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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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### 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 ...
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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 $... 1answer 152 views ### Modification of a Markov process on the real line Consider a real-valued Markov process$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 &... 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 \... 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_{... 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 ... 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. ... 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:... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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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