Stability of the stochastic heat equation in $L^1([0,1])$

We consider the white-noise driven stochastic heat equation on $[0,\infty)\times[0,1]$ with Lipschitz-continuous drift and diffusion coefficients $b$ and $\sigma$. We derive an inequality for the $L^1([0,1])$-norm of the difference between two solutions. Using some martingale arguments, we show that this inequality provides some {\it a priori} estimates on solutions. This allows us to prove the strong existence and (partial) uniqueness of weak solutions when the initial condition belongs only to $L^1([0,1])$, and the stability of the solution with respect to this initial condition. We also obtain, under some conditions, some results concerning the large time behavior of solutions: uniqueness of the possible invariant distribution and asymptotic confluence of solutions.


The equation
Consider the stochastic heat equation with Neumann boundary conditions: Here b, σ : R → R are the drift and diffusion coefficients and u 0 : [0, 1] → R is the initial condition. We write formally W (d t, d x) =Ẇ (t, x)d t d x, for W (d t, d x) a white noise on [0, ∞) × [0, 1] based on d t d x, see Walsh [14]. We assume in this paper that b, σ are Lipschitz-continuous: for some constant C b,σ , for all r, z ∈ R, |b(r) − b(z)| + |σ(r) − σ(z)| ≤ C b,σ |r − z|.
( ) Our goals in this paper are the following: • prove some new stability estimates of the solution with respect to the initial condition u 0 ∈ L 1 ([0, 1]); • study the uniqueness of invariant measures and the asymptotic confluence of solutions.
Let us mention that our results extend without difficulty to the case of Dirichlet boundary conditions and to the case of the unbounded domain R (with u 0 ∈ L 1 (R)).
This equation has been much investigated, in particular since the work of Walsh [14]. In [14], one can find definitions of weak solutions, existence and uniqueness results, as well as proofs that solutions are Hölder-continuous, enjoy a Markov property, etc. Let us mention for example the works of Bally-Gyongy-Pardoux [1] (existence of solutions when the drift is only measurable), Gatarek-Goldys [7] (existence of solutions in law), Donati-Pardoux (comparison results and reflection problems), Bally-Pardoux (smoothness of the law of the solution), Bally-Millet-Sanz [3] (support theorem), etc. Sowers [13], Mueller [10] and Cerrai [4] have obtained some results on the invariant distributions and convergence to equilibrium. We

Mild solutions
We now define the classical notion of weak solutions we will use, see Walsh [14]. When we refer to predictability, this is with respect to the filtration ( t ) t≥0 generated by W , that is t = σ(W (A), A ∈ ([0, t] × [0, 1])). We denote by G t (x, y) the Green kernel associated with the heat equation ∂ t u = ∂ x x u on R + × [0, 1] with Neumann boundary conditions, whose explicit form can be found in Walsh [14]. Here we will only use that for some constant C T , for all x, y ∈ [0, 1], all t ∈ (0, T ], see [14], Definition 1. Assume ( ) and consider u 0 ∈ L 1 ([0, 1]). We say that a predictable process u = (u(t, x)) t≥0,x∈ [0,1] with values in R is a mild solution to (1) and if a.s., for a.e. t ≥ 0, x ∈ [0, 1], Da Prato-Zabczyk [5] have checked that for any u 0 ∈ L 2 ([0, 1]), there exists a unique mild solution u to (1). Using a similar approach, together with the estimate one easily checks that for any u 0 ∈ L 1 ([0, 1]), there exists a unique mild solution to (1). Furthermore, for any pair u, v of mild solutions to (1) starting from u 0 , v 0 ∈ L 1 ([0, 1]), for any T > 0, where C T,σ,b depends only on T and on the Lipschitz constants of σ and b. We will prove (5) and (6) for the sake of completeness, which implies uniqueness. A similar approach allows one to prove existence.

