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      Questions tagged [stochastic-processes]

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

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      8
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      2answers
      579 views

      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
      2answers
      783 views

      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 $...
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      votes
      1answer
      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
      2answers
      1k views

      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
      1answer
      410 views

      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
      1answer
      439 views

      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
      2answers
      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
      0answers
      466 views

      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
      1answer
      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
      1answer
      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
      3answers
      527 views

      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
      2answers
      2k views

      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
      1answer
      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
      4answers
      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
      4answers
      15k views

      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=...

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