# Questions tagged [stochastic-processes]

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

1,415
questions

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votes

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### Probabilistic Solution of the Porous Medium Equation

It is well known that the transition density for standard Brownian motion $B_t$ in $\mathbb{R}^d$ yields a solution to the global Cauchy problem for the heat equation $$u_t = \Delta u$$ with initial ...

**4**

votes

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### law of iterated logrithm

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which
$\mathcal{F}_t$ is filtration satisfying general conditions. $W_t$ is
a standard Brownian motion. By law of iterated logarithm, one has
$...

**0**

votes

**1**answer

152 views

### Modification of a Markov process on the real line

Consider a real-valued Markov process $X$ with a transition density $f(x,y)$, i.e.
$$
\mathsf P[X\in A|X_0 = x] = \int\limits_A f(x,y)\,dy.
$$
For this process I want to find
$$
u(x) = \mathsf P[X_n &...

**4**

votes

**2**answers

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### BM and interpretation of stopping time sigma algebra

Suppose $H$ and $K$ are open subsets of $\mathbb{R}^d$ containing the origin with $H\subset K$, $B_t$ a standard Brownian motion starting at the origin, $\mathcal{F}_t$ its canonical filtration, and $\...

**2**

votes

**1**answer

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### The finite-dimensional distributions of infinite-dimensional limit of finite-dimensional vectors

Suppose we have the process $\{\varepsilon_t,t\in \mathbb{N}\}$. Suppose that this the finite-dimensional distributions of this process are Gaussian, i.e. for any $t_1,...,t_n$, vector $(\varepsilon_{...

**8**

votes

**1**answer

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### Extending state space to make a process Feller

Let $X$ be a locally compact Hausdorff space, and let $Y_t$ be a continuous Markov process on $X$ with transition function $P(t, x, \Gamma) := \mathbb{P}_x (Y_t \in \Gamma)$. Let $T_t$ be the ...

**1**

vote

**2**answers

328 views

### How to determine a specific graph process is Markovian or not ?

Say, here is a min-degree graph process, which starts with G_0 = the complement of K_n. Given G_t, choose a vertex u of minimum degree in G_t u.a.r., then a vertex v not adjacent to u in G_t u.a.r. ...

**7**

votes

**0**answers

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### planar mappings that preserve elliptic measure

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction $f:...

**1**

vote

**1**answer

311 views

### A bjection between two stochastic processes

Let x(t) be a Markov process. We define the stochastic process y(t) such that :
y(t) = x(f(t))
f : T -> T
T is the parameter set of the process x(t).
If we ...

**8**

votes

**1**answer

768 views

### Is there a regular Dirichlet form with no associated Feller process?

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda (hereafter, [FOT]). In Chapter 7, where they discuss the construction of a Markov process ...

**1**

vote

**3**answers

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### convergence in distribution of stochastic gradient descent.

The stochastic gradient descent algorithm where only a noisy gradient (zero mean noise) is used to update current estimate is known to converge almost surely to the minimizer. However, if one is ...

**1**

vote

**2**answers

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### Derivative of a differentiable stationary Gaussian process

Thanks for your help in advance. I'm interested in understanding the properties of derivatives of a differentiable stationary Gaussian process. Specifically, is the derivative also a Gaussian ...

**5**

votes

**1**answer

573 views

### Ergodicity of Convoluted White Noise

I have a question regarding ergodicity in infinite dimensional spaces.
Let $\mathcal{D}$ be the space of distributions on a Schwartz space, and let $\mu$ be the white noise process which exists by ...

**39**

votes

**4**answers

3k views

### Polynomials on the Unit Circle

I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with ...

**16**

votes

**4**answers

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### Maximum of Gaussian Random Variables

Let $x_1,x_2,…,x_n$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij})_{1\leq i,j\leq n}$.
Let $m$ be the maximum of the random variables $x_{i}$
$$
m=\max\{x_i:i=...