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Norm equivalences for Gaussian random functions (Cameron-Martin space)

Preliminaries Consider the Hilbert space $H :={L^2_{\text{per}}(\mathbb{R})}$ of Gaussian random functions, $2\pi$-periodic in $\mathbb{R}$. These random functions are drawn from a Gaussian measure $\...
ares's user avatar
  • 101
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 ...
xFioraMstr18's user avatar
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<\...
Seung Hyeon Yu's user avatar
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: ...
0xbadf00d's user avatar
  • 167
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) &...
Henning's user avatar
  • 123
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*} ...
sharpe's user avatar
  • 721
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}...
esner1994's user avatar
1 vote
0 answers
169 views

A question about Stroock's notes on the Weyl lemma

On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
5th decile's user avatar
  • 1,461
0 votes
0 answers
145 views

“Chapman-Kolmogorov”-convolution vs. smoothness

Let $K:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a so-called "integral-kernel": we certainly require $K(x,.)$ and $K(.,y)$ to be Lebesgue measurable for almost all $x,y \in \mathbb{R}^n$. An ...
5th decile's user avatar
  • 1,461
9 votes
1 answer
652 views

Scaling in Mehta's integral

The following expression is known as Mehta's integral and deeply connected to random matrix theory: $$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
Pritam Bemis's user avatar
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 ...
Sascha's user avatar
  • 536
2 votes
0 answers
173 views

Weak convergence of $\mathcal{L}^2$ valued random variables

Consider two continuous functions $f,g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $f(x,\cdot), g(x,\cdot) \in \mathcal{L}^2(\mathbb{R},\mathcal{B},\lambda)$ for all $x \in \mathbb{R}$ and a sequence ...
esner1994's user avatar
4 votes
1 answer
623 views

Relation between Gaussian processes and RKHSs with tensor product kernels

For sets $\cal X$ and $\cal Y$, let $a:{\cal X}\times{\cal X}\rightarrow \mathbb{R}$ and $a:{\cal Y}\times{\cal Y}\rightarrow \mathbb{R}$ be positive definite symmetric kernels. Define the tensor ...
Wicher's user avatar
  • 63
0 votes
1 answer
123 views

"Geometric" Decomposition of Wiener Space

Let $C_0([0,1];\mathbb{R}^d)$ be the classical Wiener space (of continuous paths with initial value $0$) and let $\nu$ be the Wiener measure on this space. Does there exist a countable family $\left\{...
ABIM's user avatar
  • 5,405
0 votes
1 answer
80 views

A question about positive operator pregenerator [closed]

Thank you for reading. My question was raised up when I tried to prove an example in the book of Liggett(1985), which is in P13 Example 2.3(a). Here is a link of the page: https://books.google.com/...
Chennes's user avatar
  • 385
-1 votes
1 answer
122 views

Approximation of function in general measure space

Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with $$ \int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
Wenguang Zhao's user avatar
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 ...
YT_learning_math's user avatar
1 vote
0 answers
67 views

Angle between Fleming-Viot type 3-particle system

Consider $(X^1,X^2,X^3)\in (0,\infty)^3$ with each particle starting at $1$ and moving independently according to Brownian motion until random time $\tau_1:=\min \lbrace t>0: X_{t-}^1\wedge X_{t-}^...
maliesen's user avatar
  • 284
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\...
user78370's user avatar
  • 891
0 votes
0 answers
153 views

Equivalent Definitions of Gaussian Process?

The Gaussian process $\{X_t\}_{t \in T}$ ($T=[0,1]$ for example) is usually defined using its finite-dimensional distribution. I came across this statement many times: linear operator (not necessarily ...
jwyao's user avatar
  • 103
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 ...
fast_and_fourier's user avatar
1 vote
0 answers
127 views

Gradient bound for the Markov semigroup generated by the solution to an Langevin SDE

Let $h\in C^2(\mathbb R)$ with $$h''\ge\rho\tag1$$ for some $\rho>0$ and $$\int\underbrace{e^{-h}}_{=:\:\varrho}\:{\rm d}\lambda=1$$ $\mu$ be the measure with density $\varrho$ with respect to the ...
0xbadf00d's user avatar
  • 167
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 ...
Cal's user avatar
  • 59
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 ...
Abani Sarma's user avatar
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 $...
Viktor B's user avatar
  • 724
5 votes
0 answers
242 views

Spectral gap for the Brownian motion with drift on a compact manifold

Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...
Pierre PC's user avatar
  • 3,669
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 ...
user2379888's user avatar
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 ...
Creator's user avatar
  • 495
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{...
Kira G.'s user avatar
  • 161
4 votes
0 answers
322 views

Compactness of semigroups of one-dimensional diffusions

I have a question about semigroups of one-dimensional diffusions. Let $X$ be the Ornstein-Uhlenbeck process on $\mathbb{R}$. The generator is expresses as $$\frac{d^2}{dx^2}-x\frac{d}{dx}.$$ It is ...
sharpe's user avatar
  • 721
1 vote
0 answers
134 views