Stability in
Our first goal is to prove the following stability estimates in L 1 ([0, 1]).
, consider the two mild solutions u and v to (1) starting from u 0 and v 0 .
where C γ depends only on γ.
Point (i), which shows some stability with respect to the initial condition, is complementary to (6): we can consider a supremum in time, but we can only deal with the L 1 -norm. Point (ii) is quite strong, since it provides some estimates on the time interval [0, ∞). This will allow us to study the large time behavior of solutions when b is non-increasing. To prove these estimates, we rather use the weak form of (1). It seems difficult to obtain such results working directly with the mild equation.

Large time behavior
We now wish to study the uniqueness of invariant measures.  (1) starting from u 0 , (u(t)) = Q for all t ≥ 0. To prove the asymptotic confluence of solutions, we need to strengthen the injectivity assumption.
There is a strictly increasing convex function ρ : (ii) Assume additionally that (1) admits an invariant distribution Q. Then for any u 0 ∈ L 1 ([0, 1]), for u the mild solution to (1) starting from u 0 , u(t) goes in law to Q as t → ∞.
. One may also combine conditions on b and σ. But ( ) also holds if b is C 1 and if b ≤ 0 vanishes reasonably. For example if b(z) = −sg(z) min(|z|, |z| p ) for some p ≥ 1, choose ρ = ερ p with ε small enough and ρ p (z) = z p for z ∈ [0, 1] and ρ p (z) = pz − p + 1 for z ≥ 1. If b(z) = −z − sin z, choose ρ = ερ 3 with ε small enough. One may also consider the case where σ is monotonous with σ vanishing reasonably.
Observe that the asymptotic confluence of solutions may hold even if there is no invariant distribution for (1). Consider e.g. the case where b(x) ≥ c > 0 is non-increasing and σ(x) = x. Then ( ) holds, while solutions should tend to infinity (due to the uniformly positive drift).
Let us now compare Theorems 4 and 5 with known results. The works cited below sometimes concern different boundary conditions, but we believe this is not important.
• Sowers [13] has proved the existence of an invariant distribution supported by C([0, 1]), assuming ( ), that σ is bounded and that b is of the form b(z) = −αz + f (z), for some bounded f and some α > 0. He obtained uniqueness of this invariant distribution when σ is sufficiently small and bounded from below.
• Mueller [10] has obtained some surprising coupling results, implying in particular the uniqueness of an invariant distribution as well as a the trend to equilibrium. He assumes ( ), that σ is bounded from above and from below and that b is non-increasing, with |b(z) − b(r)| ≥ α|z − r| for some α > 0.
• Cerrai [4] assumed that σ is strictly monotonous (it may vanish, but only at one point).
(i) She obtained an asymptotic confluence result which we do not recall here and concerns, roughly, the case b(z) −sg(z)|z| m as z → ±∞, for some m > 1.
(ii) Assuming ( ), she proved uniqueness of the invariant distribution as well as an asymptotic confluence property, under the conditions that for all Thus the main advantages of the present paper are that the uniqueness of the invariant measure requires very few conditions, and we allow σ to vanish (it may be compactly supported).
Example 1. Assume ( ) and that b strictly decreasing. Then there exists at most one invariant then we have asymptotic confluence of solutions. Here to apply [13,10] one needs to assume additionally that σ is bounded from above and from below, while to apply [4], one has to suppose that σ is strictly monotonous.
Example 2. Assume ( ), that b is non-increasing and that σ is strictly monotonous. Then there exists at most one invariant distribution. If furthermore σ is C 1 with 0 < c < σ < C, then we get asymptotic confluence of solutions using [4] or Theorem 5 (here [13,10] cannot apply, since σ vanishes). But now if σ ≥ 0 reasonably vanishes then Theorem 5 applies, which is not the case of [4]: take e.g.
then the invariant measure Q (that exists due to Sowers [13]) is unique and one may show, using the comparison Theorem of Donati-Pardoux [6], that Q is supported by [−1, 1]-valued continuous functions on [0, 1].
However, there are some cases where [13,4] provide some better results than ours.

