# Questions tagged [stochastic-processes]

A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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### Bounding Brownian motion and an Ito process simultaneously

Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...
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### Log Sobolev inequality for Wiener space

I am reading https://arxiv.org/pdf/1003.1649.pdf and saw equation 10.2.3 that said that on Wiener space $$E\left[f^2\log\left[\frac{f^2}{E[f^2]}\right]\right]\leq 2 E[|\nabla f|_H^2],$$ where $\nabla$ ...
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### Bound moments wrt. known initial and final moments

Let $X$ be an $L^p$ random variable, where $p\in (0,1)$ and $W_t$ usual Brownian motion (with $W_t$ independent from $X$). I'd like to bound $$\mathbb E|X+W_t|^p$$ purely in terms of $\mathbb E|X|^p$ ...
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### 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 ...
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### Sticking martingales together

Let $(\Omega, \mathcal F, P)$ be a probability space with sigma algebras $\mathcal F_{n, t}$ for $n \in \mathbb N$, $t \in [0, 1]$, where for all $n$, $\mathcal F_{n, t} \subset \mathcal F_{n, h}\$ ...
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### Probability to cross an envelopp for 1D random walk?

Imagine we have an evolving sequence composed of 1 and -1 (ex: -1-11-111...) where the probability to get -1 or 1 is 1/2. n is the lengh of my sequence. I can make an analogy with random walk: let ...
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### Perron-Frobenius and Markov chains on countable state space

The following question naturally arises in the theory of Markov chains with countable state space to which I would be curious to know the answer: Let $A:\ell^1 \rightarrow \ell^1$ be a contraction, i....
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### Is a linear combination of Markov generator a Markov generator?

Let $X$ be a compact metric set and $\mathcal{C}(X)$ be the set on continuous real functions over $X$ endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega_1$ and $\Omega_2$ be two Markov ...
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### Limit of a linear discrete-time stochastic process with uniform noise

I have posted this in the math and stats sites, but I am not sure where the proper forum for this question is. If it is not here, please go on and delete it. Suppose we have a stochastic linear ...
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### Continuous-time random walk on $\mathbb{R}$ that never stays still

Consider a walker on the real line $\mathbb{R}$ and two probability density functions $w$ and $j$ defined over $\mathbb{R}$. A walker starts at $0$ and iterates the following: it samples a waiting ...
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### Translation of Dellacherie's Capacités et Processus Stochastiques

I have been studying the Strasbourg school's general theory of processes from Dellacherie and Meyer's Probabilities and Potential, and I really like it. I have heard very good reviews about another ...
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### Infinite-dimensional Gaussian measure vs finite-dimensional Wiener measure

I'm trying to figure out the connections between two contructions of Gaussian measure. Let $(U, \langle\cdot,\cdot\rangle_U)$ be a seprable Hilbert space, and $\mathcal{B}(U)$ be the Borel sigma-...
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### Continuity of the random variable defining the occupation measure of a continuous Gaussian process

Suppose $Z:\Omega \times [0,1] \to \mathbb{R}$ is a continuous Gaussian process with mean $\mu(t)$ and covariance kernel $C(t,s)$. Consider the random variable  X_\alpha = \lambda( \{t \; : \; Z(t) &...
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 ...