Localization of solutions to stochastic porous media equations: finite speed of propagation ∗

It is proved that the solutions to the slow diffusion stochastic porous media equation $dX-{\Delta}( |X|^{m-1}X )dt=\sigma(X)dW_t,$ $ 1< m\le 5,$ in $\mathcal{O}\subset\mathbb{R}^d,\ d=1,2,3,$ have the property of finite speed of propagation of disturbances for $\mathbb{P}\text{-a.s.}$ ${\omega}\in{\Omega}$ on a sufficiently small time interval $(0,t({\omega}))$.


Introduction
Let O be a bounded and open domain of R d , d = 1, 2, 3, with smooth boundary ∂O.
Consider the stochastic porous media equation where m ≥ 1, W t is a Wiener process in L 2 (O) of the form (1.2) k=1 is a sequence of independent Brownian motions on a filtered probability space {Ω, F, F t , P} while {e k } k∈N is an orthonormal system in L 2 (O) and σ(X where {µ k } is a sequence of nonnegative numbers.We assume that e k ∈ C 2 (O) and σ(X(s))dW s . (1.7) Here we use the standard notation L p (E; B), p ∈ [0, ∞], for a measure space (E, E, µ) and a Banach space B, i.e., L p (E; B) denotes the space of all B-valued measurable maps f : The main result of this work, Theorem 2.3 below, amounts to saying that if 1 < m ≤ 5, which is the case of slow diffusion under stochastic perturbation, then the process X = X(t, •) has the property of finite speed propagation of disturbances in the following sense (see [4] , for P-a.e. ω ∈ Ω.In this sense, we speak about finite speed of propagation of X(t).This localization property for stochastic porous media equations has resisted its proof for quite some time, because the stochastic perturbation is a serious obstacle to adapt localization proofs and techniques from the known deterministic case.This lock was broken by the results in [6], and, particularly [9], which allow to transform the problem to a deterministic partial differential equation (PDE) with random coefficients.This latter PDE, however, is not of porous media or any other known type, so that the necessary estimates become much more complicated, but eventually lead to success.
We mention that in the case 0 < m < 1 (fast diffusion) and if d = 1, also for m = 0, the solution X = X(t, x) has a finite extinction property with positive probability (see [7], [6] respectively) which also can be seen as a localization property of stochastic flows associated with equation (1.1).
The main result, Theorem 2.3, is formulated in Section 2 and proved in Section 3 via some arguments inspired by the local energy method of S.N.Antontsev [1] (see also [2], [3], [4], [11], [12], [18] for some recent results on the localization of solutions to deterministic porous media equations).However, the overlap is not large.In a few words, the idea of the proof is to reduce equation (2.5) to a random partial differential equation on (0, T ) × O and combine the energy method from [1]- [3], with some sharp L ∞ estimates obtained in the authors' work [9].
Here, the discussion is confined to stochastic porous media equations with Dirichlet homogeneous boundary conditions because the previous existence theory we invoke and use here was developed so far in this case only.However, one might expect that everything extends mutatis mutandis to the Neumann reflection conditions on boundary.
As regards the case O = R d , this still remains open.We shall use standard notations and results for spaces of infinite dimensional adapted stochastic processes (see [10], [15]).

The main result
Proposition 2.1.Assume that x ∈ L m+1 (O).Then equation (1.1) has a unique strong solution X.If x ≥ 0 a.e. in O, then X ≥ 0 a.e. in Ω×(0, T )×O and (2.1) Remark 2.2.Existence and uniqueness, as well as nonnegativity of solutions to equation (1.1) has been discussed in several papers (see [5], [6], [17]).But the notion of solution was different.More precisely, solutions were not required to satisfy (1.6), but only that is a continuous process in H 1 0 (O), and that (1.7) holds with the Laplacian in front of the ds-integral.We refer to [16] for a detailed discussion.In the present paper, we need the stronger notion of solution as in (1.5)-(1.7).For very recent results on existence of such "strong" solutions for general SPDE of gradient type, including our situation as a special case, we refer to [13].
. By a suitable regularization, we apply Itô's formula for ϕ in (2.5) and obtain that Taking into account that (2.4) Localization of solutions to stochastic porous media equations: finite speed of propagation Moreover, arguing as in [5], [6], that is, by applying the Itô formula to the function This yields via the Burkholder-Davis-Gundy inequality (see the proof of Theorem 2.2 in [6]) that for some α > 0 and, therefore, and weakly star in L ∞ (0, T ; Moreover, by (2.4) it follows that where η ∈ β(X) a.e. in Ω × (0, T ) × O. Below, we are only concerned with small T > 0, so we may assume that T ≤ 1.Furthermore, for a function g : [0, 1] → R, we define its α-Hölder norm, α ∈ (0, 1), by Now, we are ready to formulate the main result.
As explicitly follows from the proof, the function t → r(t) is a process adapted to the filtration {F t }.
It should be mentioned also that the assumption x ≥ 0 in O was made only to give a physical meaning to the propagation process.
The conditions m ≤ 5 and x ∈ L ∞ (O) might seem unnatural, but they are technical assumptions required by the work [9] on which the present proof essentially relies.

Proof of Theorem 2.3
For the proof we shall take ξ 0 = 0 ∈ O and set B r = B r (0).The method of the proof relies on some sharp integral energy type estimates of X = X(t) on arbitrary balls B r ⊂ O.
Before we introduce our crucial energy functional φ in (3.14) below and explaining the idea of the proof subsequently, we need some preparations by a few estimates on the solution y to (3.2).Everywhere in the following we fix α ∈ (0, 1  2 ), α > 0 and assume that x ≥ 0 so that (3.4) holds and fix T ∈ (0, 1].By Green's formula, it follows from (3.2) that for all ψ ∈ C ∞ 0 (O).Fix r > 0 and let ρ ε ∈ C ∞ (R + ) be a cut-off function such that ρ ε (s) = 1 for 0 ≤ s ≤ r + ε, ρ ε (s) = 0 for s ≥ r + 2ε and for χ ε = 1 [r+ε,r+2ε] , uniformly in s ∈ [0, ∞).Roughly speaking, this means that ρ ε is a smooth approximation On the other hand, we have where , the above calculation is justified.) Everywhere in the following, the estimates are taken P-a.s. on the set Ω α H,R ∩ Ω δ(R) T .
We set B ε r = B r+2ε \ B r+ε .Then, by (3.10), (3.11), we see that On the other hand, we have We introduce the energy function In order to prove (2.6), our aim in the following is to show that φ satisfies a differential inequality of the form where 0 < θ < 1 and 0 < δ < 1 and from which (2.6) will follow.Taking into account that function φ is absolutely continuous in r, we have by (3.9), a.e. on (0, r 0 ), Then, letting ε → 0 in (3.12), (3.13), we obtain that by the definition of δ(R).

.21)
In order to estimate the surface integral from the right-hand side of (3.21), we invoke the following interpolation-trace inequality (see, e.g., Lemma 2.2 in [12]) for all σ ∈ [0, 1] and θ = (d(1 We shall apply this inequality for z = (y m e −µ ) m and σ = 1 m .We obtain, by (3.17) that , where, as will be the case below, C is a positive function , independent of t and r, which may change below from line to line.

(3. 15 )
In order to estimate the right-hand side of (3.15), we introduce the following notations