All Questions
Tagged with fa.functional-analysis pr.probability
616 questions
4
votes
0
answers
95
views
When the Jacobian of unstable measure converges
Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
- 1,043
1
vote
0
answers
65
views
Normalizing constants preserve metric entropy
Suppose $\mathcal{F}=\left\{f\in L^2([a,b]): 0<\underline{c}\leq f\leq\overline{c} \right\}$. Consider the following transformation
$$\tilde{\mathcal{F}} := \left\{\frac{f}{\int f d\mu}: f\in \...
- 11
2
votes
0
answers
52
views
Reference Request: Dirichlet operators with singular coefficients
Let $d\geq 2$, $\delta \in (0,1)$ and let $\mathcal{L}_{d,\delta}$ be the second order differential operator defined by
\begin{align*}
\mathcal{L}_{d,\delta}(f)(x) = \Delta(f)(x)-\delta \|x\|^{\delta-...
- 778
4
votes
3
answers
801
views
Quick derivation of classical probability theory from von Neumann algebraic framework
Watching (the begining of) a lecture on free probability theory by Dimitri Shlyakhtenko https://www.youtube.com/watch?v=F8Urtr39jM0, I'm led to consider the following question
Question. How can one ...
- 6,853
1
vote
1
answer
82
views
Local inverse bound of Cameron Martin and Banach norms
Let $X$ be a Banach space with a centered Gaussian measure $\mu_0$. Let $E$ be the Cameron-Martin space of $X$. Let the respective norms be $\|\cdot \|_X$ and $\|\cdot \|_E$. It is well known (see ...
- 133
6
votes
1
answer
641
views
Bochner-Minlos for moment-generating functions?
It is well-known that the Bochner-Minlos theorem characterises measures on duals of nuclear spaces by their characteristic functions. Is there a similar version for moment-generating functions?
I have ...
- 651
2
votes
0
answers
104
views
Weak convergence rates for integral operators
Suppose $q=\sum_{i=1}^m\pi_i\delta_{x_i}$ is a discrete measure on $\mathbb{R}^n$ and let $q\ast \varphi_\epsilon$ denote the convolution of $q$ with some mollifier $\varphi_\epsilon$, so that $q\ast\...
- 75
30
votes
1
answer
1k
views
Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments
It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...
- 6,853
1
vote
1
answer
368
views
Does the almost sure convergence of absolutely continuous r.v.'s imply the weak convergence of the pdf's in $(L^\infty)^*$?
The following question was asked in a comment at Almost sure convergence vs convergence of probability density functions :
Suppose that $(X_n)$ is a sequence of random variables (r.v.'s) converging ...
- 128k
2
votes
1
answer
393
views
Is there a Hilbert space approach to commutative probability theory on locally compact spaces?
I was recently made aware (thanks to the answers on Why does Riesz's Representation Theorem apply in quantum mechanics?) that the $C^*$ algebra approach and the Hilbert space approach to quantum ...
- 2,071
1
vote
0
answers
305
views
Gaussian measures on infinite dimensional spaces
On Zabczyk & Da Prato book about infinite dimensional SDEs they introduce the idea of Gaussian measures in infinite dimensional Banach spaces. They do so by means of Fernique theorem.
In the ...
- 515
10
votes
2
answers
2k
views
Why does Riesz's Representation Theorem apply in quantum mechanics?
$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$.
It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
- 2,071
3
votes
0
answers
60
views
Existence, Uniqueness, and "ODE Characterization" of Minimizers for Variational Functionals from Large Deviations
A [classical result][1] of E. Lieb is that the functional
$$\mathcal E(\phi):=\int_{\mathbb R^3}|\nabla\phi(x)|^2~dx-\int_{(\mathbb R^3)^2}\frac{|\phi(x)|^2|\phi(y)|^2}{|x-y|}~dx~dy$$
for $\phi\in W^1(...
- 891
0
votes
0
answers
144
views
Optimization over the set of all bounded probability measures
Given $X$ finite, fix a continuous function $\theta \in \Delta^+ (X) \to [0,1]$, fix a probability measure $\mu^*$, and a $\varepsilon > 0$. Consider:
$$
\max_{\mu \in \Delta^+ (X)} \theta (\mu), \...
- 67
1
vote
1
answer
137
views
Embeddings of spaces of probability measures
What is the relationship between the spaces $X_1\triangleq \mathscr{P}(C([0,1],\mathbb{R}))$ and $X_2\triangleq C([0,1],\mathscr{P}(\mathbb{R}))$; where $\mathscr{P}(\cdot)$ denotes the Borel ...
