All Questions
Tagged with fa.functional-analysis stochastic-processes
157 questions
4
votes
0
answers
109
views
How fast is discrete-time diffusion on a continuous set?
This question is inspired by Joseph O'Rourke's beautiful answer to my previous question.
Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns have unit norm and sum ...
- 7,633
4
votes
0
answers
1k
views
The spectrum of a Markov Operator and Invariant Measures
Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...
- 509
3
votes
2
answers
307
views
Random matrix is positive
This is a follow up question on my previous question here that was on solved in the deterministic setting by Denis Serre, when the perturbation can be separated. Therefore, I decided to split the ...
- 536
3
votes
1
answer
228
views
Are Holder Condition and signal to noise ratio (SNR) related?
This question was posted in https://math.stackexchange.com but I got hardly any view. If posting here is an objection please let me know I would delete it immediately.
This question has evolved from ...
- 495
3
votes
1
answer
308
views
$f: [0,1]\rightarrow L^1(\Omega)$ as a (measurable?) function from $[0,1]\times \Omega\rightarrow \mathbb{R}$
Given a map from $\big([0,1], \mathcal{B}[0,1], m\big)$ to a Banach space $(X, \|\cdot \|)$. There are strong measurable functions (they are the point wise a.e. limit of simple functions) and weak ...
- 485
3
votes
1
answer
199
views
Markov-semigroup Sobolev inequality
I have a question about the following definition:
A probability measure $\mu$, such that the Markov semigroup $e^{Lt} \in \mathcal{L}(L^2)$ exists and is symmetric, satisfies the Sobolev inequality ...
- 259
3
votes
1
answer
397
views
Fractional Brownian motion via Hilbert space
The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms:
Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, \...
- 273
3
votes
2
answers
987
views
Regarding sample continuity of Gaussian Processes
Suppose we have a Gaussian Process $X_t$ on $\mathbb{R}^n$ with mean function $m(t)$ and covariance function $K(t,s)$. Then is $X_t$ being sample continuous (i.e. the sample paths of $X_t$ are almost ...
- 151
3
votes
1
answer
439
views
Strong continuity of the Ornstein-Uhlenbeck operator
It's well known that the Ornstein-Uhlenbeck semigroup defined by
$$
P_tf(x)=\int_{\mathbb{R}}f\left(xe^{-t}+\sqrt{1-e^{-2t}}z\right)\frac{e^{-z^2/2}}{\sqrt{2\pi}}\,dz
$$
is not strongly continuous on ...
- 617
3
votes
1
answer
91
views
Conditional Expectation in Diffusion Process
Consider a $d$-dimensional diffusion process $\mathbf{X}=(\mathbf{X}_t)_{t\in [0,T]}=([X^1_t,...,X^d_t])_{t\in [0,T]}$ that is the unique strong solution of the following SDE:
$$\left\{\begin{matrix}
...
- 143
3
votes
1
answer
180
views
Are the paths of the Brownian motion contained in a suitable RKHS?
Let $H_B$ be the reproducing kernel Hilbert space (RKHS) of the Brownian Motion $(B_t)$ on $[0,1]$. It is well known that with probability 1 the paths of $(B_t)$ are not contained in $H_B$.
But is ...
- 31
3
votes
1
answer
275
views
Sum of two parts of a continuous stochastic process
Let $X$ be a centered continuous stochastic process which is square integrable on $[0,2]\times \Omega$ and the basis of $L^2(0,2)$ is $\{e_i\}$. By using Karhunen-Leove Theorem one can write for all $...
- 213
3
votes
1
answer
99
views
Regularity of finite variation kernels in the (intersection) of the semimartingale spaces $H^p$
Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| \,...
- 235
3
votes
0
answers
98
views
Algebra core for generator of Dirichlet form
This is a question about the existence of a core $C$ for the generator $A$ of a regular Dirichlet form $\mathcal{E}$ having a carré du champ $\Gamma$, so that $C$ is an algebra with respect to ...
- 143
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
3
votes
0
answers
148
views
Markov semigroups and resolvents, difference of continuity
Let $(E,d)$ be a locally compact separable metric space. We have a Markov process $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in E})$ on $E$. For bounded measurable function $f$ on $E$, we define
\begin{align*}
...
- 721
3
votes
0
answers
569
views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
3
votes
1
answer
281
views
Covariation of the stochastic integral and the Wiener process
Let$^1$
$T>0$
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint operator with finite trace $\operatorname{tr}Q$
$(e^n)_{n\in\mathbb N}$ be an ...
