# All Questions

Tagged with stochastic-processes dg.differential-geometry

15
questions

**4**

votes

**1**answer

206 views

### An application of Itô's formula to an SDE on a Lie group

I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.
Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE
$$dg(t)...

**2**

votes

**0**answers

105 views

### Is there a distinct Ito-Sasaki version of Riemannian stochastic development?

Given a smooth manifold $M$ with a linear torsion-free connection on its tangent bundle, the Eells-Elworthy-Malliavin stochastic development provides a way of transforming a semimartingale $X$ defined ...

**2**

votes

**0**answers

100 views

### Ito formula between manifolds

I have seen many Ito formulae giving dynamics for $f(X_t)$ where $f:M\to \mathbb{R}$ is a smooth function from a manifold $M$ and $X_t$ is a (continous) manifold-valued semi-martingale.
My question ...

**7**

votes

**2**answers

347 views

### Intuition for the Drift Term of the Laplace-Beltrami Operator

In coordinates, the Laplace-Beltrami operator on a Riemannian manifold $(M,g)$ can be written as:
$$
\Delta_g = g^{ij}\partial_{ij} - g^{jk}\Gamma^\ell_{jk}\partial_\ell
$$
The second term:
$$
\mu^\...

**2**

votes

**0**answers

130 views

### Ito lemma for manifold semimartingales

I'm looking for a generalization of the usual Ito lemma to manifolds $M$, preferably not under the assumption that $M$ is embedded in $\mathbb{R}^d$. Unfortunately any reference I've found either ...

**5**

votes

**1**answer

290 views

### Reference: Stochastic Analysis on Hilbert Manifolds

I'm looking for a reference to a book which develops an It\^{o} lemma for semi-martingales with values in infinite dimensional Hilbert-Manifolds. I expect the techniques to be the same but still I ...

**1**

vote

**2**answers

176 views

### Conditional Wiener measure continuous

consider a complete Riemannian manifold $M$ with heat kernel $p_M$ and let $U\subset M$ be an open set. Let $W_{x,t}^{y}$ be the Wiener measure associated to the Brownian motion starting at $x$ and ...

**4**

votes

**0**answers

84 views

### Feynman-Kac formula and time-ordering for vector bundles

Let $M$ be a compact Riemannian manifold and let $\mathrm{d}\mathbb{W}^{yx;T}(\gamma)$ denote the Brownian Bridge measure, i.e. the Wiener measure on the paths that travel from $x$ to $y$ in time $T$ (...

**4**

votes

**1**answer

475 views

### Carre du Champ, Subunit Paths and CC-metrics

Let the operator $L$ be given by $Lf(x):=\nabla\cdot (A\nabla f(x))$, where $f:\mathbb{R}^d\rightarrow \mathbb{R}$ belongs to a suitable class of functions $\mathcal{A}$. The carre du champ operator $\...

**4**

votes

**1**answer

133 views

### Exponential of approximate quadratic variation of Brownian motion

Let $X_t$ be a Brownian motion or a Brownian Bridge on a (\edit: compact) Riemannian manifold. Let $T>0$ be given.
The question is: Does there exists a constant $C>0$ such that for all ...

**1**

vote

**2**answers

275 views

### Principal bundles and Subriemannian Geometry

In sub-Riemannian geometry, one considers manifolds $P$ equipped with a subbundle $\mathcal{H}$ of $TP$, the horizontal distribution. One then has a Riemannian metric only on this distribution $\...

**4**

votes

**1**answer

508 views

### Stochastic interpretation of heat kernel on fiber bundle

I'm looking for a stochastic interpretation of the heat equation for vector valued function.
The classical set up is the following :
If $(M,g)$ is a riemannian manifold then we could consider the ...

**3**

votes

**1**answer

91 views

### Density for Translated Process

Let $M$ be a (compact) Riemannian manifold. Let $v$ be a smooth vector field on $M$ with flow $\Theta_t$. Let $L$ be an elliptic second order differential operator on $M$ that generates the Ito ...

**2**

votes

**0**answers

79 views

### Almost sure transversality of smooth random maps

I still am novice as far as probability is concerned and after fruitlessly Googling for an answer for a few days I thought I might have a better chance with MO.
Let me first formulate the ...

**3**

votes

**2**answers

273 views

### Ito Diffusions with low regularity?

I would like to have an It? Diffusion
$$ X_t = \int_0^t b(s) \mathrm{d}s + \int_0^t \sigma(s) \mathrm{d}B_s.$$
where the (vector- and matrix-valued, respectively) functions $b$ and $\sigma$ have lower ...