Example 5.
If σ is small enough and bounded from below and if b(z) = −αz + h(z), with α > 0 and h bounded, then Sowers [13] obtains the uniqueness of the invariant distribution even if b is not non-increasing.

Plan of the paper
In the next section, we briefly prove (5) and (6). Section 3 is devoted to the proof of Theorem 2. Theorems 4 and 5 are checked in Section 4. We briefly discuss the multi-dimensional equation in Section 5 and conclude the paper with an appendix containing technical results.

Mild solutions
In the whole section, T > 0 is fixed. For u 0 ∈ L 1 ([0, 1]) and t ∈ [0, T ], we easily get, using the Cauchy-Schwarz inequality in y and integrating in x using (2): This implies (5). Consider now two mild solutions u, v to (1) Integrating in x using (7) and (2), we deduce that for all t ∈ [0, T ], Iterating once this formula and using that for all 0 ≤ r ≤ t, . We have checked (6).
Thus we can pass to the limit as ε → 0 in (10) and get, a.s., Step 3. Using ( ), there holds, for all r 1 , z 1 , r 2 , z 2 in R, Indeed, it suffices, by symmetry, to check that sg( Using (9), it is thus routine to make n tend to infinity in (11) and to obtain, a.s., For the last term, we used that, by the Plancherel identity, setting for simplicity Using (12) and then (9), We used here again that (e k ) k≥1 is an orthonormal basis of L 2 ([0, 1]).

where C b,γ,T depends only on b, γ, T . (ii) Assume that b is non-increasing. Then for all
where C γ depends only on γ.
Proof. Let us first prove point (i). Let C b be the Lipschitz constant of b. Denote by L t the RHS of (8). The Itô formula yields The right hand side M t of this inequality is a nonnegative continuous local martingale starting from Point (i) immediately follows (with C b,γ,T = C γ e C b γT ).
We now check point (ii). Since b is non-increasing, Proposition 6 yields which is a nonnegative continuous local martingale starting from This ends the proof.
Finally, we extend the previous estimates to general initial conditions.
Proof of Theorem 2. We divide the proof into two steps.
Step 2. We now prove the desired estimates. For u 0 and v 0 in L 1 ([0, 1]), we consider u n 0 and v n 0 continuous with ||u n We denote by u, v, u n , v n the corresponding mild solutions to (1). For each n ≥ 1, we may apply Corollary 7-(i) to u n , v n . Using the Fatou Lemma and Step 1, we deduce that for all T > 0, all γ ∈ (0, 1), If furthermore b is non-increasing, we get similarly, using Corollary 7-(ii), for any T > 0, any γ ∈ (0, 1), Since C γ does not depend on T , we can let T increase to infinity. Next, using again Step 1, Corollary 7-(ii) and ( ), we deduce that for any T > 0, any γ ∈ (0, 1), To conclude the proof, it suffices to let T increase to infinity.

Large time behavior
We now prove the uniqueness of the invariant measure.
Finally, we give the Proof of Theorem 5. Point (ii) is immediately deduced from point (i). Let thus u 0 , v 0 ∈ L 1 ([0, 1]) be fixed and let u, v be the corresponding mild solutions to (1). We know from ( ), the Jensen inequality and Theorem 2-(ii) that a.s., Using Lemma 9, one may thus find an increasing (deterministic) sequence (t n ) n≥1 such that ρ(||u(t n ) − v(t n )|| L 1 ([0,1]) ) tends to 0 in probability, so that ||u(t n ) − v(t n )|| L 1 ([0,1]) also tends to 0 in probability (because due to , ρ is strictly increasing and vanishes only at 0). Next, we use Theorem 2-(ii) with e.g. γ = 1/2 to get, We used here that conditionally on t n , (u(t n + t, x)) t≥0,x∈[0,1] is a mild solution to (1), starting from u(t n ) (with a translated white noise). Thus for any ε > 0, using the Markov inequality which tends to 0 as n → ∞ by dominated convergence. Consequently, as n tends to infinity, ∆ t n tends to 0 in probability.

