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      All Questions

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      4
      votes
      1answer
      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
      0answers
      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
      0answers
      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
      2answers
      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
      0answers
      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
      1answer
      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
      2answers
      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
      0answers
      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
      1answer
      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
      1answer
      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
      2answers
      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
      1answer
      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
      1answer
      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
      0answers
      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
      2answers
      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 ...

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