All Questions
8 questions
4
votes
1
answer
107
views
Identify an SDE on the sphere from its generator
I have a diffusion on the 2-sphere with expression:
$$
(L\phi)(u):=\frac{1}{2{N(u)}}\Big(f(u)\Delta_{\mathbb S^2}\phi+
2g\left( \nabla_{\mathbb S^2}\phi, \nabla_{\mathbb S^2}f\right)\Big)
$$
...
- 73
2
votes
0
answers
81
views
Assumptions for uniform measure of SDE on manifolds
Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
3
votes
1
answer
390
views
Reference request for a Riemannian Fokker-Planck equation
The original post is in StackExchange but no one has answered it yet. I personally think it is more related to the research area so I put it in MathOverflow. Below is the question in the original post:...
- 187
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 ...
- 177
1
vote
0
answers
157
views
The stochastic parallel transport as a limit of piecewise geodesic parallel transports
Let $(M,g)$ be a Riemannian manifold, and $E \to M$ be a vector bundle endowed with a connection $\nabla$. If $c:[0,1] \to M$ is a continuous curve, and if $\Delta = \{t_1, \dots, t_m\} \subset [0,1]$,...
- 5,407
2
votes
0
answers
101
views
The Itō isometry for Riemannian manifolds
If $\alpha$ is a real smooth $1$-form, and if $\mathcal C$ is the space of continuous functions $c : [0,1] \to \mathbb R^n$, endowed with the Wiener measure $w$, and if $I_\alpha : \mathcal C \to \...
- 5,407
0
votes
0
answers
88
views
Independent increments for the Brownian motion on a Riemannian manifold
In am not a probabilist, but I must do some stochastic-flavoured work on a connected Riemannian manifold $M$. A nice thing about the Brownian motion on $\mathbb R^n$ is that we may talk about its ...
- 5,407
5
votes
0
answers
275
views
stochastic control / geometric mean
Consider the following problem:
Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...
- 427