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

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Let $G, \partial G$ be the interior and the boundary of simple polygon respectievly. For $p \in \partial G$ let $$ D_p=\left\{ d \in S^1: \exists\varepsilon > 0 \text{ s.t. } p+\delta d \in G, 0 &...
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I am looking for related constructions in spectral theory / random matrix theory / dynamical systems where a triadic (three-vertex) geometric criterion acts as a sharp switch for an order–chaos ...
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Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,\infty)},\mathbb{P})$ be a filtered probability space, and let $\tau \colon \Omega \to [0,\infty]$ be an $(\mathcal{F}_t)$-stopping time. We will say ...
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Say I have an SDE $$dx_t = b(x_t)dt + dB_t$$ where $B_t$ is a Brownian motion. I would like to compute the most probable path between two points $y_0$ and $y_1$. I know the Onsager–Machlup functional ...
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Suppose $X \in \mathbb R^d$ is a Markov process with cadlag paths and a stationary distribution $\mu$. Let $\tau_n : =\inf\{t\ge 0: |X(t) | \ge n\}$ and assume that $X$ is non-explosive and hence $\...
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In spatial statistics, the Matérn Gaussian field on $\mathbb{R}^d$ is often defined as the (weak) solution to the SPDE $$ (\kappa^2 - \Delta)^{\alpha/2} u \;=\; W, $$ where $W$ is Gaussian spatial ...
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Let $d_1,d_2\in \mathbb{N}_+$, a stopping time $\tau$, and consider a system of BSDEs \begin{align} Y_{\tau}^1 & = \xi^1+\int_{t\wedge \tau}^{\tau}\, f_1(t,Y_t^1,Z_t^1,Y_t^2,Z_t^2)dt - \int_{t\...
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The separability that is used in the context of stochastic processes is typically already defined specifically for stochastic processes. I will define this for deterministic functions instead such ...
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Let $ \mathcal{F} = \{ f_\theta : \theta \in \Theta \} $ be a class of functions indexed by $ \theta $. If $ \mathcal{F} $ is a Donsker class, then the empirical process $ \mathbb{G}_n(f_\theta) = \...
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Let $S_n$ be a symmetric simple random walk starting at $0$. Numerically, I verified that for any $j\geq -2$ such that $n+j$ is even, we have $$\Phi \left(\frac{j}{\sqrt{n}}\right)\leq \mathbb{P}(S_n\...
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Define an operator $B_\theta$ by $B_{\theta}f(x) = \theta\int_{y} (f(y)-f(x))q_{\theta}(x, dy)$ where $q_\theta(x, dy)$ is the law of a normal $\mathcal{N}(x, \frac{1}{\theta})$ r.v., so that $B_\...
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Consider a filtered probability space in which there exist two independent Brownian motions $W$ and $B$. For every progressively measurable process $u=(u_t)_{t\ge 0}$ taking values in $[0,1]$, define ...
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Background and motivation Let $(\Omega, \pi)$ be a finite state space with a stationary distribution $\pi$. Consider an ergodic Markov chain on $\Omega$ with transition matrix $P$ that is irreducible ...
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In a two-dimensional Minkowski spacetime patch with light-cone coordinates $(U,V)\in(0,1)^2$, consider the timelike foliation defined by $$ V(U)=e^{s/\ln U},\qquad s>0 $$ Randomizing the global ...
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Consider a branching random walk given by the collection of i.i.d random variables $X(i_{0},\ldots,i_{t})$. Here $t \in \mathbb{N}$ and $i_{k} \in \{1,\ldots,n\}$ for any $k \in \mathbb{N}$. Each $X(...
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Consider the path of a 3d Brownian motion. Is its complement homeomorphic to some fixed $3$-manifold with probability one? If not what can we say about the complement? What about 4d?
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Let $(x_s)_{s\in[0,T]}$ be a stochastic process such that, for every $s\in[0,T]$, $$ x_s \in \mathbb{D}^{1,2}(\mathbb{R}), $$ and assume $$ \mathbb{E}\left[\int_0^T |x_s|^2\,ds\right] < \infty, \...
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Let $ E $ be the Banach space of continuous functions on $[0, T]$, equipped with the supremum norm. Let $ \sigma : E \to \mathbb{R} $ be a function of class $ C^1 $ (Fréchet differentiable). Let $ X = ...
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Let $W=(W_t)$ be a real-valued Brownian motion on some filtered probaiblity space. Let $H=(H_t)$ be a progressively measurable process taking values in some Hilbert space $\mathcal H$, endowed with ...
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I wonder if a Localized Yamada–Watanabe theorem up to a stopping time exists. Here is more details: Let $(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb P)$ be a filtered probability space ...
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Let $f$ be some discrete series where $f[m]=0\space \forall m \geq 0$, and $S_{XX}^- (z)$, be the anti-causal part of the spectrum of a regular process (a process whose spectrum can be written as $\...
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I am going through some basic stochastic process weak convergence theory and really need your help. Thanks! The question might be very naive. Suppose $B(t)$ is the standard Brownian Motion. If $f(x)$ ...
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Let $\mathscr G⊆\mathscr F$ be a sub-$\sigma$-algebra, $X\in \mathbb{L}^0(\mathscr G,\mathbb R^d)$, $\varphi : \mathbb R^d×\Omega→\mathbb R$ be bounded and $B(\mathbb R^d)×F$ -measurable. Assume for ...
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Background: An $\mathbb{R}^d$-valued process $Y_{\cdot}=(Y_t)_{t \ge 0}$ is called a $\sigma$-martingale if there exists some $\mathbb{R}^d$-valued martingale $M_{\cdot}$ and a $\mathcal{F}_{\cdot}^M :...