- 167
3
votes
0
answers
473
views
textbook of measure theory abstracted as functional analysis [closed]
Background
I have studied intro functional analysis, probability theory based on measures, and some elementary connection between them e.g. that weak conversion of random variables correspond to weak*...
- 163
3
votes
0
answers
231
views
I've found a representation of the Itō-Stratonovich correction term and don't understand the used notion of a "trace"
Consider a Stratonovich SPDE $$X_t=X_0+\int_0^tb(s,X_s)\:{\rm d}s+\int_0^t\sigma(s,X_s)\circ{\rm d}W_s\tag 1$$ in a separable $\mathbb R$-Hilbert space $H$ with $W$ being a $Q$-Wiener process on a ...
- 167
3
votes
0
answers
78
views
Perscribed/Inverting Conditional Expectation
I'm having difficulty finding papers which deal with the following inversion problem.
Suppose I have a stochastic process $Y_t$ (which is described by a certain Hilbert-Space-valued SDE). I want to ...
- 5,405
3
votes
0
answers
113
views
Stationarity of Brownian motion with drift
Suppose the following SDE for $X_t$ is well-posed:
$$dX_t = \sqrt{2}\, dB_t - \nabla\Phi(X_t)\,dt.$$
For what $\Phi\in C^1(R^d)$ will $X$ have stationary distribution $u_{\infty}$? For what $\Phi$ ...
- 165
3
votes
0
answers
134
views
The distribution of maximum of fraction Brownian motion over finite time interval
Suppose that $\{B_t^H,\ t\geq 0\}$ is a fractional Brownian motion with Hurst exponent $H$, I wonder if there are explicit expressions for the joint distribution of
$(\sup_{0\leq t\leq T}B_t^H,B_T^H)$...
- 71
2
votes
1
answer
773
views
On the continuity of map $\Gamma$
Let $M$ be the space of right continuous functions $\ell: \mathbb R_+\to [0,1]$ that are non increasing s.t. $\ell(0)=0$. Define the map $\Gamma : M\to M$ by $\Gamma[\ell](t):=\mathbb P[\tau^{\ell}>...
- 1,334
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
2
votes
1
answer
336
views
Is this a "contradiction" on stochastic Burgers' equation? How to understand it?
For the stochastic Burgers' equation with linear noise, I can deduce two results. Both of them can be applied to same initial data, but the first result means the global existence with high ...
2
votes
1
answer
300
views
Reverse martingale convergence theorem in Banach spaces
In section 1.5 of a course given by Gilles Pisier, the author is claiming that in the excerpt below $\operatorname E[\varphi_i\mid\mathcal A_{-n}]\to\operatorname E[\varphi_i\mid\mathcal A_{-\infty}]$ ...
- 167
2
votes
1
answer
440
views
Can we extract information from signature (rough path theory) to construct part of signal?
This question is related to rough path theory. Consider we have obtained signature obtained from a set discrete data points postulating linear from one data point to another. Such signature are used ...
- 21
2
votes
2
answers
351
views
Weak convergence for discrete-time processes using characteristic functions
I am looking for a good reference about the analogues of the Bochner Theorem and the Lévy Continuity Theorem
for probability measures on $\mathbb{R}^{\mathbb{N}}$ with the product topology.
...
2
votes
1
answer
149
views
On a core for Neumann Laplacian on $C(\overline{D})$
Let $D \subset \mathbb{R}^d$ be a bounded $C^1$ domain. We consider a reflected Brownian motion $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in \overline{D}})$ on $\overline{D}$. Let $\{p_t\}_{t>0}$ denote ...
- 721
2
votes
1
answer
756
views
Functional representation of adapted jointly measurable stochastic processes
It seems like the question stated here in MSE has no answer yet and seems therefore for me to be not of a basic question type. For this reason I move it to MO.
Let $X_t : \Omega \to E, \ t \geq 0$ be ...
- 1,773
2
votes
1
answer
236
views
Self-adjointness of generator and semigroup of an SDE
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\...
- 835
2
votes
1
answer
89
views
Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances
Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$:
$$
d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|.
$$
Let $\rho$ be the Levy-Prokhorov metric on the ...
- 177
2
votes
1
answer
86
views
Smoothness of resolvent of the infinitesimal generator of an Ito diffusion acting on bounded continuous function
Let $dX_t=\sigma(X_t)\,dW_t+\mu(X_t)\,dt$ be an Ito diffusion with Lipschitz coefficients and $\sigma(x)>0$. Let $f(x)$ be a continuous and bounded and non decreasing function. Can we prove that ...
