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Stochastic ordinary and partial differential equations generalize the concepts of ordinary and partial differential equations to the setting where the unknown is a stochastic process.

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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 …
gregarki khayal's user avatar
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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 …
gregarki khayal's user avatar
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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, \ …
gregarki khayal's user avatar