Vanishing Stochastic Perturbations of

Hamiltonian Systems

CASA Day, Nov 2013

Work with M. Hong Duong and Mark Peletier

Upanshu Sharma

“It has been said that something as small as the flutter

of a butterfly's wing can ultimately cause a typhoon

halfway around the world.”

-The Butterfly Effect

1

Toy Problem

2

Toy Problem

2

Toy Problem

2

Hamiltonian – Conserved Quantity

3

Celestial Mechanics

Magnetic Traps

etc…

Hamiltonian – Conserved Quantity

3

Problem

4

What are the dynamics

as ?

Problem

4

SDE to PDE

5

Diffusion on a Graph

6

In the limit :

Diffusion on a Graph

6

Show me the MATH !

7

Show me the MATH !

7

Result…

8

• x\

• A

• C

Conclusions

9

Result…