Finite time extinction for the 1D stochastic porous medium equation with transport noise

We establish finite time extinction with probability one for weak solutions of the Cauchy-Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate $O(t^\frac{1}{2})$ whereas the support of solutions to the deterministic PME grows only with rate $O(t^{\frac{1}{m{+}1}})$. The rigorous proof relies on a contraction principle up to time-dependent shift for Wong-Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.


Introduction
We study the stochastic porous medium equation with Stratonovich transport noise du = ∆u m dt + ν∇u • dB t (1) in the space-time cylinder Q = D×(0, T ) for some bounded convex domain D ⊂ R d with C 2 boundary ∂D and time horizon T ∈ (0, ∞]. Here u ≥ 0 denotes the socalled density variable, B is a d-dimensional standard Brownian motion and m > 1 as well as ν > 0 are parameters. Moreover, the Stratonovich differential ν∇u • dB is an abbreviation for the expression 1 2 ν 2 ∆u dt + ν∇u · dB. We are in particular interested in studying the corresponding Cauchy-Dirichlet problem, i.e., we supplement the stochastic PDE (1) with u(x, 0) = u 0 (x), x ∈D, u(x, t) = 0, (x, t) ∈ ∂D × (0, T ]. The initial condition is given by a non-negative smooth function u 0 ∈ C ∞ cpt (D) with compact support in D. In particular, (2) and (3) are compatible to any order.
The main result of the present work is that weak solutions (in the precise sense of Definition 1 below) to the 1D Cauchy-Dirichlet problem of the SPME (1)- (3) become extinct in finite time on a set of full probability. For a mathematically precise statement we refer the reader to Theorem 3 below. To the best of the author's knowledge, this is the first result establishing finite time extinction in This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie Grant Agreement No. 665385 . I am very grateful to M. Gerencsér for the suggestion of this problem. I want to thank J. Maas and M. Gerencsér for helpful discussions on the subjects of this paper, as well as valuable feedback on an earlier version of this manuscript. Special thanks go to F. Cornalba for suggesting the additional κ-truncation in Proposition 5. the slow diffusion regime m > 1 for the stochastic porous medium equation. In contrast, finite time extinction in the fast diffusion regime 0 < m < 1 is well known in several cases. For instance, Barbu, Da Prato and Röckner establish finite time extinction for porous medium type models with multiplicative noise in the works [2], [3] and [4]. Gess treats the case of stochastic sign fast diffusion equations in [15]. In a very recent work of Turra [20], finite time extinction in the fast diffusion regime is established for transport noise. The slow diffusion case with transport noise, however, is left open.
The underlying idea of the present work is to exploit that the support of solutions to the deterministic porous medium equation grows only with rate O(t 1 m+1 ) whereas Brownian motion spreads on average with a faster rate O(t 1 2 ). Hence one may expect that the effect of the random advection term in (1) eventually dominates the non-linear diffusion term, thus leading to finite time extinction on a set of full probability. Let us briefly mention the three main ingredients which provide a rigorous justification for this heuristic (for a more detailed and mathematical account on the strategy, we refer the reader to Section 3). Along the way we provide connections to the-for the purposes of this work-relevant parts of the by now extensive literature for the stochastic porous medium equation.
In a first step, we recover the unique weak solution (in the sense of Definition 1 below) to the Cauchy-Dirichlet problem of the SPME (1)-(3) by means of a suitable Wong-Zakai approximation. We then perform, on the level of the approximate solutions, a simple stochastic flow transformation yielding smooth solutions to a deterministic porous medium equation, but now with two copies of a mollified Brownian path as the lateral boundary. In principle, we would then like to remove the mollification parameter to formally obtain a solution to the deterministic PME with two copies of a Brownian path as the lateral boundary. We may view this solution-again at least on a formal level-as a solution to the full space porous medium equation by trivially extending it outside of the domain. We finally put a Barenblatt profile above the solution at the initial time. It follows by the comparison principle that finite time extinction happens once the domain with two copies of a Brownian path as the lateral boundary is pushed outside the support of such a Barenblatt solution. Since the support of the Barenblatt solution has finite speed of propagation of order O(t 1 m+1 ), this actually occurs with probability one. However, two technical issues being linked to each other arise in the above strategy. The first is concerned with the proposed limit passage after the stochastic flow transformation. The problem is that the domains on which the approximate and transformed solutions are supported are not monotonically ordered as one sends the mollification scale to zero. This fact together with the desire to make use of the comparison principle motivated us to perform this limit passage in the framework of viscosity theory by means of the technique of semi-relaxed limits. This allows us to rigorously construct a maximal subsolution for the deterministic porous medium equation with two copies of a Brownian path as the lateral boundary (to be understood in the precise sense of Definition 14 below). To the best of the author's knowledge, the present work seems to be the first instance to make use of viscosity theory to study the Dirichlet problem for the stochastic porous medium equation after stochastic flow transformation.
A notion of viscosity solution for the full space deterministic porous medium equation has been developed by Caffarelli and Vázquez [6] (see also the work of Vázquez and Brändle [22]). However, as a consequence of the degeneracy of the porous medium equation the usual notion of supersolutions from [7] had to be adapted. But since we only need to talk about subsolutions for our purposes, this is actually of no concern for us.
