3. Probability Essentials 21
3.5. Stochastic Partial Differential Equations
<∞. (3.12) Then it holds P-almost surely for any 0≤t≤T:
Z
A
Z
Rq×[0,t]
f(x, s,·, u)M(ds dx)
!
µ(du) = Z
Rq×[0,t]
Z
A
f(x, s,·, u)µ(du)
M(ds dx).
This result can be shown using elementaryf first and the full proof can be found on page 296 of [Wal86].
The reader may have noticed that for the Gaussian noises the letters W and W˙ are used both. The precise notation and distinction between the two is cho-sen according to the following heuristic. If B is Brownian motion, we denote its temporal increments by dBt. The noises defined in Definition 3.3.15 are martin-gale measurese Wt(dx) = W(t, dx). We could denote their spatial increment by
∂tWt(dx), but using physics notation, it is more convenient to write ˙W(t, dx). In a weak setting of an SPDE the notation ˙W(t, x) will be used, but note that it is only a formal notation. What we have well-defined in this section is the integration w.r.t. time and space simultaneously and the notation will be RT
0
R
RqW(dt dx).
3.5. Stochastic Partial Differential Equations
There is already quite a large body of introductory literature on SPDE and its integration theory. However, there are several ways to approach the topic. The
3.5 Stochastic Partial Differential Equations 43 first one, which we follow here is the approach of Walsh using martingales mea-sures, as presented in [Wal86]: it is a more analytic treatment of the subject using martingale measures. The second treatment based on a Hilbert-space approach is more functional analytic, formulating SPDE as Hilbert-space valued SDE. This is comparable to the abstract setting of PDE as in Section 2.3. Some good references are [DPZ08], [PZ00] and [PR07]. Another approach to SPDE theory is Kryloy’s Lp-theory as presented in [Kry96].
Before defining SPDE rigorously, let us consider another example (see [Wal86]).
Imagine you left a guitar outside your house and a sandstorm came. What would be the melody the guitar strings are playing then? The amplitude u of the guitar string can be modeled as a solution to the wave equation. The sandstorm can then be modeled as an added inhomogeneity ˙W to the system:
(∂t2u(t, x) =κ2∂2xu(t, x) + ˙W(t, x), (t, x)∈R+×[0, L], u(0,·) =∂tu(0,·) = 0, u(t,0) =u(t, L) = 0,
whereL is the length of the guitar string andκ2 is the speed of wave propagation.
We will make that more rigorous in a moment.
More abstractly, assume that A and C are partial differential operators (PDO) as in Section 2.1 on R+×Rq, then we look for solutions of
A(u) =C(u) ˙W
with specific initial conditions (suitable for the equation A(u) = 0). As for PDE it is not possible to treat these equations in full generality if one wants to go to finer statements. We will single out one variable which will be called the “time” and call Rq “space”. Consider the following evolution equation:
∂tu(t, x) =A(t, x)u(t, x) +b(t, x, u(t, x)) +C(t, x, u) ˙W(t, x), u(0,·) =u0(·) (3.13) where A(t, x) only acts on spatial derivatives of u; b : R+×Rq+m → Rm and u:R+×Rq→Rm for a certainm∈N.Usually,C can be chosen quite general (see e.g. (6.7) in [DPZ08]), but we restrict toC(t, x, u) =σ(t, x, u) for a certain function σ :R+×Rq+m →R:
∂tu(t, x) =A(t, x)u(t, x) +b(t, x, u(t, x)) +σ(t, x, u) ˙W(t, x), u(0,·) =u0(·) (3.14) The last term in (3.14) will be called thenoise term. We say that the equation has additive noise ifσ does not depend onu andmultiplicative noise otherwise. In the additive noise case, the use of the mild solutions below allows an explicit solution of (3.14) and thus is well understood. However, we will focus on the multiplicative noise case.
44 Probability Essentials Given the concepts developed in the last section we can now make the notion of a solution to (3.1) precise. Let us fix A a PDO, b, σ, u0 and L( ˙W) and denote by A∗ the formal adjoint ofA, see (2.10). Here ˙W is a Gaussian noise with correlation functional Lk as defined in Definition 3.3.15.
Definition 3.5.1. We say that aweak solution for (3.14) exists, if there is a filtered probability space (Ω,F,Ft,P) supporting a noise ˙W and an adapted random field u:R+×Rq×Ω→Rm s.t. for allφ∈Cc(Rq) itP-almost surely holds that The random field willu will be called a weak solution.
From the discussion in Section 2.3, we recall that there is also another important concept of solutions in the PDE setting, the concept of mild solutions. We will provide a description for this concept only in the following special case (for the general Hilbert-space setup see [DPZ08]): Assume that A is the generator of a strongly continuous semigroup (S(t))t≥0 on Cb(Rq) with densityst:Rq →Rm×m, i.e. S(t)f =R
Rqst(· −y)f(y)dy, f ∈Cb(Rq).
Definition 3.5.2. We say that amild solution for (3.14) exists, if there is a filtered probability space (Ω,F,Ft,P) supporting a noise ˙W and an adapted random field u :R+×Rq×Ω →Rm s.t. (3.15), (3.16) hold and for all t >0, x∈ Rq, P-almost The random field uwill be called a mild solution.
The following theorem relates these two notions:
Theorem 3.5.3 (Thm 6.5 of [DPZ08]). Any weak solution is a mild solution. Any mild solution satisfying
3.5 Stochastic Partial Differential Equations 45 Both of these notions were stochastically weak concepts, see Definition 3.2.10.
There is also a stochastically strong concept of solutions:
Definition 3.5.4. We say that astochastically strong solution for (3.14) exists, if for any filtered probability space (Ω,F,Ft,P) supporting a noise ˙W we have an adapted random field u :R+×Rq×Ω → Rm s.t. (3.15), (3.16) and (3.17) holds P-almost surely.
There is an essential difference between stochastically weak and stochastically strong solutions: in the latter case, u needs to be adapted, independent of which probability space, including the filtration, is chosen. Choose a probability space supporting a noise ˙W and its canonical filtration (Ft)t≥0 (see (3.9)). Then, ut is measurable w.r.t Ft and hence needs to be a function of the noise ˙W up to time t (see the considerations after Definition 3.2.10). Obviously, strong existence implies weak existence.
Oftentimes, it will be convenient that we restrict our attention to certain sub-classes U of functions (continuous, vanishing at infinity, etc.) of functions. On these subclasses one can ask for uniqueness of solutions to (3.14). As for stochastic ordinary differential equations there are several notions of uniqueness:
Definition 3.5.5 (Uniqueness).
(a) We say that uniqueness in law inU holds for (3.14) if for any weak solution u∈ U the f.d.d. of the random field uare equal.
(b) We say that pathwise uniqueness (PU) in U holds for (3.14) if any weak solutions in U defined on the same probability space are indistinguishable.
A certain set of functions will be of special interest to us for the question of uniqueness. This class was first used in that context by Shiga in [Shi94]. Define for a real-valued functionu:Rq→R:
kukλ,∞:= sup
x∈Rq
e−λ|x||u(x)|.
We consider the following functions spaces.
Ctem={f :Rq→R:f continuous,kfkλ,∞<∞ ∀λ >0},
Crap ={f :Rq→R:f continuous,kfkλ,∞<∞ ∀λ <0}. (3.19) We could say that functions are increasing subexponentially (tempered) or decreas-ing subexponentially (rapid), if they are in Ctem or Crap, respectively. Note that we writeCtem+ orCrap+ if we restrict to non-negative functions.