Stochastic partial differential equations are partial differential equations with random processes, much like stochastic ordinary differential equations are ordinary differential equations with random processes. The topic is an active research area, and there exist several different approaches.
An example of a SPDE is the Burgers' equation with stochastic forcing, which is a model for e.g. turbulence.
Solutions of SPDE are stochastic processes in continuous time in infinite dimensional spaces like spaces of (generalized) functions, Banach and Hilbert spaces or infinite dimensional manifolds, depending on the approach taken.
On this page we will concentrate on the stochastic Burgers' equation and the stochastic Navier-Stokes equations. Both are examples of nonlinear evolution equations, with a notion of a time variable and spatial variables. Both equations can be handled with Hilbert spaces as solution spaces. We fix two separable, real Hilbert spaces $U, H$. Every integrals in Banach spaces on this page are Bochner integrals.
In a first step we will need to construct the stochastic Itô integral
where $W$ is a Wiener process on $U$, and $\phi$ is a process which takes values in $L(U, H)$, the space of unbounded linear operators from $U$ to $H$.
For a topological space $X$ let $\mathbb{B}(X)$ be its $\sigma$-algebra.
Random variables are functions on a probability space $(\Omega, F, P)$, (set of events, $\sigma$-algebra, measure), which we will mostly suppress as is commonly done in the literature.
A probability measure $\mu$ on $(U, \mathbb{B}(U))$ is called a Gaussian probability measure if for all $u \in U$ we have that the mapping
is a Gaussian real random variable.
As in the finite dimensional situation, Gaussian probability measures are completely characterized by their characteristic function(al), this is the Bochner-Minlos theorem.
A measure $\mu$ on $(U, \mathbb{B}(U))$ is a Gaussian probability measure iff
for all $u \in U$, with a $m \in U$ (the mean) and $Q$ a selfadjoint, positive operator of trace class (the covariance operator).
We will say that a $U$-valued Gaussian random variable $X$ is a $N(m, Q)$ random variable just like in the finite dimensional case.
Since $Q$ is selfadjoint and of trace class, it has a orthonormal basis of eigenvectors $e_k, k \in \mathbb{N}$, this is the Hilbert-Schmidt theorem:
with $0$ as the only accumulation point of the sequence $(\lambda_k)$.
It is possible to characterize $N(m, Q)$ random variables as follows:
A $U$-valued random variable is a $N(m, Q)$ random variable iff
with independent, real $N(0, 1)$ random variables $\beta_k$, the sum converges in the $L^2$ sense.
Let’s note as a remark that we can construct in this way a $N(m, Q)$ random variable for every $m$ and every selfadjoint, positive, trace-class operator $Q$. We are now ready to define the standard $Q$-Wiener process given such an operator $Q$:
As usual we will mostly suppress the underlying probability space $(\Omega, F, P)$, (set of events, $\sigma$-algebra, measure).
A $U$-valued random process $W_t$ is called a standard $Q$-Wiener process iff
$W(0) = 0$,
$W$ has a.s. continuous trajectories,
the increments are independent and
the increments $W_t - W_s$ are $N(0, (t-s) Q)$ random variables with $t \gt s$.
So, the definition of Wiener processes in infinite dimensions is essentially the same as in finite dimensions.
The Q-Wiener process can be characterized similar to Gaussian random variables:
A $U$-valued random process $W_t$ is a $Q$-Wiener process iff
where the $\beta_k(t)$ are real valued Wiener processes. The sum converges in $L^2(C([0, T], U), P)$ and has a P-a.s. continuous modification, where the space $C([0, T], U)$ has the sup-norm.
As a corollary we see that there is to every suitable covariance operator $Q$ a $Q$-Wiener process.
We need the concept of martingales in Banach spaces analog to the situation in finite dimensions. In a first step we note that the conditional expectation exists also in Banach spaces:
Let E be a real separable Banach space and X be a Bochner-integrable E-valued random variable on the probability space $(\Omega, F, P)$, let $G$ be a $\sigma$-field contained in $F$.
Then there exists a $G$-measurable random variable $Y$ such that for all $g \in G$ we have:
All such random variables are equal P-almost everywhere. As in the finite dimensional case we write $Y = E(X | G)$ and say that $Y$ is a conditional expectation of $X$ given $G$. Further we have:
The definition of a martingale with respect to a filtration is the same as in the finite dimensional case. We can get real-valued submartingales from $E$-valued martingales:
If $M_t$ is an $E$-valued martingale and $p \in [1, \infty)$, then $\| M_t \|^p$ is a real-valued submartingale.
This means that we can translate Doob’s maximal inequality directly to $E$-valued martingales.
Let $\mathcal{M}^2_T (E)$ for a fixed $T$ be the space of all $E$-valued square integrable martingales, then we have the
$\mathcal{M}^2_T (E)$ is a Banach space when equipped with the norm
Note: Every $Q$-Wiener process with respect to a normal filtration (which is defined in the same way as in the finite dimensional case) is a continuous square integrable martingale.
The definition of the Itô integral proceeds essentially in the same way as in the finite dimensional case, but one has to do some extra exercises in linear functional analysis along the way.
…to be continued.
Most content of this page is based on:
See also:
Stochastic partial differential equation, Wikipedia
Helge Holden, Bernt Øksendal, Jan Ubøe, Tusheng Zhang: Stochastic partial differential equations. A modeling, white noise functional approach. 2nd ed. (ZMATH)