However, by taking a semi-relaxed limit it is by no means clear anymore how this maximal subsolution and the transformed unique weak solution to (1)-(3) relate to each other. In order to close the argument, we need to make sure that the unique weak solution of the original problem (1)-(3) is dominated after stochastic flow transformation by the constructed maximal subsolution in a suitable sense. It turns out that we can justify this once we provided convergence of the Wong-Zakai approximations in a strong enough topology like L 1 . Let us mention at this point the recent work of Fehrman and Gess [12], who prove rough path well-posedness for nonlinear gradient type noise using a kinetic formulation which in particular implies Wong-Zakai type results, but with the equation being posed on the torus.
There has been recently a lot of focus on providing well-posedness for stochastic porous medium equations in an L 1 framework. The corresponding notion of solution is the one of entropy solutions first put forward in the work of Dareiotis, Gerencsér and Gess [8], and the theory is based on a quantitative L 1 contraction principle for such solutions. In the work [8], the authors consider the case of nonlinear multiplicative noise with periodic boundary conditions. The subsequent work of Dareiotis and Gess [10] treats the regime of nonlinear conservative gradient noise (again with periodic boundary conditions), whereas the recent work [11] by Dareiotis, Gess and Tsatsoulis provides the corresponding theory of [8] for the Dirichlet problem on smooth domains.
Following ideas developed in [8], we roughly speaking aim to establish L 1 convergence of the sequence of Wong-Zakai approximations. However, as we are concerned in this work with the Cauchy-Dirichlet problem for the SPME with transport noise, the actual implementation differs in some aspects to the above mentioned works. Most importantly, since two different approximate solutions are advected with different speeds (as a consequence of the two associated different mollification scales) it is natural to introduce a time-dependent, random shift function and to compare the Wong-Zakai approximations only after shifting. We then essentially establish quantitative L 1 contraction for this sequence of shifted densities, see Proposition 5 below for the mathematically precise formulation. This result may be of independent interest.
Notation. Throughout the paper, we fix a filtered probability space (Ω, F, F, P) with filtration F = (F t ) t∈[0,∞) which is subject to the usual conditions, i.e., the filtration F is right-continuous and F 0 is P-complete. By B = (B t ) t∈[0,∞) we denote a standard d-dimensional F-Brownian motion. For a given time horizon T ∈ (0, ∞] we let P T be the predictable σ-field on Ω T := Ω × [0, T ]. Let D ⊂ R d be an open domain. (In the one-dimensional setting, we will instead use the notation I ⊂ R for a bounded open interval.) The space of all smooth functions with compact support in D is denoted by C ∞ cpt (D). The space H 1 0 (D) is the usual space of Sobolev functions with zero trace on the boundary. Its topological dual space will be denoted by H −1 . The space of H −1 valued, continuous and F-adapted stochastic processes intersected with L q (Ω T , P T ; L q (D)) is given by H −1 q (D). The space of L 2 (D) valued, continuous and F-adapted stochastic processes intersected with the space L 2 (Ω T , P T ; H 1 0 (D))) (resp. the space L 2 (Ω T , P T ; H 1 (D))) is L 2 0 (D) (resp. L 2 (D)). If we want to specify the target values for a function space, we will include this in the notation. E.g., C ∞ cpt (D; [0, ∞)) is the space of all non-negative smooth functions with compact support in D. Finally, the space of functions of bounded variation is denoted by BV .
The Lebesgue measure on R d is denoted by L d , the s-dimensional Hausdorff measure by H s . We write χ A for the indicator function of a measurable set A (with respect to any measure space). For two numbers a, b ∈ R we abbreviate a ∨ b for their maximum respectively a ∧ b for their minimum. Finally, we define a + := a ∨ 0 and sign + (a) := χ (0,∞) (a) for a ∈ R.

Main result
The main result of the present work establishes that weak solutions of the onedimensional stochastic porous medium equation with Stratonovich transport noise (1)-(3) almost surely have the finite time extinction property. Before we provide the precise statement, let us introduce the notion of weak solutions.
be a time horizon as well as u 0 ∈ C ∞ cpt (D; [0, ∞)) be an initial density. A nonnegative stochastic process u ∈ H −1 m+1 (D) is called a weak solution to the Cauchy-Dirichlet problem (1)-(3) of the stochastic porous medium equation (SPME) with initial density u 0 on the space time cylinder for all T ∈ (0, T * ).
Remark 2. Existence and uniqueness of weak solutions in the sense of Definition 1 was established, for instance, in [9,Theorem 3.3]. In fact, the authors consider a much more general class of degenerate quasilinear stochastic PDE, allowing for more general initial conditions and, in particular, for signed solutions.
Being equipped with the notion of weak solutions we may now formulate the main result of the present work, which is finite time extinction with probability one for weak solutions of the 1D stochastic porous medium equation with Stratonovich transport noise.

Outline of the strategy
Let us comment on the strategy for the proof of the main result, Theorem 3. In a nutshell the argument works as follows. By stochastic flow transformation, which in our case of pure transport noise is of particularly easy form, we may consider first the transformed densities Formally, the transformed density v is then subject to the deterministic porous medium equation with initial density v(·, 0) = u 0 and Dirichlet boundary conditions on the lateral boundary t∈[0,∞) (νB t +∂I) × {t}. Extending v trivially to the entire space-time domain R × (0, ∞), and denoting this extension byv, we obtain, at least formally, a solution to the Cauchy problem of the deterministic porous medium equation ∂ tv = ∂ xxv m with initial densityv(·, 0) = u 0 ≥ 0. Now, we may choose a Barenblatt profile B with free boundary ∂I at t = 0 and which strictly dominates the initial density, i.e., {B(·, 0) = 0} = ∂I and u 0 < B(·, 0) on I. By comparison it followsv ≤ B on R × (0, ∞). Since the support of the Barenblatt solution has finite speed of propagation of order O(T 1 m+1 ) (with the implicit constant only depending on the initial density), we deduce the validity of (6). However, since Brownian motion has average spread rate of order O(T 1 2 ) and we assume that m > 1, this entails that finite time extinction happens with probability one: P(T extinct < ∞) = 1.
