Formulæ for the heat kernel of an elliptic operator exhibiting small-time asymptotics

1988 ◽  
pp. 167-180
Author(s):  
Keith D. Watling
Keyword(s):  
Elliptic Operator ◽  
Heat Kernel ◽  
Small Time ◽  
Time Asymptotics ◽  
Small Time Asymptotics

scholarly journals Small-time fluctuations for the bridge of a sub-Riemannian diffusion

2021 ◽  
Vol 54 (3) ◽  
pp. 549-586
Author(s):  
Ismäel BAILLEUL ◽  
Laurent MESNAGER ◽  
James NORRIS
Keyword(s):  
Diffusion Processes ◽  
Energy Functional ◽  
Small Time ◽  
Minimal Energy ◽  
Minimal Path ◽  
Time Asymptotics ◽  
Unique Path ◽  
Small Time Asymptotics ◽  
Conditioned Diffusion ◽  
Minimal Energy Path

We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-Riemannian cut locus, then the fluctuations of the conditioned diffusion from the minimal energy path, suitably rescaled, converge to a Gaussian limit. The Gaussian limit is characterized in terms of the bicharacteristic flow, and also in terms of a second variation of the energy functional at the minimal path, the formulation of which is new in this context.

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