For part iii we suppose that µ
n s
≻ ν
n
and µ
n
→ µ, ν
n
→ ν. Choose pairs X
n
, Y
n
with L X
n
= µ
n
and L Y
n
= ν
n
and X
n
≻ Y
n
almost surely. The laws of X
n
, Y
n
are tight so that we may find a subsequence n
k
and versions ˆ X
n
k
, ˆ Y
n
k
that converge almost surely to a limit X , Y . Now pass to the limit as k
→ ∞ to deduce that µ
s
≻ ν.
3 Existence of the stochastic travelling wave
3.1 The solution from Hx = Ix
0 stretches stochastically
It is straightforward to extend the basic stretching lemma from McKean [5] to deterministic equa- tions with time dependent reactions, as follows. Since it plays a key role in this paper, we present
the proof with the small changes that are needed.
Lemma 7. Consider the deterministic heat equation
u
t
t, x = u
x x
t, x + Hut, x, t, x, t
0, x ∈ R, where H : [0, 1]
× [0, T ] × R → R is a measurable function that is Lipschitz in the first variable, uniformly over t, x.
Suppose u and v are mild solutions, taking values in [0, 1], and u, v ∈ C
1,2
0, T ] × R. Suppose that u0 crosses v0. Then ut crosses vt at all t
∈ [0, T ].
Proof Consider w : [0, T ]
× R → [−1, 1] defined as w = u − v. Define Rt, x =
Hut, x, t, x − Hvt, x, t, x
ut, x − vt, x
I ut, x 6= vt, x. R is bounded and w is a mild solution to w
t
= w
x x
+ wR. We now wish to exploit a Feynman-Kac representation for w. Let Bt : t
≥ 0 be a Brownian motion, time scaled so that its generator is the Laplacian, and defined on a filtered probability space Ω,
F
s
: s ≥ 0, P
x
: x ∈ R, where under
P
x
, B starts at x. Fix t 0. Then for s ∈ [0, t,
M s = wt − s, Bs exp
Z
s
Rt − r, Br d r
is a continuous bounded F
s
martingale and hence has an almost sure limit M t as s ↑ t. As s ↑ t one has wt
− s, Bs → w0, Bt use the fact that wr, x → w0, x as r ↓ 0 for almost all x. For any
F
s
stopping time τ satisfying τ ≤ t we obtain from E
x
[M 0] = E
x
[M τ ∧ s] by letting s
↑ t, that wt, x = E
x
[M τ] = E
x
wt
− τ, Bτ exp Z
τ
Rt − r, Br d r
.
10 Suppose wt, x
1
0 for some x
1
, in particular x
1
≥ θ wt. Consider the stopping time τ =
inf
≤s≤t
{s : |Ms = 0} ∧ t. Then E
x
1
[M τ] = E
x
1
[M tIτ = t] = wt, x
1
0. From this we can
445
construct a deterministic continuous path ξs : s ∈ [0, t] such that ξ0 = x
1
and wt − s, ξs
0 for 0 ≤ s ≤ t. Now take x
2
x
1
. Consider another stopping time defined by τ
∗
= inf
≤s≤t
{s : Bs = ξs} ∧ t. We claim M
τ
∗
≥ 0 almost surely under P
x
2
. Indeed, on {τ
∗
t} this is immediate from the construction of
ξ. On {τ
∗
= t} we have Bτ
∗
= Bt ≥ ξt. Since w0, ξt 0 we know that ξt ≥ θ
u0 − v0 and the assumption that u0 crosses v0 ensures that w0, Bt ≥ 0 and hence M
τ
∗
≥ 0. Applying 10, with x = x
2
and τ replaced by τ
∗
, we find that wt, x
2
≥ 0 when x
2
≥ x
1
. If θ
ut − vt ∈ −∞, ∞ then we can choose x
1
arbitrarily close to θ
ut − vt and the proof is finished. In the cases
θ ut − vt = −∞ we may pick x
1
arbitrarily negative and in the case
θ ut − vt = +∞ there is nothing to prove.
By using a Wong-Zakai result for approximating the stochastic equation 1 by piecewise linear noises, we shall now deduce the following stretching lemma for our stochastic equations with white
noise driver.
Proposition 8. Suppose that u, v are two solutions to 1 with respect to the same Brownian motion. Then, for all t
0, i if u0 crosses v0 almost surely then ut crosses vt almost surely;
ii if u0, v0 ∈ D and u0 ≻ v0 almost surely then ut ≻ vt almost surely.
Proof Define a piecewise linear approximation to a Brownian motion W by, for ε 0,
W
ε t
= W
k ε
+ ε
−1
t − kεW
k+1ε
− W
k ε
for t ∈ [kε, k + 1ε] and k = 0, 1, . . .
Then the equation du
ε
d t = u
ε x x
+ f u
ε
+ gu
ε
dW
ε
d t ,
u
ε
0 = u0 can be solved succesively over each interval [k
ε, k + 1ε], path by path. If u solves 1 with respect to W then we have the convergence
u
ε
t, x → ut, x in L
2
for all x ∈ R, t ≥ 0.
