation of a probability measure space $(X,\mathcal{B}(X),\mu)$ and
$\varphi ,\phi \in L^2(\mu)$ (so-called observable) . The correlation function of $\varphi ,\phi$ (a ...
2
votes
0
answers
94
views
Local martingale for a (two-dimensional) diffusion
Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)\,dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda ...
2
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0
answers
75
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Random walks on randomly evolving graphs
I am interested in analyzing a random walk on a growing tree with vertices labelled on a tree with following properties.
The number of nodes at depth $k$ is a an exponential function of $k$. One can ...
2
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0
answers
51
views
Trouble understanding a Lemma in Pastur's Paper
I'm having trouble understand Eq 3.51 Lemma 3.3 in https://arxiv.org/pdf/2001.06188.pdf
The basic premise is
$$\begin{align}
&\eta _{j}(t)=t^{1/2}\eta _{j}+(1-t)^{1/2}q_{n}^{1/2}\gamma _{j},
\;t \...
2
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0
answers
55
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Regularity on $\mathbb{T}^3$ of the "functional average" of a map $S : C^\infty(\mathbb{T}^3, \mathbb{R}) \to L^2(\mathbb{T}^3, \mathbb{R})$
For simplicity, let $C^\infty(\mathbb{T}^3, \mathbb{R})$ be the real Frechet space of periodic smooth functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus.
For a fixed ...