User 0xbadf00d

40 votes

3 answers

10k views

What is "white noise" and how is it related to the Brownian motion?

asked Feb 3, 2016 at 21:05

20 votes

1 answer

6k views

(Elementary) Markov property of the Brownian motion

asked Jul 3, 2015 at 15:05

18 votes

3 answers

1k views

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

asked Jan 27, 2014 at 11:17

17 votes

0 answers

382 views

If $X_i∼fλ$, $Z∼\mathcal N(0,I_d)$ and $Y=X+\ell d^{-α}Z$ with $α<1/2$, then $\liminf_{d→∞}\text E\left[1∧\prod_{i=1}^d\frac{f(Y_i)}{f(X_i)}\right]=0$

asked Mar 8, 2019 at 12:55

15 votes

2 answers

5k views

When is the automorphism group $\text{Aut }G$ cyclic?

asked Jan 30, 2014 at 13:54

13 votes

1 answer

2k views

$5$ questions on the definition of the Gelfand triple

asked Jan 8, 2016 at 20:01

12 votes

1 answer

1k views

Why is a predictable stochastic process called predictable?

asked Jul 6, 2015 at 17:01

12 votes

4 answers

9k views

Laplacian of a radial function

asked Dec 2, 2014 at 22:30

12 votes

2 answers

2k views

If $f∈C^1$ and $\{∇f=0\}$ has Lebesgue measure $0$, then $\{f∈B\}$ has Lebesgue measure $0$ for all Borel measurable $B⊆ℝ$ with Lebesgue measure $0$

asked May 6, 2019 at 15:49

9 votes

2 answers

504 views

If $h$ is twice differentiable, then $|h|$ is twice differentiable except on a countable set

asked May 1, 2019 at 17:06

9 votes

3 answers

1k views

Is there a version of the Arzelà–Ascoli theorem capturing $C([0,\infty))$?

asked Apr 13, 2019 at 16:13

9 votes

2 answers

418 views

If $Y\sim\mu$ with probability $p$ and $Y\sim\kappa(X,\;\cdot\;)$ otherwise, what's the conditional distribution of $Y$ given $X$?

asked Jul 26, 2019 at 10:20

8 votes

2 answers

159 views

When is an operator $T$ on $L^1(\mu)$ of the form $(Tf)(x)=\int\mu({\rm d}y)p(x,y)f(y)$?

asked Jun 3, 2021 at 10:23

8 votes

0 answers

108 views

Show convergence of measure of a sequence of sets

asked Dec 30, 2021 at 8:06

8 votes

1 answer

308 views

If $X^{(n)},X$ are càdlàg and $X^{(n)}\to X$ in distribution, do the corresponding transition semigroups strongly converge?

asked Nov 4, 2018 at 18:01

8 votes

1 answer

792 views

Convergence of the distribution of the Langevin diffusion to its invariant measure

asked Dec 31, 2018 at 19:41

8 votes

1 answer

89 views

$A:=\left\{y\in\mathbb{R}:\mu\left(f^{-1}(\left\{y\right\})\right)>0\right\}$ is countable, if $\mu$ is a finite measure and $f$ has compact support

asked Dec 27, 2014 at 20:24

8 votes

4 answers

2k views

What did Johann Bernoulli wrong in his proof of $\ln z=\ln (-z)$?

asked May 17, 2014 at 15:19

8 votes

0 answers

198 views

Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.

asked May 6, 2016 at 19:22

7 votes

0 answers

315 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

asked Apr 16, 2016 at 18:51

7 votes

1 answer

316 views

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

asked Apr 27, 2016 at 16:55

7 votes

1 answer

7k views

Natural matrix norm of an inverse matrix

asked Feb 28, 2014 at 17:59

7 votes

2 answers

410 views

How to prove $\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}$

asked Jan 13, 2014 at 14:14

7 votes

1 answer

3k views

What is the distribution of a stochastic process?