We were surprised not to be able to find such a result in the literature that covered our assumptions. The closest papers that we found were [8], whose assumptions did not cover Nemitski operators
for the reaction and noise, and [12], which proves convergence in distribution for our model on a finite interval. Nevertheless this Wong-Zakai type result is true and can be established by closely
mimicking the original Wong-Zakai proof for stochastic ordinary differential equations. The details are included in section 2.6 of the thesis [13]. We note that the proof there, which covers exactly
equation 1, would extend easily to equations with higher dimensional noises. Also it is in this proof that the hypothesis that f , g have continuous thrid derivatives is used.
In a similar way we construct v
ε
with v
ε
0 = v0. For all k, all paths of u
ε
and v
ε
lie in C
1,2
kε, k + 1ε] × R. By applying Lemma 7 repeatedly over the intevals [kε, k + 1ε] we see that u
ε
t crosses v
ε
t for all t ≥ 0 along any path where u0 crosses v0. We must check that this is preserved in the limit. Fix t
0. There exists ε
n
→ 0 so that for almost all paths u
ε
t, x → ut, x and v
ε
t, x → vt, x for all x ∈ Q. 446
Fix such a path where in addition u0 crosses v0. Suppose that θ
ut − vt ∞. Arguing as in part iii of Lemma 5 we find that lim sup
n →∞
θ u
ε
n
t − v
ε
n
t ≤ θ ut − vt. Now choose
y ∈ Q with y θ
ut − vt. Taking n large enough that y θ u
ε
n
t − v
ε
n
t we find, since u
ε
n
t crosses v
ε
n
t, that u
ε
n
t, y ≥ v
ε
n
t, y. Letting n → ∞ we find ut, y ≥ vt, y. Now the continuity of the paths ensures that ut crosses vt. For part ii it remains to check that
τ
a
ut crosses vt. But this follows from part i after one remarks that if u solves 1 then so too does
τ
a
ut : t ≥ 0.
Corollary 9. i Let u, v be solutions to 1 satisfying
L u0
s
≻ L v0. Then L ut
s
≻ L vt for all t ≥ 0.
ii Let u be the solution to 1 started from Hx = I x 0. Then L ˜ut
s
≻ L ˜us for all ≤ s ≤ t.
Proof For part i, we may by Strassen’s theorem find versions u0 and v0 that satisfy u0 ≻ u0
almost surely. The result then follows by from Lemma 8 ii and uniqueness in law of solutions. For part ii we shall, when
µ ∈ M D, write Q
µ t
for the law of ut for a solution u to 1 whose initial condition u0 has law
µ. We write ˜ Q
µ t
for the centered law of ˜ ut. We write Q
H t
and ˜ Q
H t
in the special case that
µ = δ
H
. Since H is less streched than any φ ∈ D we know that Q
H s
s
≻ Q
H
= δ
H
for any s ≥ 0. Now set µ = Q
H s
and apply part i to see that Q
H t+s
= Q
µ t
s
≻ Q
H t
where the first equality is the Markov property of solutions. This shows that t → Q
H t
is stochastically increasing. By Lemma 6 ii the family t
→ ˜ Q
H t
is also increasing. The stochastic monotonicity will imply the convergence in law of ˜
ut on a larger space, as explained in the proposition below. Define
D
c
= {decreasing, right continuous φ : R → [0, 1]}. 11
Then D
c
is a compact space under the L
1 l oc
topology: given a sequence φ
n
∈ D
C
then along a suitable subsequence n
′
the limit lim
n
′
→∞
φ
n
′
x exists for all x ∈ Q; then φ
n
′
→ φ where φx = lim
y ↓x
ψ y is the right continuous regularization of ψx = lim sup
n
′
→∞
φ
n
′
x.
Proposition 10. Let u be the solution to 1 started from Hx = Ix 0. Then ˜
ut, considered as random variables with values in
D
c
, converge in distibution as t → ∞ to a limit law ν
c
∈ M D
c
.
Proof Choose t
n
↑ ∞. Then by Strassen’s Theorem 9 we can find D valued random variables U
n
with law L U
n
= L ˜ut
n
and satisfying U
1
≺ U
2
≺ . . . almost surely. Note that U
n
0 = a and that U
n
has continuous strictly negative derivatives by Theorem 3 ii. The stretching pre-order, together with Lemma 5 v, implies that almost surely
U
n+1
x ≥ U
n
x for x ≥ 0 and U
n+1
x ≤ U
n
x for x ≤ 0. Thus the limit lim
n →∞
U
n
x exists, almost surely, and we set U to be the right continuous modifi- cation of lim sup U
n
. This modification satisfies U
n
x → Ux for almost all x, almost surely. Hence U
n
→ U in D
c
, almost surely, and the laws L ˜u
t
n
converge to L U in distribution. We set ν
c
to 447
be the law L U on D
c
. To show that L ˜u
t
→ ν it suffices to show that the limit does not depend on the choice of sequence t
n
. Suppose s
n
is another sequence increasing to infinity. If r
n
is a third increasing sequence containing all the elements of s
n
and t
n
then the above argument shows that
L ˜u
r
n
is convergent and hence the limits of L ˜u
s
n
and L ˜u
t
n
must coincide.
Remark We do not yet know that the limit ν
c
is supported on D. We must rule out the possibility
that the wavefronts get wider and wider and the limit ν
c
is concentrated on flat profiles. We do this by a moment estimate in the next section. Once this is known, standard Markovian arguments in
section 3.3 will imply that ν = ν
c
|
D
, the restriction to D, is the law of a stochastic travelling wave.
3.2 A moment bound
