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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 ...
user avatar
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
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5 votes
0 answers
178 views

Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mathscr{P}$ be a probability ...
Goulifet's user avatar
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4 votes
0 answers
330 views

Book recommendation in functional analysis and probability

I am interested by functional analysis and probability. I would like to know if you have any books that deal with these two subjects (at a graduate level) to recommend? I'm looking for a book that has ...
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
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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
4 votes
0 answers
282 views

Markov operators and existence of ergodic measures

My question refers to the yesterday's question (see here) of John Learner and goes as follows: Can we deduce the existence of an ergodic measure if we know that an invariant measure exists, but the ...
Almost sure's user avatar
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 ...
Dustin G. Mixon's user avatar
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 ...
Jeremy Voltz's user avatar
3 votes
0 answers
91 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
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
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
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 $(...
mathex's user avatar
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2 votes
0 answers
75 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 ...
Jorkug's user avatar
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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 ...
Uriel Herrera's user avatar
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 ...
RiezFrechetKolmogorov's user avatar
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
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
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
149 views

Question about continuity in the "complete Skorohod Topology"?

I am reading the book in progress of Timo Seppäläinen about the "Translation Invariant Exclusion Process" https://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf In one of the exercises, exercise 8.9 ...
Mario Antonio Ayala Valenzuela's user avatar
2 votes
0 answers
166 views

Must rows of a transition matrix be distinct?

Is it true that for all continuous time Markov processes on a countable state space $S$, we have all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ? This ...
Fantastic's user avatar
  • 165
2 votes
0 answers
619 views

Laplace transform of a integral function of CIR/CEV process

The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...
KNN's user avatar
  • 323
2 votes
0 answers
188 views

Equivalence of two non-degenerate Gaussian measures on Banach space

The motivation of this question is to show that two probabilities on $C_{0}^{n}(0,1)$ (the space of continuous $\mathbb R^{n}$ valued process on $[0,1]$ starting from zero) induced by two non-...
kenneth's user avatar
  • 1,399
1 vote
0 answers
70 views

On calculating the second quantization operator $\Gamma(A)$ of the Ornstein-Uhlenbeck operator $A$

Let $A$ be a self-adjoint operator on a Hilbert space , and let $d\Gamma(A)$ be the generator of the second quantization of $A$. Consider the following theorem from Segal's "Non-Linear Quantum ...
matilda's user avatar
  • 90
1 vote
0 answers
177 views

A question on Gaussian small ball probability

Consider the random variable $$ G = \sum_{j=1}^{\infty} \lambda_j Z_j^2 $$ where $Z_j \sim_{\substack{i.i.d}} N(0,1)$ and $\lambda_j$ some non increasing sequence of positive numbers with $\sum_{j=1}^{...
Exc's user avatar
  • 119
1 vote
0 answers
306 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 ...
Chaos's user avatar
  • 515
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
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
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
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
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
1 vote
0 answers
417 views

Defining density of a random function using Radon-Nikodym Theorem

Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$. Let $X$ be random function defined ...
Janak's user avatar
  • 213
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
0 votes
0 answers
81 views

Measurable Extension

Let $(\Omega, \mathcal{F})$ be a measurable space and $X$ some metric space (probably Polish) with the Borel $\sigma$-algebra and a function $f: \Omega \times X \to \mathbb{R}$. Usually, functions ...
Mrcrg's user avatar
  • 136
0 votes
0 answers
78 views

Different measurability of Hilbert-space valued random variable

My question is motivated by this link. Let $(\Omega,\mathcal{F})$ and $(Y,\mathcal{B})$ be measurable spaces, a measurable map $T:\Omega\to Y$ is called a $Y$-valued random variable. Now let $H$ be a ...
John's user avatar
  • 503
0 votes
0 answers
46 views

Independence of variables predicted by the generator

Let $X$ be en compact metric set, and denote by $\mathcal{C}(X)$ the set of real continuous functions defined on $X$, endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega$ be the generator ...
G. Panel's user avatar
  • 449
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
0 votes
0 answers
252 views

Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
Obriareos's user avatar
  • 195
0 votes
0 answers
322 views

Comparison of Parameter estimation using maximum likelihood and Maximum entropy

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
Creator's user avatar
  • 495