The Wiener measure is the probability law on the space of continuous functions $g$ with $g(0)=0$, induced by the Wiener process.
Questions tagged [wiener-measure]
85 questions
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Has the Hamiltonian Path Integral Been Made Rigorous?
It is well known that the Lagrangian formulation of the path integral has been made rigorous, via the Wiener measure and/or the Trottier product formula. I haven't seen mathematicians discuss the hamiltonian version though, where one integrates over…
JLA
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Ito Isometry against non-Brownian SDE
Suppose $X_t$ is a Semi-martingale and $H_t$ is $X_t$-predictable.
I know that if $X_t=W_t$ is a Wiener process then
$$
\mathbb{E}[H\cdot W_T^2] = \mathbb{E}\bigg[\int_0^TH_t^2dt\bigg],
$$
where $H\cdot W_T$ denotes the stochastic integral of $H_t$…
user355356
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Embedding $L^p \subset L^q$ compact? And relation to abstract Wiener spaces
I am currently reading Hui Hsiung Kuo's book "Gaussian Measures in Banach Spaces" and there is an exercise (Exercise 21, p. 86) in which you are asked to show that for $1 \leq p < 2$, $(i, L^{2}[0,1], L^{p}[0,1])$ is not an abstract Wiener space (I…
Andre
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Can a Wiener process be obtained as the limit of a "memoryless collision time" model?
Let $(N_t)_{t \geq 0}$ be a Poisson process of intensity $1$, and for each $\lambda>0$ and $t \geq 0$ let
$$ W^{(\lambda)}_t = \sqrt{\lambda} \int_0^t (-1)^{N_{\lambda s}} \, ds = \frac{1}{\sqrt{\lambda}} \int_0^{\lambda t} (-1)^{N_s} \, ds. $$
Is…
Julian Newman
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Approximation of Poisson by Wiener
Recently I learned that it is a widespread idea in applied math to approximate high rate Poisson processes by a Wiener process. I.e. take $N$ to be a homogeneous Poisson with rate $\lambda$, then for a large enough $\lambda$ one can select some time…
demitau
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Functional root of Wiener process
Does there exist a stochastic process $X$ such that when two such trajectories are sampled, their composition is Wiener-distributed? It would be natural to call such a process a functional square root of Wiener process, and take a look at some…
Alexey Slizkov
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Obtaining Classical Wiener Space from abstract Wiener measure
The question
I'm working on understanding the Abstract Wiener Space construction and wanted to rederive the defining property of the classical counterpart,
$$\require{cancel} \xcancel{\xi_{t+s} - \xi_t}\ B_{t+s} - B_t\sim \mathcal{N}(0, s)…
MrArsGravis
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Is there a relation between cylinder set measures and discretization of path integrals?
Path integral via discretization
So let me start with what seems to be the point of view of physicists (corrections are highly appreciated since this is what I understood!). Let a quantum system with coordinates $q_a$ and momenta $p_b$ be given…
Gold
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Schwartz space, Gaussian measures and integration over paths
I'm studying the Wiener measure motivated by the path integral in quantum mechanics. For that I'm using the book by Glimm & Jaffe "Quantum Physics: a Functional Integral Point of View" that deals with it from that perspective.
Now, I'm having a…
Gold
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3
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How to show the following definition gives Wiener measure
On the first page of Ustunel's lecture notes, he defines the Wiener measure in the following way:
Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by $\mathcal{B}_t = \sigma\{W_s; s\leq t\}$, then there…
Petite Etincelle
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Feynman–Kac formula: conditional expectation vs. Wiener integral
The Feynman–Kac formula for the solution $u(t,x)$ of the one-dimensional heat equation
\begin{align*}
\partial_t u &= \frac{1}{2}\Delta_x u,\\
u(0,x) &= f(x)
\end{align*}
is given…
3
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Deriving trapezoid rule from conditional expectation of Brownian motion
I have read here and in P. Diaconis' paper Bayesian Numerical Analysis that, in particular,
$$\mathbb{E}\left(\int_0^1 B_t dt | B_{t_0}, B_{t_1}, \dotsc, B_{t_{n-1}}, B_{t_n}\right)$$
yields the trapezoid rule for approximating the integral…
Nap D. Lover
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The convergence of the disintegrations of a sequence of measures
Let $C = \{c : [0,1] \to \mathbb R ^n \mid c \text{ is continuous and } c(0)=0 \}$ be endowed with the Wiener measure $P$. Consider an exhaustion $\mathbb R^n = \bigcup _{k \ge 0} U_k$ where each $U_k \ni 0$ is a connected open subset with smooth…
Alex M.
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Topology on $C(\mathbf{R}_+,\mathbf{R})$ to get Wiener measure
Below is an extract from Le Gall's Brownian Motion, Martingales, and Stochastic Calculus, p27. I am having trouble seeing why "$\mathscr{C}$ coincides with the Borel $\sigma$-field on $C(\mathbf{R}_+,\mathbf{R})$ associated with the topology of…
Raphaël
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Construction of the probability space of the Wiener process
I am reading a number of papers by different authors which are introductions to stochastic differential equations. All of these papers define the Wiener process $W_t$ (Brownian motion) quite simply by a few properties such as
$W_0=0$ with…
Lars Ericson
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