Abstract
In Chapter 2, Section 2.2, we showed how one can start with a transition probability function P(s, x; t, ·) and end up with a Markov process. The problem is: where does P(s, x; t, ·) come from? The example we gave there, namely:
is a natural one from the probabilistic point of view because of its connection with independent increments and Gaussian processes. It turns out to be natural from another point of view as well: the theory of second order parabolic partial differential equations. The connection between the P(s, x; t, ·) in (1.1) and partial differential equations is well-known and easy to derive. Namely, if ϕ ∈ C b (Rd) and
then
where Δ is Laplace’s operator
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Stroock, D.W., Varadhan, S.R.S. (2006). Parabolic Partial Differential Equations. In: Multidimensional Diffusion Processes. Classics in Mathematics / Grundlehren der mathematischen Wissenschaften. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28999-2_4
Download citation
DOI: https://doi.org/10.1007/3-540-28999-2_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-22201-0
Online ISBN: 978-3-540-28999-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)