Toward the multi-dimensional case?
Consider now a bounded smooth domain D ⊂ R d , for some d ≥ 2. Consider the (scalar) equation with some Neumann boundary condition.
It is well known that the mild equation makes no sense in such a case, since even if σ(u) is bounded, G t−s (x, y)σ(u(s, y)) does not belong to L 2 ([0, t] × D). The existence of solutions is thus still an open problem. See however Walsh [14] when σ ≡ 1, b(u) = αu and Nualart-Rozovskii [9] when σ(u) = u, b(u) = αu. In these works, the authors manage to define some ad-hoc notion of solutions, using that the equations can be solved more or less explicitly. In the literature, one almost always considers the simpler case where the noise W is colored, see Da Prato-Zabczyk [5].
However the weak form makes sense: a predictable process u = (u(t, x)) t≥0,x∈D is a weak solution if a.s., and if for all function ϕ ∈ C 2 b (D) (with Neumann conditions on ∂ D), all t ≥ 0, a.s., Assume now that σ(0) = b(0) = 0. Then v ≡ 0 is a weak solution. Furthermore, the estimate of Theorem 2-(i) a priori holds. Choosing u 0 ∈ L 1 (D) and v 0 = 0, this would imply (19). Unfortunately, we are not able to make this a priori estimate rigorous.
But following the proof of Proposition 6 and Corollary 7, one can easily check rigorously the following result. For (e k ) k≥1 an orthonormal basis of L 2 (D), set B k t = t 0 D e k (x)W (ds, d x). For u 0 ∈ L ∞ (D) and n ≥ 1, consider the solution (see Pardoux [11]) to Then if σ(0) = b(0) = 0, for any γ ∈ (0, 1), any T > 0, where the constant C b,γ,T depends only on γ, T, b (the important fact is that it does not depend on n). Passing to the limit formally in (20) would yield (19). Unfortunately, (20) is not sufficient to ensure that the sequence u n is compact and tends, up to extraction of a subsequence, to a weak solution u to (18). But this suggests that, when σ(0) = b(0) = 0, (generalized) weak solutions to (18) do exist and satisfy something like (19), possibly in an extended sense. Recall that when D = R d , b ≡ 0 and σ(z) = z, the (generalized) solution is known to exist and is called the super Brownian motion, see e.g. Krylov [8]. The super Brownian motion takes its values in the set of finite measures, which is the closure of L 1 in some sense. Such a construction is possible, essentially because in this case, ||u(t)|| L 1 = ||σ(u(t))|| 2 L 2 .
Next, we state a technical result on a.s. converging integrals.
Lemma 9. Let (K t ) t≥0 be a nonnegative process. Assume that A ∞ = ∞ 0 K t d t < ∞. Then one may find a deterministic sequence (t n ) n≥1 increasing to infinity such that K t n tends to 0 in probability as n → ∞.
Proof. Consider a strictly increasing continuous concave function φ : R + → [0, 1] such that φ(0) = 0. Using the Jensen inequality, we deduce that which tends to 0 as T → ∞ by the dominated convergence Theorem. As a consequence, we may find a sequence (t n ) n≥1 such that lim n E[φ(K t n )] = 0. The conclusion follows.
Finally, we prove a technical result on coupling.

Lemma 10.
Consider two probability measures µ, ν on a Polish space . Let Ψ : × → R + be continuous and assume that Ψ(x, y) > 0 for all x = y. If there is a sequence of × -valued random variables (X n , Y n ) n≥1 such that for all n ≥ 1, (X n ) = µ and (Y n ) = ν and if lim n E[Ψ(X n , Y n )] = 0, then µ = ν.