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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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      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 ...
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      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 ...
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      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 ...
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      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?
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      votes
      1answer
      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 ...
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      1answer
      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\}...
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      votes
      1answer
      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 ...
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      1answer
      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 = \...
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      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^{-...
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      vote
      1answer
      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 ...
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      vote
      1answer
      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 \...
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      1answer
      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 ...
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      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=...
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      1answer
      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 $\{\Phi(t,x,\sigma)\}_{\sigma \in [0,1]}$ ...
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      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 ...

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