Operator-valued stochastic integral and quadratic variation for operator-valued processes

Let $U$ be a separable $\mathbb R$-Hilbert space and $W$ be a $Q$-Wiener process on a complete and right-continuous filtered probability space. Let $H$ be a separable $\mathbb R$-Hilbert space and $X$ ...
0xbadf00d's user avatar
  • 167
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 ...
Xiao's user avatar
  • 485
6 votes
1 answer
898 views

Injectivity of a Fredholm operator

While doing my study on the boundary-crossing time of a stochastic process, I happened to deal with the following question which is somehow related to Fredholm theory. Question : Suppose $K$ is ...
Taro Tokyo's user avatar
0 votes
0 answers
90 views

criterions for polar set of Feller processes

Suppose $X_t$ is the solution to $$ d X_t=b(X_t)dt+dL_t,\quad X_0=x. $$ where $L$ is a rotational symmetric $\alpha-$stable process with $\alpha\in (0,1]$, $b$ is Lipchitz. Assume $\Gamma\subseteq ...
Guohuan Zhao's user avatar
2 votes
0 answers
169 views

Stochastic Approximation in Reproducing Kernel Hilbert Space

Consider an iterative algorithm with incremental updates \begin{align} x_{t+1} = x_t + \alpha_t \cdot [ h(x_t) + M_{t+1}], \end{align} where $\{x_t \}_{t \geq 0}$ is in a reproducing kernel Hilbert ...
Steve's user avatar
  • 1,127
2 votes
0 answers
169 views

How can we show that a $Q$-Wiener process on a Hilbert space $U$ takes values in $Q^{1/2}U$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$ $U$ be an infinite-...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
235 views

Associative law of the stochastic integral in Hilbert spaces

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $T>0$ $I:=(0,T]$ $(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$ ...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
100 views

Convergence and boundedness in $L^\infty([0,T]\times \Omega)$ of Karhunen-Loeve expansion

Let $X:[0,T]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process in $L^2([0,T]\times\Omega)$. Consider the Karhunen-Loeve expansion of $X$: $$ X(t,\omega)=\mu_X(t)+\sum_{n=1}^\infty \sqrt{\nu_n}\...
user39756's user avatar
  • 141
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. ...
Abdelmalek Abdesselam's user avatar
1 vote
1 answer
175 views

Stochastic operator on $\ell^1$ has dense range

Let $P:\ell^1(\mathbb{Z}^d) \rightarrow \ell^1(\mathbb{Z}^d)$ be given by $$(Pz)(x)=\sum_{y \tilde \ x} \frac{1}{2d} z(y)$$ where the tilde indicates that $y$ is a neighboured vertex of $x.$ I ...
BaoLing's user avatar
  • 329
1 vote
0 answers
159 views

Construction of the quadratic variation process in infinite dimensions

Let $H$ be a separable $\mathbb R$-Hilbert space $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a ...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
113 views

Outer product $\sum_i |k_{x_{i}}(\cdot)\rangle\langle k_{x_{i}}(\cdot)|$ of reproducing kernel functions as identity operator in RKHS?

In a separable Hilbert space $\mathcal{H}$, given a complete orthonormal basis $\{|e_i\rangle\}$, the identity operator can be written as $\mathbb{1} = \sum_i |e_i\rangle\langle e_i|$. Now if this ...
foo_bar's user avatar
  • 11
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 ...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
87 views

Linear evolution equation $u'(t)=A(t,\omega)u(t)$ with time-dependent random operator

I have had some previous knowledge on evolution equations in a Banach space of the form $$u'(t)=Au(t),$$ where $A$ generates some strongly continuous operator semigroup. Now I am looking at a problem ...
Chuwei Zhang's user avatar
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*...
diadochos's user avatar
  • 163
6 votes
1 answer
386 views

Reference Request: Vector-Valued Ito Formula

I know that there exist Ito formulae to understand $ f(X), $ where $f: H\rightarrow \mathbb{R}$ is sufficiently nice, $H$ is a Hilbert space and $X$ is an $H$-valued semi-martingale. However I'm ...
ABIM's user avatar
  • 5,405
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 ...
0xbadf00d's user avatar
  • 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 ...
ABIM's user avatar
  • 5,405
4 votes
0 answers
414 views

Definition of the Stratonovich integral in Hilbert spaces

Let $T>0$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$ $B$ be a (standard, real-...
0xbadf00d's user avatar
  • 167
1 vote
1 answer
223 views

Stochastic integral is a continous or closed operator?

The Setup Let $\xi_t$ be a process adapted to the filtration $\mathfrak{F_t}$ of the semi-martinagale $X_t$, such that both are square integrable. Then is the map \begin{align} F_T: L^2(\mathfrak{...
ABIM's user avatar
  • 5,405