To make this argument rigorous, we rely on a suitable approximation procedure. To this end, we first consider the d-dimensional setting and fix a finite time horizon T * ∈ (0, ∞). Moreover, choose ρ ∈ C ∞ cpt ((0, 1); [0, ∞)) with´R ρ(r) dr = 1 and define ρ ε := 1 ε ρ( · ε ) for ε > 0. We then introduce the smooth approximations B ε = (B ε t ) t∈[0,∞) to Brownian motion defined via Since the mollifier ρ is supported on positive times, we note that B ε is F-adapted. We also have almost surely that as well as Fix α ∈ ( 1 3 , 1 2 ). For purely technical reasons, we make also use of the classical fact that there is a square integrable random variable C α such that with probability one In particular, for each integer M ≥ 1 and each δ > 0 we can find an ε = ε (M, δ) > 0 such that for all ε ≤ ε it holds We then proceed by considering the inhomogeneous Cauchy-Dirichlet problem Note that the regularization not only comes from the Wong-Zakai approximation but also by choosing strictly positive initial and boundary data. In this way we circumvent the degeneracy at u = 0 of the porous medium operator at the level of the approximations, i.e., we can solve the problem (12)- (14) in a classical and pathwise sense such that the maximum principle applies. Proofs will be provided in the subsequent Section 4.
A key ingredient for the rigorous justification of the comparison argument outlined at the beginning of this section is that the Wong-Zakai type approximations u ε from Lemma 4 recover the unique weak solution u to the Cauchy-Dirichlet problem (1)-(3) in a certain pointwise sense. We obtain this by establishing a sort of contraction principle for the sequence of u ε by means of Kružkov's device of variable doubling [16]. To this end, the proof loosely follows the strategy of Dareiotis, Gerencsér and Gess [8] or Dareiotis and Gess [10] who study entropy solutions for stochastic porous medium type equations posed on the torus. The Cauchy-Dirichlet problem in the framework of entropy solutions was recently studied in the work of Dareiotis, Gess and Tsatsoulis [11]. However, they do not consider the case of gradient type noise. Let us also mention the recent work of Fehrman and Gess [12], who prove rough path well-posedness for nonlinear gradient type noise using a kinetic formulation which in particular implies Wong-Zakai type results, but with the equation being posed on the torus..
As we are concerned in this work with the Cauchy-Dirichlet problem for the SPME with transport noise, and roughly speaking aim to establish L 1 convergence for the Wong-Zakai type approximations u ε , the actual implementation of the doubling of variables technique differs in some aspects to the above mentioned works. First, since two different solutions u ε and uε are advected at different speeds, it seems to be natural to introduce for our purpose a (time-dependent, random) shift function and to compare the solutions only after shifting, i.e., we are led to study L 1 convergence for the sequence . Second, since the introduction of this shift in turn changes the domain on which the equation for u ← ε is posed it is necessary to create a boundary layer in order to apply the doubling of variables method up to the boundary. This is done by means of an additional truncation as follows.
Let ζ δ κ (r) := κ + ζ δ (r−κ). The idea then is to study basically L 1 contraction for the sequence ζ ε κ •u ← ε for ε ≤ κ 2 . Note that for each time slice t ∈ [0, T * ] the function , and for ε ≤ κ 2 it is actually constant in a neighborhood of ∂D − ν(B t −B ε t ). In particular, by introducing such a boundary layer there is no jump of the Neumann data for ζ ε κ • u ← ε across the boundary ∂D − ν(B t −B ε t ). This turns out to be absolutely essential in order to apply the doubling of variables technique up to the boundary.
Proposition 5 (Contraction principle up to time-dependent shift for truncated Wong-Zakai type approximations). For each ε > 0 let u ε denote the unique classical solution to (12)- (14) in the sense of Lemma 4. We extend u ε to a function defined on R d ×[0, T * ] by setting it equal to ε outside of D×[0, T * ]. Denoting this extension again by u ε we then define for all ε > 0 the shifted densities Then for all κ ∈ (0, T * ∧ 1) and all compact sets K ⊂ D there exists a constant C > 0 independent of κ and K, some ϑ > 0 and some ε 0 = ε 0 (κ, K) such that for all ε,ε ≤ ε 0 it holds We may lift this quantitative stability estimate for the truncated and shifted densities κ ∨ u ← ε to qualitative L 1 convergence of the shifted densities u ← ε .
Corollary 6. Let the assumptions and notation of Proposition 5 be in place. Then the sequence of shifted densities u ← ε is Cauchy in C([τ, T * ]; L 1 (Ω×D, P ⊗ L d )) for all positive times τ ∈ (0, T * ). The sequence of shifted densities is moreover Cauchy denote the corresponding limit in this space. Then, u is also the weak limit of the Wong-Zakai type approximations u ε from Lemma 4 in the space L m+1 (Ω T * , P T * ; L m+1 (D)) The final issue concerning the Wong-Zakai type approximations u ε from Lemma 4 is the identification of the limit object u in Corollary 6 as the unique weak solution to the Cauchy-Dirichlet problem (1)-(3) of the stochastic porous medium equation.