asked May 2, 2015 at 22:15

7 votes

0 answers

164 views

Implementation of a simulation of an incompressible Newtonian fluid with uniform density

asked Dec 10, 2015 at 16:41

7 votes

1 answer

319 views

If $\mu$ has a density with respect to the Lebesgue measure, is $C_c(\mathbb R)$ dense in $L^p(\mu)$?

asked Jan 17, 2019 at 0:02

7 votes

1 answer

152 views

If $f$ is a measurable random field, then $(ω,x)↦E[f(x)\mid F](ω)$ has a measurable version $g$ and $E[f(X)\mid F]=g(X)$ for all $F$-measurable $X$

asked May 20, 2018 at 11:31

7 votes

2 answers

1k views

Show that the carré du champ operator is nonnegative

asked Feb 5, 2019 at 23:00

7 votes

1 answer

178 views

Show $\int\frac{f(u+\varepsilon v)-f(u)}\varepsilon\:{\rm d}\mu\xrightarrow{\varepsilon\to0}\int f'(u)v\:{\rm d}\mu$ for a large class of $f,u,v$

asked Aug 5, 2019 at 17:45

7 votes

1 answer

230 views

Convolution of tight measures is tight

asked Dec 25, 2020 at 8:35

Stack Exchange Network

1,803 Questions

What is "white noise" and how is it related to the Brownian motion?

(Elementary) Markov property of the Brownian motion

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

If $X_i∼fλ$, $Z∼\mathcal N(0,I_d)$ and $Y=X+\ell d^{-α}Z$ with $α<1/2$, then $\liminf_{d→∞}\text E\left[1∧\prod_{i=1}^d\frac{f(Y_i)}{f(X_i)}\right]=0$

When is the automorphism group $\text{Aut }G$ cyclic?

$5$ questions on the definition of the Gelfand triple

Why is a predictable stochastic process called predictable?

Laplacian of a radial function

If $f∈C^1$ and $\{∇f=0\}$ has Lebesgue measure $0$, then $\{f∈B\}$ has Lebesgue measure $0$ for all Borel measurable $B⊆ℝ$ with Lebesgue measure $0$

If $h$ is twice differentiable, then $|h|$ is twice differentiable except on a countable set

Is there a version of the Arzelà–Ascoli theorem capturing $C([0,\infty))$?

If $Y\sim\mu$ with probability $p$ and $Y\sim\kappa(X,\;\cdot\;)$ otherwise, what's the conditional distribution of $Y$ given $X$?

When is an operator $T$ on $L^1(\mu)$ of the form $(Tf)(x)=\int\mu({\rm d}y)p(x,y)f(y)$?

Show convergence of measure of a sequence of sets

If $X^{(n)},X$ are càdlàg and $X^{(n)}\to X$ in distribution, do the corresponding transition semigroups strongly converge?

Convergence of the distribution of the Langevin diffusion to its invariant measure

$A:=\left\{y\in\mathbb{R}:\mu\left(f^{-1}(\left\{y\right\})\right)>0\right\}$ is countable, if $\mu$ is a finite measure and $f$ has compact support

What did Johann Bernoulli wrong in his proof of $\ln z=\ln (-z)$?

Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Natural matrix norm of an inverse matrix

How to prove $\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}$

What is the distribution of a stochastic process?

Implementation of a simulation of an incompressible Newtonian fluid with uniform density

If $\mu$ has a density with respect to the Lebesgue measure, is $C_c(\mathbb R)$ dense in $L^p(\mu)$?

If $f$ is a measurable random field, then $(ω,x)↦E[f(x)\mid F](ω)$ has a measurable version $g$ and $E[f(X)\mid F]=g(X)$ for all $F$-measurable $X$

Show that the carré du champ operator is nonnegative

Show $\int\frac{f(u+\varepsilon v)-f(u)}\varepsilon\:{\rm d}\mu\xrightarrow{\varepsilon\to0}\int f'(u)v\:{\rm d}\mu$ for a large class of $f,u,v$

Convolution of tight measures is tight