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19 votes
Accepted

Is there any reason to use paracontrolled calculus over regularity structures?

I don't think that the reason given in the paper by Bailleul and Bernicot is a good one. Basically, they treat an example which is simple enough so that it is still manageable to describe the various ...
Martin Hairer's user avatar
18 votes

What phenomena are better modelled by SDE instead of ODE?

This is a really broad question, but in general noise terms become important if there are few degrees of freedom; for example, chemical reaction kinetics can be accurately described by coupled ODE's, ...
Carlo Beenakker's user avatar
16 votes
Accepted

Existence of normal number except random numbers

Computable, absolutely normal numbers do actually exist. See V. Becher, S. Figueira: An example of a computable absolutely normal number, Theoretical Computer Science 270 (2002), 947-958.
Francesco Polizzi's user avatar
15 votes

Why do stochastic integrals depend on the choice of partitioning points?

First, note that the right comparison is not with the Riemann integral but rather with the Riemann-Stieltjes integral. To be concrete, consider $\int_0^1 X_s dW_s$ where $W$ is Brownian motion and $...
ofer zeitouni's user avatar
11 votes

What phenomena are better modelled by SDE instead of ODE?

Brownian motion is an obvious example. Brownian motion described particles dispersed in a liquid that are large enough that the random jossling of the water molecules becomes important. Being one of ...
AccidentalTaylorExpansion's user avatar
9 votes
Accepted

Kolmogorov continuity theorem and Holder norm

One can apply a deterministic result, called Garsia--Rodemich--Rumsey inequality, to estimate $\mathrm{E}[||X||^\alpha_{\gamma;[0,T]}]$. Here is a particular form of this result, which is most ...
zhoraster's user avatar
  • 1,533
9 votes
Accepted

Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?

The reason a Langevin diffusion leaves $\nu(x)=e^{-U(x)}$ invariant is because it is symmetric or reversible with respect to $\nu$. In comparison to general diffusion processes, the ergodic ...
Nawaf Bou-Rabee's user avatar
8 votes
Accepted

For a stable matrix $B$ and anti-symmetric $T$, such that $B(I+T)$ is symmetric, show that $\mbox{tr}(TB)\leq0$

First let us check that $T$ exists and is unique. Let $\mathrm{Sym}_n$ be the space of symmetric matrices (with real coefficients), $\mathrm{M}_n$ the space of all matrices and $\mathrm{Alt}_n$ the ...
alqp's user avatar
  • 96
8 votes

A singular stochastic differential equation

This is sticky reflecting Brownian motion, see for example this relatively recent paper. You can alternatively construct it by taking a reflected Brownian motion and then "stretching out" ...
Martin Hairer's user avatar
8 votes
Accepted

SDE driven by fractional Brownian motion

When $H > 1/2$, one can interpret the integral as a Riemann-Stieltjes integral and one gets solutions by classical Picard iteration in $C^\alpha$ for $1/2 < \alpha < H$. It's not hard to see ...
Martin Hairer's user avatar
7 votes

Does a theory of stochastic differential algebras exist?

Yes. A systematic study of stochastic (differential) algebra could be found in Grenander, Ulf. Probabilities on algebraic structures. Dover Books, 1981. Grenander studied the operation of ...
Henry.L's user avatar
  • 8,031
7 votes

Why do stochastic integrals depend on the choice of partitioning points?

To complement the excellent answer by Ofer Zeitouni, let me offer a functional analysis perspective. We want to define an integral of the following form: $\int F(W_t)dW_t=\int F(W_t)W'_tdt$, say, for ...
Kostya_I's user avatar
  • 8,947
7 votes
Accepted

How to make sense of recursively defined SPDE solutions, like in Hairer's "Solving the KPZ equation" paper?

What I mean is that $$ X_\epsilon^\tau(t) = \int_{-\infty}^t P_{t-s} \Pi_0^\perp (\partial_x X_\epsilon^{\tau_1}(s)\, \partial_x X_\epsilon^{\tau_2}(s))\,ds\;, $$ where $P_t$ denotes convolution with ...
Martin Hairer's user avatar
7 votes
Accepted

Path integral presentation of solutions of Dirac equation

There are several relevant papers: Path Integral Approach to Relativistic Quantum Mechanics: Two-Dimensional Dirac Equation (1987) Path Integral for Relativistic Equations of Motion (1997) Path ...
Carlo Beenakker's user avatar
7 votes
Accepted

A comparison of diffusions

The inequality is not true in general — additional assumptions are needed. I think some kind of monotonicity of $a_1$ and $a_2$ should help, but this is merely a guess. Here is a counterexample. ...
Mateusz Kwaśnicki's user avatar
6 votes

Asymptotic behavior of an integral of OU process

Using the notation of the OP, let $I(t)=\int_0^t X_s^2 ds$ where $X$ solves the above SDE. By Chebyshev's inequality, we have that $$ P(I(t) > \alpha \mid X_0 = x) \le \frac{E\left\{ I(t) \mid X_0 ...
Nawaf Bou-Rabee's user avatar
6 votes

Walker whose Velocity is a Brownian Bridge

As Kwaśnicki remarked, the velocity process $v_t$ is a Brownian bridge, which can be represented as: $$ v_t = v_0 (1 - \frac{t}{T}) + v_T \frac{t}{T} + (T - t) \int_0^t \frac{1}{T-s} d B_s \;. $$ (...
Nawaf Bou-Rabee's user avatar
6 votes
Accepted

Under what condition we get back path from signatures in rough path theory?