Proposition 7. Let u ∈ L m+1 (Ω T * , P T * ; L m+1 (D)) be the limit of the Wong-Zakai approximations u ε in the sense of Corollary 6. Then it holds that u ∈ H −1 m+1 (D), and u is the unique weak solution to the Cauchy-Dirichlet problem (1) From now on we will restrict ourselves to the one-dimensional setting d = 1. We still have to make rigorous the outlined comparison argument. The key ingredient for this will be provided in Section 5.1. It consists of the construction of a maximal subsolutionp max (in the sense of viscosity theory [7], precise definitions will follow in Section 5.1) for the Cauchy-Dirichlet problem of the porous medium equation after stochastic flow transformation The pressure variable p is obtained from the density variable u via the transformation g(u) with g : [0, ∞) → [0, ∞) given by g(r) := m m−1 r m−1 . To the best of the author's knowledge, the present work seems to be the first instance to make use of the pressure formulation to study the stochastic porous medium equation after stochastic flow transformation in the setting of viscosity theory.
The main difficulty for solving (24)-(26) comes from the fact that the lateral boundary consists of two translates of a Brownian trajectory. To overcome the lack of regularity of the lateral boundary, we first consider the approximate initialboundary value problem (see Lemma 15) and then pass to the limit ε → 0 by means of the technique of semi-relaxed limits, see for instance [7,Section 6]. In this way we obtain a maximal subsolution to the problem (24)-(26) in the sense of viscosity theory [7], see Proposition 16. The main motivation for working in the framework of viscosity theory is the non-monotonicity of the underlying space-time domains t∈(0,T * ) (νB ε t +I) × {t} as ε → 0 which necessitates the usage of a relaxed limit. In this way, however, the interpretation of the lateral boundary condition (26) in a strong sense is lost in the limit. It is well-known that boundary regularity for solutions to stochastic PDEs with gradient type noise and given Dirichlet data proves to be a delicate issue. This already shows up in the linear case, see, for instance, the works by Krylov [17] and [18]. For a recent work in the semilinear regime, we refer the reader to [14]. On the other side, "continuity up to the lateral boundary" is inessential for our purposes and anyway not expected, if at all, to be obtained by the methods in this work. Note that the maximal subsolutionp max of course depends on the realization of Brownian motion, and is therefore random. However, since the construction ofp max is ultimately a purely deterministic consequence of the probabilistic facts (8)-(11), we obtain the maximal subsolution in a pathwise sense on a set of full probability.
The proof of Theorem 3, which is the content of Section 5.5, then roughly speaking proceeds as follows. Denoting by p the map which we obtain from the unique weak solution u of (1)-(3) in the sense of Definition 1 by first applying a stochastic flow transformation and then a density-to-pressure transformation, we have the estimate p ≤p max , see Proposition 17. This bound is essentially a combination of the following facts: i) The solution to (27)-(29) may in fact be obtained from the Wong-Zakai type approximations u ε of (12)-(14) by first applying a stochastic flow transformation and then a density-to-pressure transformation. ii) The Wong-Zakai approximations u ε (or more precisely, their shifted counterparts (22)) converge on each positive time slice in L 1 to the unique weak solution of the Cauchy-Dirichlet problem (1)-(3), see Corollary 6. iii) The maximal subsolution to (24)-(26) dominates the upper semi-relaxed limit (with respect to parabolic space-time cylinders) of the transformed u ε . However, taking a semi-relaxed limit allows to compare with the transformed density u by means of the previous two items. Finally, we compare the maximal subsolutionp max to a Barenblatt profile (written in the pressure variable) as outlined in the heuristic argument. We make use of the comparison principle in the framework of viscosity solutions for the deterministic porous medium equation as developed by Caffarelli and Vázquez [6] resp. Vázquez and Brändle [22]. The remaining argument after comparing with the Barenblatt profile, in particular the derivation of (6), then works as already sketched before.

Recovering weak solutions by Wong-Zakai type approximations
4.1. Proof of Lemma 4 (Wong-Zakai type approximation). We make use of a usual trick of avoiding the degeneracy of the porous medium operator, see for instance [21, Proof of Theorem 5.5]. Let ε > 0 be fixed and choose a bounded smooth function a ε : By the choice of a ε and since B ε t as defined in (7) is smooth on [0, T * ] almost surely, we can make use of standard quasilinear theory [19] to solve the Cauchy-Dirichlet problem (12)- (14) in a classical sense, i.e., we obtain with probability one a classical solution u ε ∈ C 2,1 x,t (D×[0, T * ]) ∩ C ∞ (D×(0, T * )) such that the equations (12)- (14) are satisfied pointwise everywhere.
We can infer from the maximum principle that ε ≤ u ε (x, t) ≤ ε + u 0 L ∞ (D) holds true for all (x, t) ∈D×[0, T * ] as it is asserted in (15). The derivation of the energy estimate (16) is standard: multiply the equation with u ε , integrate over D and use the regularity of u ε to integrate by parts in the spatial differential operators. Note in this respect that as a consequence of (15) and (14) it holds n ∂D · ∇u m ε ≤ 0 on ∂D, where n ∂D is the exterior unit normal vector field along the C 2 manifold ∂D. This is the only reason for the inequality sign in (16) as we may compute for the second term We proceed with the bound (17) for the time derivative. Multiplying the equation (12) with ∂ t u m ε , integrating over D, performing an integration by parts in the term with the porous medium operator and estimating the transport term by Hölder's and Young's inequality yields for all t ∈ (0, T * ) the estimatê Multiplying this bound with t and integrating the resulting estimate over (0, T ) we may infer using also (16) and (15) for all T ∈ (0, T * ). Moreover, it follows from (7) and Doob's maximal inequality that E sup 0≤t≤T * | d dt B ε t | 2 ≤ Cε −β T * E|B T * | 2 for some absolute constant C > 0. This establishes the estimate (17). The bound (18) is now a consequence of plugging in the equation (12), then using the triangle inequality, estimating the term with the time derivative by means of (17) and bounding the transport term similarly as at the end of the proof of (17). This concludes the proof of Lemma 4.