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So here's a problem that has tormented me for years: You have $N$ rocks, and need to distribute them into $K$ piles, potentially some of them empty. After the initial distribution into piles of sizes ...
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We are interested in the piecewise linear approximation of the $\Phi^4_d,d=2,3$ model, interpreted in the mild sense: $(\partial-\Delta)\phi=\phi^3+\xi.$ for this kind of approximation, how to define ...
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Note: This question is motivated by some work on tug-of-war games in connection with the infinity Laplacian. If further details on the background of the problem are desired, please let me know and I ...
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I am interested in any knowledge on this question: The process starts with a single particle at time $0$, then after a random time of distribution $G$ which has a density on $(0, \infty)$ it gives ...
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This question concerns lattice & piecewise linear approximation (version of Wong-zakai theorem). Regularity structures allowed to tackle these topics for several SPDE, for example: https://arxiv....
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Let $n,m\in \mathbb{N}_+$, $f:\mathbb{R}^n\to \mathbb{R}^m$ be a continuous function, and $X_{\cdot}=(X_t)_{t\ge 0} := M_{\cdot}+A_{\cdot}$ be a semimartingale. Is there a characterization of when $...
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I'm trying to understand some things about this theorem which comes from the triangle inequality for the transportation metric $\rho_K$: Suppose the state space $\mathcal{X}$ of a Markov chain is the ...
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Let $X_{n+1} = U_n X_n + V_n X_{n-1}$ where $X_0 = X_1 =1$ and $(U_n), (V_n)$ are sequences of random variables with $E[U_n] = E[V_n] = 1$, both independent, identically distributed and independent ...
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Kolmogorov's continuity theorem states that if for a $\mathbb{R}$--valued process $(X_t)_{t\geq 0}$ we have $$\mathbb{E}\left[\vert X_t-X_s\vert^\alpha \right] \leq K \vert t-s\vert^{1+\beta},$$ then ...
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Let $\{W_t\}_{t\in [0,1]}$ be a Wiener process (Brownian motion), with initial condition $W_0=0$. And let $(\Omega,\mu)$ be the probability space on which this stochastic process is defined, so that $...
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Suppose I have a random measure $\mu$ on $\mathbb{R}$. I am looking for a criterion to check if $\mu$ has a density with respect to the Lebesgue measure. (Using Radon-Nikodym “directly” doesn’t seem ...
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Let $(X_t)_{t \in [0,2\pi]}$ be a continuous Ito martingale. Let $ t_1, \ldots, t_N, t_{N+1} $ be a discretization grid on $[0,2\pi]$. Let s $\in [0,2\pi]$. I would like to show the following ...
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Let $Y$ be a continuous stochastic process on $[0,T]$ with a complete filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ satisfying the usual condition. Let $\tau$ be a ...
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I briefly recall the definition of Functionnal Ito Calculus, developped by Bruno Dupire in https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1435551. The paths we consider are càdlàg (right ...
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I have been trying to prove the following without success. Let $f$ and $g$ be two IFR ($\log$-concave) PDF's which intersect at one and only one point, with $f>g$ for left half of real axis, and $...
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$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bE}{\mathbb{E}} \newcommand{\coloneq}{:=} \newcommand{\colon}{:} \newcommand{\rD}{D} \newcommand{\...
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In this article Koehler and Mossel discuss a spin system with spin values from symmetric group $S_q$ for some $q$. They define the Hamiltonian as $$ H(\sigma)=\sum_{i\sim j}d_\tau(\sigma_i,\sigma_j) $$...
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In my professor’s lecture notes, a Markov process (referred to as Definition 1.3) is defined as follows: Let $(X_t)_{t \ge 0}$ be an $\mathcal{F}_t$-adapted process with values in a measurable space $...
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Let $(W_t)_{t\ge 0}$ be a one-dimensional standard Brownian motion on its natural filtration $(\mathcal{F}_t)$. For fixed constants $$0 < \underline{\sigma} < \overline{\sigma} < \infty,$$ ...
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$$Ee^{\frac{\lambda}{2}\int_{0}^{1}B_{s}^{2} ds} $$ Bt is a brownian motion, λ is a postive number, my question whether this integral would diverge when λ≥1. I also want to know could we find the ...
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Let $(\mathcal{X},d)$ be a metric space and let $\operatorname{Homeo}(\mathcal{X})$ denote the set of compactly supported homeomorphisms between $\mathcal{X}$ and itself, i.e. which means that these ...
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Let $(B_t)_{t\ge 0}$ be standard Brownian motion in $\mathbb R^{2}$ started at the origin. Fix a rate $\lambda>0$ and let $(N_t)_{t\ge 0}$ be an independent Poisson process with jump times $$0=\...
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My coauthor and I are currently writing a paper on Gabor frames, and we are trying to make our main result more appealing, basically instead of saying that the function has to satisfy some crazy ...
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Let $\mathcal{B}$ be critical binary branching Brownian motion on $\mathbb{R}$ started from a single particle at the origin. Fix $k \ge 1$ and a constant $\lambda > 0$. Condition on the atypical ...
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Suppose $(X_n)_{n \in \mathbb{Z}}$ follows a stationary ergodic distribution, where $X_0 \in \{-1,1\}$ has mean $0$. We know that $S_n/n \to 0$ where $S_n = \sum_{i=1}^nX_n$. Can we rule out the ...
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Let $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ a probability space. Let $(M_t)_{t \le T}$ be a true martingale (for its natural filtration for example), what are the most general sufficient ...
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