- 5,405
1
vote
0
answers
123
views
Derivatives of measures of bounded variation on intervals
Investigating an abstract Cauchy problem on the space of measures with bounded variation I came up with the following space:
Let $\operatorname{BV}[a,b]$ the space of all functions $f:[0, 1] \to \...
- 301
3
votes
1
answer
688
views
Positive definite kernels involving the $\min$ function
I am interested in the positive kernels of the form $k(x,y) = \min\{a(x,y), b(x,y)\}$ (assuming $k(x,y) = k(y,x)$). Some examples including $\min\{x,y\}$ and $\min\{f(x)g(y), f(y)g(x)\}$, but are ...
- 41
6
votes
2
answers
333
views
Is there a way to reconstruct the convolution $(f * g)(x)$ of $f$ with a Gaussian $g$ from sampled values, $(f*g)(a), a \in A$?
Suppose that $f: \mathbb{R} \to \mathbb{C}$ is a function which has support in $[-1,1]$. Let $g = g_\sigma$ be a centered Gaussian with variance $\sigma^2$. Is there a way to reconstruct the ...
- 437
0
votes
0
answers
302
views
Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem
In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
- 167
4
votes
3
answers
2k
views
Duality of finite signed measures and bounded continuous functions
Let $E$ be a metric space, $C_b(E)$ denote the space of bounded continuous functions $E\to\mathbb R$ (equipped with the supremum norm), $\mathcal M(E)$ denote the space of finite signed measures on ...
- 167
2
votes
0
answers
142
views
Radon-Nikodým-like theorem for Radon measures
Let $(E,d)$ be a metric space, $\mu$ be a nonnegative Radon$^1$ measure on $\mathcal B(E)$ and $\nu$ be a finite (signed) Radon measure on $\mathcal B(E)$.
I'm searching for a Radon-Nikodým-like ...
- 167
1
vote
1
answer
96
views
Laguerre convolution truncation error
Suppose i have extended two d-variate functions $f$ and $g$ (two densities: positives and integrate to one) supported on $\mathbb{R}_{+}^d$ into the following (tensorised) Laguerre($\alpha = 0$) ...
- 686
5
votes
1
answer
283
views
Malliavin derivative of stopped Brownian motion
Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion"
I have a small question concerning the Malliavin derivatives. It could ...
- 393
2
votes
1
answer
183
views
Does set of finitely additive probability measures embed linearly into a strictly convex dual Banach space?
I am trying to better understand a condition that appears in Theorem 1 of this paper.
Let $K$ be a convex and compact subset of a locally convex tvs. The condition is:
$K$ embeds linearly into a ...
- 869
0
votes
0
answers
44
views
Solving nonlinear equations involving expectations
Let $X$ be a random variable and $g(x,y)$ be a function of two variables. Consider the equation
$$
\mathbb{E}_Xg(X,y) = 0
$$
Are there any specialized techniques for solving such equations (...
- 11
0
votes
1
answer
212
views
Expressing the measure of a set in terms of the characteristic function of the measure
Let $\mu$ be a discrete, finitely supported probability measure in $\mathbb{R}^d$ and denote by $\phi$ be the characteristic function of $\mu$, i.e. $\phi(t)=\mathbb{E}e^{i<t,X>}$, where $X$ is ...
- 2,288
4
votes
0
answers
160
views
Can we show equivalence of two distributions based on their statistics?
Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \...
- 57
1
vote
1
answer
342
views
Concentration inequality for the supremum of $L_2$ norm of a vector-valued Gaussian process with iid components
Let $\Omega$ be a compact subset of $\mathbb R^p$ and let $f_1,\ldots,f_k$ be zero mean identically distrubuted Gaussian processes on $\Omega$ such that $f_1(x),\ldots,f_k(x)$ are independent $x \in \...
- 6,853
5
votes
1
answer
774
views
Question/References on the Skorokhod M1 topology
Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
user128095
1
vote
0
answers
56
views
Moduli of continuity and Wasserstein differentiability of functions between measures
Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
- 11
2
votes
0
answers
176
views
Uniqueness for measure valued ode
Morning! Basically I'm working on a mean field scaling for some measure valued process (valued on $M_F(\mathbb{N})$). The limit turns up as a (deterministic) solution to a measure valued ODE. Let's ...
4
votes
3
answers
250
views
A functional integral inequality
Suppose $f:I=(0,1)\to \mathbb R$ is a continuous function that satisfies
$$ \int_I f(t) e^{at}\,dt \geq 0\quad \text{for all $a \in \mathbb R$}.$$
Does it follow that $f\geq 0$ on $I$?