We aim to derive an estimate for sup T ∈[0,T * ] E v ε −vε L 1 (K) of the same type as the asserted bound (23). The proof of this proceeds in several steps. For the sake of better readability, let us occasionally break the proof into intermediate results.
Then the following "entropy estimate" holds true ) and all (y, s) ∈ R d ×(0, T * ). A corresponding estimate holds true switching the roles of v ε and vε, see (48) below. Proof.
Step 1 (Equation for u ← ε and u ← ε ): The first step is to derive the equation for the shifted densities u ← ε and u ← ε , respectively. To this end, we aim to apply ) and we thus obtain from an application of Itô's formula for each test function η ∈ C ∞ cpt (D × (0, T * )) with probability one Since u ε solves the equation (12) classically in D × (0, T * ), and is by definition constant outside of it, we may computê Integrating by parts in the second term on the right hand side of the latter identity as well as performing a change of variables Note that there is no boundary integral appearing from the integration by parts in the second term on the right hand side of (35) since u ε ∈ C(R d × (0, T * )). We compute analogouslŷ and From (34), (36), (37) and (38) we infer that for each η ∈ C ∞ cpt (R d × (0, T * )) it holds with probability one Analogously one derives the equation for u ← ε .
Step 2 (Convex approximation to r → r ∨κ as test function): In the next step we aim to derive the equation for v ε = ζ ε q κ • u ← ε based on the equation for the shifted density derived in (39). The idea is to test the equation (39) with the test function is arbitrary. However, since the shifted density u ← ε is only Hölder continuous in the time variable we have to regularize first. To this end, we make use of the Steklov average η h (x, t) := η(x, s) ds which is an admissible test function for (39) for all sufficiently small h > 0 (depending only on the support of ξ). Since ∂ t η h (x, t) = η(x,t+h)−η(x,t) h we obtain by a simple change of variables for each h > 0 and each ξ ∈ C ∞ cpt (R d ×(0, T * ); [0, ∞)) almost surely. We then deduce from the smoothness and convexity of ζ ε q κ as well as by reverting the change of variables the bound for each h > 0 and each ξ ∈ C ∞ cpt (D×(0, T * ); [0, ∞)) almost surely. Hence, we may infer from this, (39) and standard properties of the Steklov average after letting which is valid for each ξ ∈ C ∞ cpt (R d ×(0, T * ); [0, ∞)) on a set with probability one. Note that on R d \ (−ν(B t −B ε t ) + ∂D) we may apply the chain rule to compute that . This in turn means that we can integrate by parts in the first term on the right hand side of (41) without producing an additional surface integral. Taking all of these information together yields the bound Exploiting the sign (ζ ε q κ ) ≥ 0 and making use of the abbreviation v ε = ζ ε q κ • u ← ε we arrive at the estimate surely. An analogous estimate also holds true for the pair (u ← ε , vε = ζε (y, s)∇vε(y, s) · ∇ξ(y, s) dy ds for allξ ∈ C ∞ cpt (D×(0, T * ); [0, ∞)) almost surely.
Step 3 (Convex approximation η δ to r → |r| as test function): We proceed by testing the inequality (43) ) and (y, s) ∈ R d ×(0, T * ) are arbitrary. This again incorporates several integration by parts which we do not want to produce any additional surface integrals. We reiterate that this will indeed not be the case since neither the Dirichlet data nor the Neumann data for v ε jump across the interfaces −ν(B t −B ε t ) + ∂D for all t ∈ [0, T * ]. Hence, arguing similar to the previous step using in particular the Steklov average and the convexity of η δ we obtain the estimate ) and all (y, s) ∈ R d ×(0, T * ). As a preparation for what follows, we postprocess the right hand side of the latter inequality. Integrating by parts, adding zero and using the chain rule we may rewrite the non-linear diffusion term as follows Analogously one obtains for the Stratonovich correction term Since χ supp ∇vε v ε ≤ χ supp ∇vε u ← ε by the definition of v ε = ζ ε q κ • u ← ε and the truncation ζ ε q κ we observe that the penultimate term in (45) has a favorable sign. Together with the two identities (46) and (47) the bound (45) thus yields the asserted estimate (33).
We continue with the proof of Proposition 5. The next step takes care of the proper choice of test functions ξ(x, t) resp.ξ(y, s) in the latter two estimates. After that, we start merging them by integration over the respective independent variables (and the probability space) and summing the two resulting inequalities.
