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Results tagged with stochastic-processes
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user 44590
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
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Accepted
The (infinite) invariant measure of an SPDE
I check it with the standard Garlerkin method and confirmed that it is right, in both cases (i) and (ii).
Discribe the proof briefly (under (ii)):
Take a CONS of $H$ as $h_1 = 1$ and $h_k(x) = \co …
- 199
2
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1
answer
218
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The (infinite) invariant measure of an SPDE
Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type:
\begin{equation}\left\{
\begin{aligned}
&\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(t, x …
- 199
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1
answer
319
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On the superior of generalized Ornstein-Uhlenbeck process
Let us consider a generalized O-U process $X_t \in L^2[0, 1]$ defined by the following spde:
$dX_t = \frac{1}{2}\partial_x^2X_t + dW_t, $
$\partial_x X_t(0) = \partial_x X_t(1) = 0, $
$X_0 = 0, $
…
- 199
1
vote
1
answer
372
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On the solution of a stochastic partial differential equation
Consider a simple SPDE as follows:
$\partial_t u(t,x)=\partial_x^2 u(t,x)+V(u(t,x))+\dot{W}(t,x)$, $t>0$, $x\in(0,1)$,
$u(t,0)=u(t,1)=0$,
$u(0,x)=v(x)$,
where $V$ is a bounded, smooth potentia …
- 199
2
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answers
405
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On the infinitesimal generator of a 1-dimensional stochastic heat equation: core and explici...
Denote $E = C([0, 1])$. I am consider a 1-dimensional stochastic heat equation on $h$:
$$\partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x), \quad\text{ for all } (t, x) \in (0, \ …
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