- 4,135
14
votes
2
answers
892
views
Do distance functionals separate probability measures?
Let $(\Omega,d)$ be a compact metric space and $\mathcal P(\Omega)$ its space of Borel probability measures. Let $D=\{ d_p\mid p\in\Omega\}$ where $d_p(x)=d(p,x)$ be the set of all "distance ...
- 459
4
votes
1
answer
1k
views
Functional derivative of differential entropy
I have trouble finding the derivative of the differential entropy w.r.t the probability density function, i.e. what is $\frac{\delta F[p]}{\delta p(x)}$, where $F[p] = \int_X p(x)\ln(p(x))dx$, and $p(...
- 41
1
vote
1
answer
293
views
Does the Skorokhod space with the uniform topology admit a smooth partition of unity?
Does the Skorokhod Banach space $D[0,1]$ (cadlag functions equipped with the uniform norm) admit a smooth partition of unity? I found Johanis - Smooth partitions of unity on Banach spaces, which ...
- 617
3
votes
1
answer
983
views
About the metrizability of the space of Probability measures $\mathcal{P}(S)$
It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the ...
- 309
1
vote
2
answers
597
views
Moments of complex random variables
My question is that how much information we can get form integer moments of a complex random variable?
Let $\mathcal{Z}$ be a complex value random variable, and assume that we can compute $$\int \...
- 499
2
votes
2
answers
155
views
Existence of classical solution for a parabolic equation without Hölder continuity in time for its coefficients
Consider equation
$$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$
with initial condition $u(0, x) = g(x).$
Suppose that $c(t, x)$ and $...
- 1,399
3
votes
1
answer
404
views
The sign of the tail of Fourier transform of a positive function/ characteristic function
I am interested in a specific density (positive function) and would like to prove that the tail of its characteristic function (Fourier transform) is positive ($>0$). Here is the density $f(x)=c_\...
- 178
1
vote
1
answer
173
views
Spectral gap of a Markov chain on the nonnegative integers
Let $\lambda_k,\mu_k\in\mathbb R_{\ge0}$ $(k\ge1)$ be nonnegative real numbers such that $\sum_{k=1}^\infty k\lambda_k<\infty,$ let $S=\mathbb Z_{\ge0}$ be the nonnegative integers, let $T=\mathbb ...
- 276
14
votes
0
answers
718
views
Lower bounds on analytic functions connected to Fox H
The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
- 178
2
votes
0
answers
171
views
Can Schauder's fixed point theorem apply to a metric space?
I am currently reading the existence proof of Mean Field Game equation, which is a coupled system of Hamilton-Jacobi-Bellman equation and Fokker-Planck equation, see page 42 of the paper here. The ...
- 1,399
2
votes
1
answer
122
views
How is this bound for a Wasserstein contraction coefficient in this paper obtained?
I'm trying to understand the following conclusion from this paper (see below for the relevant paragraphs):
I'm not sure whether they really mean that it follows from the statements of Lemma 3.2 (...
- 167
1
vote
0
answers
36
views
Does a total preorder on lotteries that preserves countable mixtures preserve arbitrary mixtures?
Let $X$ be a countable set. A lottery on $X$ is a function $\lambda: X \to [0,1]$ such that $\sum_x \lambda(x) = 1$. Let $\Delta X$ be the set of lotteries on $X$.
A total preorder $\preceq$ on $\...
- 869
3
votes
0
answers
90
views
How does one define the gradient of a Markov semigroup?
In the context of functional inequalities for Markov semigroups $(\mathcal P_t)_{t\ge0}$, what is one denoting by $\nabla\mathcal P_tf$? For example, I've found the following assumption in this paper:
...
- 167
5
votes
2
answers
393
views
Connections between two constructions of infinite dimensional Gaussian measures
Let me discuss two possible constructions of Gaussian measures on infinite dimensional spaces. Consider the Hilbert space $l^{2}(\mathbb{Z}^{d}) := \{\psi: \mathbb{Z}^{d}\to \mathbb{R}: \hspace{0.1cm} ...
- 3,515
2
votes
1
answer
205
views
Eigenspace of Gaussian Markov operator
Consider the (one-dimensional) Gaussian distribution $Q := N(\nu,\tau^2)$ and the (Gaussian) Markov operator
\begin{equation*}
\begin{array}{rccc}
R : & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) &...
- 123
12
votes
1
answer
1k
views
Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces
Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
- 333
2
votes
0
answers
520
views
Example of a non-reflexive Banach space and two sequences
Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$.
If $X$ is reflexive, ...
- 199
0
votes
1
answer
1k
views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
- 710