All Questions
3 questions
1
vote
2
answers
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Solving the Poisson equation using a random walk on $\mathbb Z ^d$
How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?
- 11
11
votes
1
answer
2k
views
Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE
Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...
- 271
2
votes
1
answer
319
views
Uniqueness of classical solution with degenerate boundary
Consider heat equation on the domain $\Omega = (0,1)\times (0,1)$
in the form of
$$ \partial_{t} u = \frac 1 2 x^{3} (1-x) \partial_{xx} u, \quad
(x,t) \in \Omega$$
with initial data
$u(x,0) = x$ for ...
- 1,399