- 41
2
votes
1
answer
264
views
Bounded-pointwise continuity of Markov operators / semigroups
Let $B_b(E)$ be the space of bounded measurable functions on some Polish space $E$ endowed with the supremum norm. It seems quite classical that Markov semigroups $P_t:B_b(E)\to B_b(E)$ are in one to ...
- 59
2
votes
1
answer
288
views
order of the singularity of a Green's function to the fractional Laplacian
I was looking at a problem which involves the Green's function of a fractional Poisson equation.
To fix notation, let $D\subset \mathbb{R}^n$ very nice, i.e. a hypercube, and
\begin{equation}
\begin{...
- 161
2
votes
1
answer
164
views
Is there any parameter space of Cramér–Rao_bound
It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called '...
- 495
2
votes
1
answer
404
views
Feynman-Kac formula for lattice heat equation with non-diagonal potential
Suppose that $X$ is the continuous-time simple symmetric random walk on the lattice $\mathbb Z^d$ (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let
$$u(t,x):=\mathbf E\...
- 891
2
votes
1
answer
69
views
Lyapunov-type function in a non locally-compact space and boundedness of the average
Set-up and question.
Let $\mathcal{X}$ be a complete separable metric space which is not locally-compact. Let $V: \mathcal{X} \to [0; +\infty]$ be a function and $(X_t)_{t\geq 0}$ a Markov process in $...
- 724
2
votes
1
answer
775
views
Properties of Cameron Martin Space
In the case that I'm working with a separable Hilbert space, $H$, on which I have a trace class operator, $K$, that's coming from a Gaussian (i.e., $K$ is self-adjoint, and for simplicity, has trivial ...
- 379
2
votes
0
answers
60
views
Semigroup property in SPDEs
In fact, we know that a bounded linear operators on a Banach space $X$ satisfies the semigroup property, i.e. $$S(t+s)=S(t)S(s), \text{for every}\ t,s\geq 0.$$
However, in various literatures, I ...
- 57
2
votes
0
answers
137
views
Holder-Besov space and time continuity
Let $\mathbb{T}^d$ be the $d$-dimensional torus, $\mathscr{S}:=C^\infty(\mathbb{T}^d)$ the Schwartz space, $\mathscr{S}'$ the space of tempered distributions.
We consider a dyadic partition of unity $(...
- 573
2
votes
0
answers
74
views
References for a class of Banach space-valued Gaussian processes
Let $E$ be a separable Banach space, consider a centered $E$-valued Gaussian process $\{x_t,t\ge 0\}$ that satisfies
\begin{equation}
\mathbb{E}\phi(x_s)\psi(x_t)=R(s,t)K(\phi,\psi),\quad \phi,\psi\in ...
- 121
2
votes
0
answers
193
views
If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality
Related: On a deceptively tricky calculus problem.
The way that Leonard Gross proves the log Sobolev inequality is in the following stages:
He proves that for any operator $B$ that satisfies the log ...
- 90
2
votes
0
answers
103
views
Find a function $f\geq 0$ such that $e^{-t[(x-\partial_x)\partial_x]^2} f$ is not non-negative for some $t\geq 0$
Consider the square of the Ornstein-Uhlenbeck operator $$A=[(x-\partial_x)\partial_x]^2=(x-\partial_x)\partial_x (x-\partial_x)\partial_x.$$ We know that $[(x-\partial_x)\partial_x]^2$ cannot be a ...
- 90
2
votes
0
answers
62
views
Continuous-time Wold decomposition
I'm looking for a reference for the Wold–Zasukhin decomposition in continuous time for stationary random processes on the real line.
I am aware of the classic result in the book from Rozanov, which ...
- 21
2
votes
0
answers
112
views
Is there a way to detect instantaneous states of a Feller Process from its infinitesimal generator?
I’m working with generators of Feller processes. If $C(E)$ is the space of continuous functions over $E$; with $E$ a compact metric space, I proved that an operator $G$ over $C(E)$ is the ...
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 ...
2
votes
0
answers
172
views
Non-integer conditional moment of exponential functional of Brownian motion
Let $B_t$ be a standard Brownian motion.
I want to solve the following:
$$
\mathbb{E}\left[\left(\int_0^1 e^{\sigma B_t}dt \right)^{1/(1-\beta) }\mid e^{\sigma B_1}=z \right],
$$
for some fixed $0<\...
- 284
2
votes
0
answers
109
views
Tightness of Hilbert-space-valued arrays
Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n $ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}...
- 83