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]
94 questions
8
votes
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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
- 6,932
7
votes
1 answer
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
- 519
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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
- 837
7
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1 answer
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
6
votes
1 answer
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
- 28,141
6
votes
1 answer
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
- 1,600
4
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1 answer
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
- 14,967
4
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Is the Wiener measure locally doubling?
I read here that the Lebesgue differentiation theorem may be generalized to Borel measures which are locally doubling, i.e., where the measure of a Ball is bounded my some constant (possibly depending on the location of the center of the ball) times…
4
votes
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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…
4
votes
1 answer
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,
$$\require{cancel} \xcancel{\xi_{t+s} - \xi_t}\ B_{t+s} - B_t\sim \mathcal{N}(0, s)…
MrArsGravis
- 345
4
votes
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Topology on $C(\mathbf{R}_+,\mathbf{R})$ to get Wiener measure
Below is an extract from Le Gall's Brownian Motion, Martingales, and Stochastic Calculus, p27. I am having trouble seeing why "$\mathscr{C}$ coincides with the Borel $\sigma$-field on $C(\mathbf{R}_+,\mathbf{R})$ associated with the topology of…
Raphaël
- 169
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Is there a relation between cylinder set measures and discretization of path integrals?
Path integral via discretization
So let me start with what seems to be the point of view of physicists (corrections are highly appreciated since this is what I understood!). Let a quantum system with coordinates $q_a$ and momenta $p_b$ be given…
Gold
- 28,141
3
votes
1 answer
Feynman–Kac formula: conditional expectation vs. Wiener integral
The Feynman–Kac formula for the solution $u(t,x)$ of the one-dimensional heat equation
\begin{align*}
\partial_t u &= \frac{1}{2}\Delta_x u,\\
u(0,x) &= f(x)
\end{align*}
is given…
Emily
- 1,331
3
votes
1 answer
Deriving trapezoid rule from conditional expectation of Brownian motion
I have read here and in P. Diaconis' paper Bayesian Numerical Analysis that, in particular,
$$\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
- 1,292
3
votes
0 answers
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.
- 35,927