# Questions tagged [stochastic-processes]

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

**3**

votes

**0**answers

87 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 ...

**0**

votes

**0**answers

45 views

### Mean first-passage time for a marked Poisson process

Given a marked Poisson process in one dimension
$$
Y(t)=\sum_{\{t_i,a_i\}}g(t-t_i,a_i)
$$
so that $Y(t)$ is a sum of impulses arriving as a Poisson process and the impulses $g$ belong to a family of ...

**0**

votes

**0**answers

36 views

### Stochastic process for minimize a mean

I have the next problem: consider an inventory process $\{X_{k}\}$ such that $X_{k+1}=X_k+A_k-\xi$, $X_0=25$, where $A_k$ is the number of items of items produced at the $k$th-month and $\xi$ is the ...

**1**

vote

**0**answers

104 views

### “Brownian motion” related to the $p$-Laplace operator

The link between the Brownian motion and the Laplace operator is well-known.
What stochastic process plays an analogous role with respect to the $p$-Laplace operator?

**4**

votes

**1**answer

263 views

### How are the fields of dynamical systems, stochastic processes and additive combinatorics, inter-related?

Currently I’m interested in a couple of fields, namely dynamical systems, stochastic processes, and additive combinatorics. I was wondering if it’s feasible to keep pursuing all 3, and whether I can ...

**0**

votes

**1**answer

108 views

### Continuity of harmonic functions

I have a question about harmonic functions with respect to symmetric Markov processes.
Let $E$ be a locally compact separable metric space and $\mu$ a positive Radon measure on $E$.
Let $X=(\{X_t\}...

**2**

votes

**1**answer

324 views

### Monotonicity, Convexity, and Smoothness of the KL-Divergence between Two Brownian Motions with Different Initializers

We consider the two distributions
$$
p_t = p_0 * N(0, tI),\quad q_t = q_0 * N(0, t I),
$$
where $*$ denotes the convolution between two densities, while $p_0$ and $q_0$ have the same mean and ...

**2**

votes

**1**answer

61 views

### Intensity and compensator for a jump process

Set-up and assumptions. Let $(\mathscr{F}_t, t \geq 0)$ be a right-continuous complete filtration. Let $(X_t, t\geq 0 )$ be a pure jump $\mathbb{R}$-valued process with unit jumps, that is,
$$
X_t = \...

**0**

votes

**0**answers

31 views

### Show that the transition semigroup of the strong solution to a Langevin-type SDE is immediately differentiable

Let
$\varrho\in C^1(\mathbb R)$ with $\varrho>0$
$\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$
$\mu$ denote the measure with density $\varrho$ with respect to $\lambda$
$b:=2^{-...

**1**

vote

**1**answer

79 views

### Does the following percolation model have a name?

Consider the following model for percolation in an infinite graph: each vertex has a certain region (set of vertices) associated with it, which at the beginning contains only the vertex itself, and ...

**1**

vote

**1**answer

73 views

### Expectation of random variables coincides

Let $Y_1:=(X_i)_{i \in \mathbb Z}$ be a family of random variables that are identically distributed but not necessarily independent.
We can then also define the shifted sequence $Y_2:=(X_{i+1})_{i \...

**1**

vote

**1**answer

40 views

### How can a correlation structure for a stochastic process not correspond to a Toeplitz matrix?

I was reading the following question: A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary
The following matrix is given representing the correlation ...

**0**

votes

**0**answers

72 views

### Reference for convergence to a Poisson Point Process

Edited after comment by Ofer Zeitouni
I have a sequence of discrete time stochastic processes $\big((S_n(i))_{i \geq 1}\big)_{n\geq 1}$ such that for every $n$, $i$,
\begin{equation}S_n(i)=\sum_{j=...

**0**

votes

**1**answer

58 views

### Convergence of Stochastic Flow but not Flow

Suppose that $\phi(t,x):[0,\infty)\times \mathbb{R}^d\rightarrow \mathbb{R}^d$ is a flow. Is it possible to extend $\phi$ to a family of stochastic flows $\{\Phi(t,x,\sigma)\}_{\sigma \in [0,1]}$ ...

**1**

vote

**0**answers

69 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 ...