What is a Cauchy problem (in a Brownian motion related context)?

Question

one page from the book Brownian motion and stochastic calculus

In this page, the auther said the problem in definition 4.1 is Cauthy problem. But I think the boundary condition is just a Dirichlet problem not a Cauchy problem? am I right?

Answer 1

You can consider that as an ODE $\dot v=F(v)$, $F(v)=kv-\frac12\Delta v-g$ over the vector space of functions $v(t)\in C^2(\Bbb R^d)$. As such what is given is an initial value problem $t_0=T$, $v(t_0)=v_0=f$ which is also called "Cauchy problem".

Answer 2

From my limited understanding of PDE theory, you would be correct in saying it is a Dirichlet problem if it is a PDE in space dimension only (with no time component) and the boundary is specified. For BM related PDEs, this tends to be elliptic PDEs.

The problem in the page you linked is a Cauchy problem, which is a problem with specifies initial (or terminal conditions) when the there is time component involved. BM related PDEs with time component are usually parabolic.

Normally, you cannot specify a Dirichlet type boundary for parabolic equation because often a solution does not exist, where as parabolic equations are known to have unique solutions subject to the coefficients are regular enough.