Loops don't get canceled out in the signature. (You might like to compute the signature of a circle. The second iterated integral is nonzero; in fact, by Green's theorem, it gives you the area inside ...
Nate Eldredge's user avatar
6 votes
Accepted

English translation of "Les aspects probabilistes du contrôle stochastique"

There is no English translation of El Karoui's lecture notes, however her work on Snell envelopes is described in Reflected Solutions of Backward SDE'S, and Related Obstacle Problems for PDE's. For a ...
Carlo Beenakker's user avatar
6 votes
Accepted

Intuition behind Gubinelli derivative

In a way it is very much like a usual derivative. Recall first that for a regular function $Y$, its derivative $Y'_s$ at a point $s$ is the (unique) number such that $$ Y_{t,s}=Y'_s(t-s)+ R_{s,t}, $$ ...
m7e's user avatar
  • 458
6 votes
Accepted

When are the transition densities of an SDE symmetric?

This question is addressed here. In particular, a diffusion whose transition densities are symmetric is a special case of a $\nu$-symmetric diffusion, and by itself, this symmetry does not uniquely ...
Nawaf Bou-Rabee's user avatar
6 votes
Accepted

Weighted Lebesgue space with exponential weights: smoothing effect and properties

This kind of weight reminds me a lot of Gaussian weights. Then, you can normalize it and put it into the measure to get an $L^p$ space on $\mathbb R^n$ endowed with the Gaussian measure. My intuition ...
Clara Torres-Latorre's user avatar
6 votes

Interpretation of second order term in Fokker-Planck equation

To help the interpretation you may want to rewrite$^\ast$ the Fokker-Planck equation as \begin{align*} \frac{d}{dt}p(x) =& -\nabla\cdot [\tilde{f}p -D\cdot\nabla p],\\ \end{align*} with $D=\tfrac{...
Carlo Beenakker's user avatar
5 votes

Solve SDE $dX_t=(c+\sigma_\zeta W'_t)X_tdt + \sigma_\epsilon dW_t$

Using a (random) integrating factor, the solution is explicitly: $$ X_t =\exp\left( \sigma_{\zeta} \int_0^t W'_s ds \right) \times \left[ x_0+\int_0^t \exp\left(- \sigma_{\zeta} \int_0^s W'_r dr \...
Nawaf Bou-Rabee's user avatar
5 votes

Why the term "geometric" rough path?

I agree with Martin's answer - but there were other additional and compelling reasons. A rough path (as in the original papers) was a path in the tensor algebra over V with appropriate "p-...
tjl's user avatar
  • 71
5 votes

Does there exist a stochastic time derivative?

I think the chain rule $d[f(Z_t)]=f'(Z_t)\circ dZ_t$ is valid when the product $\circ$ is defined as in Stratonovich stochastic integral (while the SDE uses Ito's). Note that $g(Z_t)\circ dt$ doesn't ...
Jean Duchon's user avatar
  • 3,065
5 votes
Accepted

Reflecting Brownian motion and its transition probability density

Since building reflected Brownian motion in smooth bounded domains is not a problem, the only potential obstruction to the existence of the transition probabilities is that it escapes to infinity in ...
Martin Hairer's user avatar
5 votes
Accepted

Reference for Feynman-Kac

Please see p.282 of the following ref, it does not call it Feynman-Kac Formula but the whole section 5 is discussing it. It formalized the diffusion using a tensor field over a manifold which is the ...
Henry.L's user avatar
  • 8,031
5 votes
Accepted

Variance of Multi-Dimensional Ornstein Uhlenbeck process

the large-$t$ limit $\rho$ of the variance of $X_t$ is given by $$\rho=\int_0^\infty \exp(-tH)SS^{\dagger}\exp(-tH^{\dagger})dt$$ where the superscript $\dagger$ indicates the transpose (or the ...
Carlo Beenakker's user avatar
5 votes

Tanaka-Meyer formula

It looks to me like nothing's wrong; the local time $\Lambda^Z_t(0)$ of $Z$ at zero is identically zero. This makes a certain amount of intuitive sense, because $Z$ should have "bounded variation at ...
Nate Eldredge's user avatar

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