Before we move on with removing the doubling in the time variable by studying the limit τ → 0 let us first perform some computations on the non-linear diffusion term R porMed and the correction term R corr . Exploiting that (η δ ) ≥ 0 and ξ ≥ 0, completing the square |∇v m ε | 2 +|∇v m ε | 2 = |∇v m ε −∇v m ε | 2 + 2∇v m ε · ∇v m ε and integrating by parts entails that porMed . The idea eventually is-after letting τ → 0, δ → 0 and removing the doubling in the spatial variables (the latter by fine-tuning the scales θ > 0 and l > 0 as suitably chosen powers of ε ∨ε)-to integrate by parts and to use the sign in condition ii) for the spatial test functionξ. We will make this precise together with all the required error estimates in a later stage of the proof. For the moment, we only wish to mention that the same argument also applies to the term R corr ; one simply substitutes v ε for v m ε and vε for v m ε , respectively. This shows that × ρ τ (t−s)ϕ t+s 2 γ θ (x−y)∆ξ l x+y 2 dx dt dy ds =: R (1) corr . The task therefore reduces to post-process the bound with the remaining three terms left unchanged from the estimate (53). In a first step we aim to remove the doubling in the time variable by letting τ → 0.
Lemma 9. Let the assumptions and notation of Subsection 4.2 until this point be in place. Define the quantities Then the estimate holds true for all (θ, l) subject to (51), all δ > 0 and all ε,ε ≤ κ 2 .
Proof. It follows from (50) that (∂ t +∂ s )ξ = ϕ ( t+s 2 )ρ τ (t−s)γ θ (x−y)ξ l ( x+y 2 ). In particular, the singular terms as τ → 0 cancel. Hence, it follows by Lebesgue's dominated convergence based on the regularity and the bounds for the Wong-Zakai approximation u ε from Lemma 4 and definition (22) of the shifted densities u ← ε that Relying again on Lebesgue's dominated convergence due to the regularity and the a priori estimates for the Wong-Zakai approximation u ε from Lemma 4 and the definition (22) of the shifted densities u ← ε , we may also easily pass to the limit τ → 0 in all the terms on the right hand side of (54) except for the noise term R noise . More precisely, we obtain We proceed with the discussion of the noise term R noise . To this end, we define for λ > 0 the cut-offρ λ := 1 λρ ( · λ ) by means of a standard even cut-off function ρ ∈ C ∞ cpt ((−1, 1); [0, ∞)) such that´Rρ(r) dr = 1. Using that the mollifier ρ is supported on the positive real axis together with adding zero yields for all τ > 0 subject to (51) noise .

Observe that ER
(1) noise = 0 since the bounded random variable vε(y, s) is measurable with respect to F s . Similarly, due to the fact that v ε (x, t−τ ) is bounded and measurable with respect to F t−τ we may actually write Hence, it follows by an application of the Burkholder-Davis-Gundy inequality and then Lebesgue's dominated convergence relying again on the regularity and bounds for the Wong-Zakai approximation u ε from Lemma 4 and the definition (22) of the shifted densities u ← ε that ER (2) noise → 0 as τ → 0.
We continue with the proof of Proposition 5 taking care in the next step of the error terms. More precisely, we may derive the following bound.
Lemma 10. Let the assumptions and notation of Subsection 4.2 until this point be in place. In particular, recall the definition of the error term R (1) error from (58). We then have the estimate for all (θ, l) subject to (51), all δ > 0 and all ε,ε ≤ κ 2 .
Proof. We start estimating by Hölder's inequality However, by the bounds (18) and (30), the convergence (31) and the fact that sign(a−b) = sign(a m −b m ) due to the monotonicity of r → r m we infer by an application of Lebesgue's dominated convergence theorem based on the regularity of the Wong-Zakai approximation u ε from Lemma 4 that the term on the right hand side of the latter bound vanishes as δ → 0.
Recall that κ ≤ v ε ≤ u ← ε on supp ∇v ε . Hence, we can estimate by means of (21) We may then estimate using also (19), the definition (22) of the shifted densities u ← ε , ∇ξ l L ∞ (R d ) ≤ ∇γ l L 1 (R d ) ξ L ∞ (R d ) ≤ Cl −1 (which follows from Young's inequality and condition i) for the spatial cut-offξ), ϕ L ∞ (0,T * ) ≤ 1, a change of variables x → x−ν(B t −B ε t ), the fact that u ← ε ≥ κ on supp ∇v ε , and finally the a priori estimate (16) Since the other two terms of R (1) error can be treated along the same lines, we obtain in total the asserted estimate (64). This concludes the proof of Lemma 10.
We continue with the proof of Proposition 5. We have by now everything in place to let δ → 0 in (59). A straightforward application of Lebesgue's dominated convergence theorem based on the convergence in (31) and the regularity and bounds for the Wong-Zakai approximations u ε from Lemma 4 shows that by letting δ → 0 in (59) and using (64) it holds for all (θ, l) subject to (51), and all ε,ε ≤ κ 2 . The updated terms in this inequality are given by respectively. We may proceed with the next step which consists of removing the doubling in the spatial variables. More precisely, the following holds true.
We continue with the proof of Proposition 5. The next step takes care of estimating the non-linear diffusion term and the correction term.
