Stochastic partial differential equations are partial differential equations with a random driving force. Please do not use this tag just because there are stochastic processes and differential equations in your question. Consider if [tag: SDE] is a better choice. This tag is only to be used for PDEs driven by noise.
Questions tagged [stochastic-pde]
152 questions
14
votes
2 answers
Characterization of $C^{k,\alpha}$ (functions with Hölder continuous derivatives) through Taylor estimates
For $k \in \Bbb{N} = \{1,2,3,\dots\}$ and $\alpha \in (0,1)$, let us define
$$
C^{k,\alpha} := \{ f : \Bbb{R} \to \Bbb{R} \,:\, f \in C^k \text{ with } f, f', \dots, f^{(k)} \text{ bounded and } [f^{(k)}]_{C^\alpha} < \infty…
PhoemueX
- 36,211
11
votes
1 answer
Infinite convolution of a smooth compactly supported function converges uniformly
$\newcommand{\R}{\mathbb{R}}$$\newcommand{\diff}{\mathrm{d}}$Let $\rho\in C^\infty_c(\mathbb R^d;\mathbb R)$ be an even function, i.e. $\rho(-x)=\rho(x)$ for all $x\in\mathbb R^d$ that furthermore satisfies $\operatorname{supp}\rho\subset B(0,1)$…
Shashi
- 9,018
10
votes
2 answers
Why is the gaussian free field a distribution but Brownian motion is a function?
As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even though it is nowhere differentiable). By contrast,…
prdnr
- 358
8
votes
0 answers
Martingale problems and SPDEs
It is a classical result that if $X$ is a process with values in $\mathbb{R}^d$ and
\begin{align*}
M_t^i = X_t^i - \int_0^t b_i(X_s) ds \\
M_t^i M_t^j - \int_0^t a_{ij} (X_s) ds
\end{align*}
are both (local) martingales then (on an enlargement of…
Rhys Steele
- 20,326
8
votes
1 answer
Hölder continuity definition through distributions.
I am trying to prove that for a given Hölder parameter $\alpha \in (0, 1)$ and a distribution $f \in \mathcal{D}'(\mathbb{R}^d)$ the following are equivalent:
$f \in C^{\alpha}$
For any $x$ there exists a polynomial $P_x$ such that $| \langle f -…
Kore-N
- 4,275
8
votes
1 answer
Intuitive explanation of a stochastic PDE
Lindgren et al 2011 connects Gaussian Markov Random Fields (which have fast calculation properties due to the Markov attribute) and Gaussian Processes (which can model many types of data). The connection rests upon the fact (from Whittle 1954) that…
cgreen
- 545
8
votes
0 answers
Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.
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…
0xbadf00d
- 14,208
7
votes
1 answer
Reference complementing Hairer's "Introduction to Stochastic PDEs"
I want to study stochastic partial differential equations, and the standard reference seems to be Hairer's notes - after all, he got the first Fields Medal for work on this topic.
However, I also noticed that his notes are rather terse, and hence I…
ertl
- 619
- 3
- 12
7
votes
0 answers
Motivation behind stochastic PDEs
I am starting to study stochastic partial differential equations, and would like to understand when and why they are used.
It is well known that in many mathematical models for physics PDEs play a central rôle. E.g. the Navier-Stokes equations in…
Kore-N
- 4,275
5
votes
1 answer
Derivative of a Stochastic Integral with respect to Limit & with respect to Integrator
I have recently come across an attempt to differentiate the following function with respect to $t$ and with respect to $W_t$:
$$F(W_t, t):=\int_{h=0}^{h=t}W_hdW_h$$
Is it possible (i.e. is it "well defined") to differentiate $F$ with respect to $t$…
Jan Stuller
- 1,279
5
votes
0 answers
May I ask why the following operator between Banach spaces is continuous please?
I am reading the paper "On the Initial Value Problem of Stochastic Evolution Equations in Hilbert Spaces" (Here is the link for electronic version). But when reading it, I meet a problem and would like to ask for help:
Background:
1.The event space…
misakaczy
- 181
- 7
5
votes
0 answers
Prove that a martingale with a spatial parameter is differentiable
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
$M:\Omega\times[0,\infty)\times\mathbb R^d\to\mathbb R$ such…
0xbadf00d
- 14,208
5
votes
0 answers
Hilbert valued martingales - help with reference
I'm currently studying the theory of SPDEs on the book "Stochastic equations in infinite dimensions" by da Prato, Zabczyk. In the book, the theory of stochastic processes with values on a Banach space $E$, and in the particular the notion of…
Lucio
- 952
- 5
- 15
5
votes
2 answers
Scaling of space-time white noise
On different sources I found different parabolic scalings for space time white noise that I believe are in contradicton one with the other.
Let $\xi(t,x)$ be space-time white noise on $\mathbb{R}\times\mathbb{R}^d$. I apply a scaling $t\to…
pollastro
- 51
5
votes
0 answers
Are there symbolic methods/computing for stochastic processes and stochastic differential equations?
Are there symbolic methods/computing for stochastic processes and stochastic differential equations? Are there some research trends along these lines? Can this be perspective and fruitful endeavour for research?
I managed to find only SYMPLER -…
TomR
- 1,459