Intereting Posts

Solutions of the congruence $x^2 \equiv 1 \pmod{m}$
Show that exist $i>0$ such that the Fibonacci number $F_{i}$ is divisible by 2015
Conjectured compositeness tests for $N=k\cdot 2^n \pm 1$ and $N=k\cdot 2^n \pm 3$
What is the proof that the total number of subsets of a set is $2^n$?
Teenager solves Newton dynamics problem – where is the paper?
Closed form for $\prod_{k=0}^n\binom{n}{k}x^ky^{n-k}$.
Has a natural transformation between functors with codomain $Cat$ that is an equivalence on each component a weak inverse?
Most useful heuristic?
Does $\lfloor \sqrt{p} \rfloor$ generate all natural numbers?
Solutions of $\arctan x = 1 – x$. Proofs?
Recursive formula for variance
Show that a function from a Riemann Surface $g:Y\to\mathbb{C}$ is holomorphic iff its composition with a proper holomorphic map is holomorphic.
Measuring $\pi$ with alternate distance metrics (p-norm).
Group actions transitive on certain subsets
Solving $x\sqrt1+x^2\sqrt2+x^3\sqrt3+…+x^n\sqrt{n}+\dots=1$ with $x\in \mathbb{R}$ and $n\in \mathbb{N}$

Let$^1$

- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $U$ be a separable Hilbert space
- $Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace
- $(W_t)_{t\ge0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$
- $H:=\mathbb R^d$ for some $d\in\mathbb N$ and $\xi:\Omega\times[0,\infty)\times\mathbb R^d\to\operatorname{HS}(U_0,H)$, where $U_0:=Q^{1/2}U$.

Suppose we’re concerned with an SPDE $${\rm d}u_t\left(\Phi_t\left(x\right)\right)=f_t\left(\Phi_t\left(x\right)\right){\rm d}t+\nabla u_t\left(\Phi_t\left(x\right)\right)\cdot\xi_t\left(\Phi_t\left(x\right)\right){\rm d}W_t\;\;\;\text{for all }t\ge 0\text{ and }x\in H\;,\tag 1$$ where

- $u:\Omega\times[0,\infty)\times H\to\mathbb R$
- $\Phi:\Omega\times[0,\infty)\times H\to H$
- $f:\Omega\times[0,\infty)\times H\to\mathbb R$

are suitable.

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- a follow up question about modeling with exponential distributions
- how to derive the mean and variance of a Gaussian Random variable?

Can we recast $(1)$ into an equation in $\tilde H:=L^2(\mathbb R^d;\mathbb R^d)$? I want to get rid of the second parameter of $u$, i.e. I want to turn the finite-dimensional multiparameter SDE $(1)$ indexed by time and space into an infinite dimensional single parameter SDE indexed by time only.

$^1$ Let $\mathfrak L(A,B)$ be the set of bounded, linear operators from $A$ to $B$. Moreover, Let $\operatorname{HS}(A,B)$ be the set of Hilbert-Schmidt operators from $A$ to $B$.

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