Lemma 12. Let the assumptions and notation of Subsection 4.2 until this point be in place. In particular, recall from the statement of Lemma 11 the definition of the quantities R (5) porMed and R (5) corr . Then there exists some l(ξ) > 0 small enough and some absolute constantC > 0 such that we have the estimate ER (5) porMed + ER (5) corr ≤Cκ + C(κ)l −2 (ε ∨ε) α for all (θ, l) subject to (51) resp. l < l(ξ), and all ε,ε ≤ κ 2 . Proof. We first aim to replace v ε resp. vε by ζ ε q κ • u ε resp. ζε q κ • uε in the two terms R (5) porMed and R (5) corr . This can be done by an estimation similar to the proof of Lemma 11, this time using in particular (7), (10) and EC 2 α < ∞, which yields the bound However, it follows from ε,ε ≤ κ 2 that (ζ ε q κ • u ε ) m −(ζε q κ • uε) m ∈ H 1 0 (D) as well as (ζ ε q κ • u ε )−(ζε q κ • uε) ∈ H 1 0 (D). An integration by parts together with condition ii) of the spatial cut-off functionξ then entails It remains to bound the two terms featuring the difference ∇ξ l −∇ξ. By continuity of translations in L 2 , however, together with the a priori estimate (16) we find some small enough l(ξ) > 0 such that holds true, and therefore in particular the desired estimate (68) for some absolute constantC > 0.

4.3.
Proof of Corollary 6 (L 1 convergence of shifted densities). Let δ > 0 be fixed but arbitrary. By the triangle inequality we may estimate for all κ ∈ (0, T * ∧ 1) and all ε,ε ≤ κ 2 . Hence, it follows from splitting D = K ∪ (D \ K) together with the bounds (15) and (23) as well as the definition (22) of the shifted densities that The term with the initial data is estimated similarly by for all κ > 0 and all ε,ε ≤ κ 2 . Choosing first κ < τ sufficiently small such that (4L d (D)+C)κ ≤ δ 2 , we may then fix a large enough compact set K ⊂ D and some small enough ε 0 (κ, K) such that the bound holds true for all ε,ε ≤ ε 0 . This proves that the sequence of shifted densities u ← ε is a Cauchy sequence in the space C([τ, T * ]; L 1 (Ω×D, P ⊗ L d )) for all τ > 0. The corresponding assertion in the space L 1 ([0, T * ]; L 1 (Ω×D, P ⊗ L d )) is proved similarly based on the additional estimate Let us denote the corresponding limit in L 1 ([0, T * ]; L 1 (Ω×D, P ⊗ L d )) by u.
On the other side, it follows immediately from the bound (15) that the sequence of densities u ε has a weak limit in the space L m+1 (Ω T * , P T * ; L m+1 (D)), which we denote byū. It remains to verify that u =ū. To this end, let φ ∈ C ∞ cpt (D × [0, T * ]) and A ∈ F T * be fixed. We then have By a simple change of variables and the definition (22) we may write Exploiting (7), (10) and (15) we further estimate Hence, we may infer that holds true for all φ ∈ C ∞ cpt (D × [0, T * ]) and A ∈ F T * . This shows that u =ū and thus concludes the proof of Corollary 6.

4.4.
Proof of Proposition 7 (Recovering the unique weak solution). Let a test function φ ∈ C ∞ cpt (D) be fixed, and define K := supp φ. Fix also an integer M ≥ 1, and let C α be the square integrable random variable of (10). Let δ ∈ (0, 1) be such that {x ∈ R d : dist(x, K) ≤ δ} ⊂ D. Let ε = ε (M, δ) be the constant from (11). By Itô's formula, the definition (22) and the fact that the u ε solve (12) classically we have for all ε ≤ ε and all measurable A ∈ F T * (cf. the argument in the first step of the proof of Proposition 5) for all T ∈ (0, T * ). Define the shifted test function , so that we obtain by a simple change of variables Note that as a consequence of (11) we have almost surely on {C α ≤ M } for all ε ≤ ε and all t ∈ [0, T * ] that supp φ ← ε (·, t) ⊂⊂ D. Hence, integrating by parts on the right hand side in (75) does not produce any boundary integrals so that after reversing the change of variables we obtain the identity We aim to pass to the limit ε 0 ≥ ε → 0 in all four terms. This is possible by the convergence of the shifted densities u ← ε in C([τ, T * ]; L 1 (Ω×D, P⊗L d )) as well as in L 1 ([0, T * ]; L 1 (Ω×D, P⊗L d )), see Corollary 6. Because of the uniform bound (15) the convergence in L 1 can actually be lifted to convergence in any L q ([0, T * ]; L q (Ω×D, P⊗L d )), q ∈ (1, ∞), which makes the limit passage possible in the non-linear diffusion term as well as the noise term (using for the latter, e.g., the Burkholder-Davis-Gundy inequality). In summary, we obtain from letting ε 0 ≥ ε → 0 the identity νu(x, t)∇φ(x) dx dB t for all T ∈ (0, T * ). Since M ≥ 1 as well as A ∈ F T * were arbitrary, and the random variable C α is integrable, we thus recover (4). This concludes the proof of Proposition 7 since the asserted bounds for u follow immediately from (15).

Finite time extinction property
5.1. Viscosity theory: Maximal subsolution in a rough domain. From now on we will restrict ourselves to the one-dimensional setting d = 1. To start we recall some language from viscosity theory (cf. [7]). In particular, we aim to make precise what we mean by a subsolution to the Cauchy-Dirichlet problem (24)-(26) of the porous medium equation with two copies of a Brownian path as lateral boundary; and analogously for the corresponding regularized problem (27)-(29). In a first step, we introduce the relevant notions for the full space problem. The porous medium operator in terms of the pressure variable is encoded by the functional Let T * ∈ (0, ∞] be a time horizon and p : R × (0, T * ) → R be a function. The parabolic semijet J 2,+ p(x, t) of p at (x, t) ∈ R×(0, T * ) is the set of all (q, a, X) ∈ R 3 such that p(y, s) ≤ p(x, t) + a(s − t) + q(y − x) + 1 2 X(y − x) 2 + o(|s−t| + |y−x| 2 ) as R × (0, t] (y, s) → (x, t).
