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What do we know about Poisson boundaries of arbitrary Riemannian manifolds?

For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
Tyrannosaurus's user avatar
2 votes
0 answers
301 views

Ito lemma for SDEs on a Lie group

I'm trying to generalize the theorem described in this paper https://arxiv.org/abs/2001.01098 to the case of a semisimple compact matrix Lie group. In doing so i'm trying to define a formula ...
Marco's user avatar
  • 293
3 votes
0 answers
201 views

Elworthy’s 1982 “Stochastic Differential Equations on Manifolds” - relevant?

In 1982, D. Elworthy published “Stochastic Differential Equations on Manifolds”. Apparently, this was quite a seminal book in the field of stochastic DE’s/processes on manifolds. Is this reference ...
Martin Geller's user avatar
5 votes
1 answer
531 views

Riemannian metric induced by a stochastic differential equation

Following this paper, a diffusion process in $\mathcal{R}^d$ $$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$ with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be ...
can't stop me now's user avatar
2 votes
1 answer
204 views

Comparing diffusion processes in different metrics

I would like to know if it is possible to compare two diffusion processes defined on the same manifold $\mathcal{M}$ but with respect to different metrics say $g_1$ and $g_2$. Is there a way to apply ...
can't stop me now's user avatar
11 votes
1 answer
553 views

A geometric law of large numbers

Question: Choose independent uniformly distributed random variables $X$ and $Y$ on a closed Riemannian manifold $(M,g).$ The geodesic midpoint of $X$ and $Y$ is well-defined a.e., and is distributed ...
L. Roy's user avatar
  • 119
0 votes
1 answer
115 views

Average over spheres finite

Let $X_1,...,X_N$ be random variables that are iid with the uniform distribution over $\mathbb S^n.$ I am curious how to see that $f(X_1,..,X_N):=\left \lvert \sum_{i=1}^N X_i \right\rvert^{-1}$ has ...
Pritam Bemis's user avatar
5 votes
1 answer
319 views

Spherical average of $\frac{1}{x}$

Let $X_1,...,X_n$ be points on $\mathbb S^1.$ We then define the expectation value $E(X)=\frac{1}{n}\sum_{i=1}^n X_i.$ Let $\frac{dS(X_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i....
Pritam Bemis's user avatar
1 vote
1 answer
170 views

Diffeomorphism for mapping one SDE into another

Let $Y_t,X_t$ be $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$-adapted Markov diffusion processes with valued in $\mathbb{R}^n$. (When) does there exist a diffeomorphism $\phi:\mathbb{R}^n\to \...
ABIM's user avatar
  • 5,405
3 votes
1 answer
474 views

Orthonormal frame bundles on a manifold

Let $(\mathcal{M},g)$ be a torsion free compact Riemannian manifold of dimension $n$. Hence from the metric we know there is an associated horizontal sub-bundle $H_u F \mathcal{M}$ of the orthonormal ...
PPR's user avatar
  • 396
4 votes
1 answer
417 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)...
eipiplusone's user avatar
2 votes
0 answers
140 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 ...
Emilio Ferrucci's user avatar
10 votes
2 answers
767 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^\...
user3658307's user avatar
2 votes
0 answers
385 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 ...
ABIM's user avatar
  • 5,405
5 votes
1 answer
372 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 ...
ABIM's user avatar
  • 5,405
2 votes
2 answers
253 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 ...
Denilson Orr's user avatar
4 votes
0 answers
112 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 $T$ (...
Matthias Ludewig's user avatar
4 votes
1 answer
736 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 $\...
mathsquestion88's user avatar
4 votes
1 answer
294 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 ...
Matthias Ludewig's user avatar
2 votes
2 answers
442 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 $\...
Matthias Ludewig's user avatar
5 votes
1 answer
647 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 ...
user avatar
3 votes
1 answer
105 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 ...
Matthias Ludewig's user avatar
3 votes
0 answers
90 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 ...
Liviu Nicolaescu's user avatar
3 votes
2 answers
325 views

Ito Diffusions with low regularity?

I would like to have an Itô 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 ...
Matthias Ludewig's user avatar