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Self-repelling diffusion on a Riemannian manifold | | Michel Benaïm
; Carl-Erik Gauthier
; | | Date: |
21 May 2015 | | Abstract: | Let M be a compact connected oriented Riemannian manifold. The purpose of
this paper is to investigate the long time behavior of a degenerate stochastic
differential equation on the state space $M imes mathbb{R}^{n}$; which is
obtained via a natural change of variable from a self-repelling diffusion
taking the form $$dX_{t}= sigma dB_{t}(X_t) -int_{0}^{t}
abla
V_{X_s}(X_{t})dsdt,qquad X_{0}=x$$ where ${B_t}$ is a Brownian vector field
on $M$, $sigma >0$ and $V_x(y) = V(x,y)$ is a diagonal Mercer kernel.
We prove that the induced semi-group is strong Feller and has a unique
invariant probability $mu$ given as the product of the uniform measure on M
and a Gaussian measure on $mathbb{R}^{n}$. We then prove an exponential decay
to this invariant probability in $L^{2}(mu)$ and in total variation. | | Source: | arXiv, 1505.5664 | | Services: | Forum | Review | PDF | Favorites | |
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