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Brownian MotionFluctuations, Dynamics, and Applications$

Robert M. Mazo

Print publication date: 2008

Print ISBN-13: 9780199556441

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199556441.001.0001

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(p.263) Appendix C An Operator Identity

(p.263) Appendix C An Operator Identity

Source:
Brownian Motion
Publisher:
Oxford University Press

We derive the operator identity given in eqn (10.4.10). Consider an operator A + B. In the case of eqn (10.4.10), A can be taken as i(1 - )L and B as i ℘L.

Define an operator J(s) by

(C.1) Appendix C An Operator Identity

Differentiate with respect to s, obtaining

(C.2) Appendix C An Operator Identity

cancelling and rearranging,

(C.3) Appendix C An Operator Identity

From the definition, it is clear that J(0) = 1, so that integration of this equation yields

(C.4) Appendix C An Operator Identity

Finally, multiplying by exp(— As) and using the definition of J, we get the final result

(C.5) Appendix C An Operator Identity

This is the operator identity that we used in Chapter 10.

Alternatively, if we had defined K(s) by

(C.6) Appendix C An Operator Identity

similar manipulations would have yielded the alternative operator identity

(C.7) Appendix C An Operator Identity