Note that the condition in (78)  Definition 13. Let T * ∈ (0, ∞] be a time horizon and p 0 ∈ C ∞ cpt (R; [0, ∞)) be an initial pressure. An upper-semicontinuous function p : R × [0, T * ) → [0, ∞) is called a subsolution for the Cauchy problem of the porous medium equation with initial pressure p 0 if it holds for all (x, t) ∈ R × (0, T * ) and all (q, a, X) ∈ J 2,+ p(x, t), as well as if it holds p(x, 0) ≤ p 0 (x) for all x ∈ R.
In the language of [7], the functional a − F (r, p, X) encoding the porous medium equation in terms of the pressure variable is not proper. But note that it is at least degenerate elliptic. Since we will only deal with subsolutions (or for the regularized problem with classical solutions), this is of no concern for us (cf. [6] or [22] for a viscosity theory of the Cauchy problem of the deterministic porous medium equation).
We continue with the notion of subsolutions to (24)-(26) (resp. (27)-(29)). Since we have to incorporate Brownian paths as the lateral boundary, the following constructions are of course random. However, all results of this section turn out to be purely deterministic consequences of the probabilistic facts (8), (9), (10) and (11). In other words, we proceed with constructions to be understood in a pathwise sense.
The functional for the lateral boundary condition of the limit problem is simply defined by C : R → R, r → r. The one for the regularized problem is given by C ε : R → R, r → r − m m−1 ε m−1 . We then introduce a lower-semicontinuous functional encoding the equations (24) and (26). The corresponding functional for the regularized problem, i.e., encoding (27) and (29), is given by else.
The reasons for relaxing (24)-(26) (resp. (27)-(29)) to the full space setting are twofold. On one hand, working in this framework turns out to be convenient when studying (any sort of) convergence of the subsolutionsp ε for the regularized problem. On the other hand, we also want to exploit the established results from the viscosity theory [6] (see also [22]) for the Dirichlet problem of the deterministic porous medium equation (79)-(80).
We proceed with a list of intermediate results needed to prove the main result of this work, Theorem 3. The corresponding proofs will be provided afterwards. The first result concerns the construction of a subsolution to the regularized Cauchy-Dirichlet problem (27)-(29).
In a next step, we construct on a set of probability one the maximal subsolution (also referred to as Perron's solution)p max of the limit Cauchy-Dirichlet problem (24)-(26) in the sense of Definition 14. Using standard arguments from viscosity theory, the main issue is to establish the existence of a subsolution to (24)-(26). This will be done by means of the technique of semi-relaxed limits.
A crucial estimate for the proof of Theorem 3 is the content of the following result. The asserted bound is important in the sense that it serves to close the loop between the pathwise constructions performed in this section and the unique weak solution of the SPME (1)-(3). Letp max be the associated maximal subsolution of the problem (24)-(26) as constructed in Proposition 16. Then, for all t ∈ (0, T * ) the bound p(·, t) ≤p max (·, t) (92) holds true almost surely almost everywhere in R.
We have by now everything in place to proceed with the proofs.
We continue with the verification of the claims regardingp max . To this end, consider first an arbitrary subsolutionp of (24)-(26) with initial pressurep 0 := is satisfied for all (x, t) ∈ R × [0, T * ). Hence, since we already established the existence of a subsolution to (24)-(26) the definition ofp max is meaningful, and the asserted bound (90) then follows at once from (98). In particular,p max < ∞ is satisfied on R × [0, T * ). Thatp max is subject to (84) is once again a classical fact from viscosity theory, see [7,Lemma 4.2]. Moreover, as we have already argued for the semi-relaxed limit p semi-rel , it then follows that the upper-semicontinuous envelope ofp max constitutes a subsolution of (24)-(26). However, by the definition ofp max we may then infer thatp max is actually equal to its upper-semicontinuous envelope; in particular, a subsolution of (24)-(26) in the sense of Definition 14. This concludes the proof of Proposition 16.

5.4.
Proof of Proposition 17 (Comparison of transformed weak solution with viscosity solution). It obviously suffices to prove that for all t ∈ (0, T * ) p(·, t) ≤p semi-rel (·, t) almost surely almost everywhere in R. By definition of p, see (91), and since we havep semi-rel ≥ 0 it moreover suffices to show that (99) is almost surely satisfied for all t ∈ (0, T * ) almost everywhere in νB t + I. We aim to exploit Corollary 6, i.e., that the unique weak solution u of (1)-(3) can be recovered by means of the Wong-Zakai approximations u ε from Lemma 4, at least after employing an additional time-dependent shift (22). So fix an integer M ≥ 1 as well as some κ > 0. Let C α be the square integrable random variable of the estimate (10). Let finally ε > 0 be fixed, and denote by ε = ε (M, ε) the constant from (11).
Finite time extinction with probability one now follows from standard properties of Brownian motion. Indeed, assume that we have P(T extinct < ∞) < 1. There would then exist a constant C < ∞ such that on a set with non-vanishing probability it holds 0 ≤ t − 1 2 |B t | < C for all t ∈ (1, ∞); a contradiction. Hence, P(T extinct < ∞) = 1 which also by (6) entails that P(T extinct < ∞) = 1.
It remains to prove that u(·, T ) vanishes almost surely almost everywhere in I from the stopping timeT extinct onwards. So let T ∈ (0, ∞) be fixed. We infer from (103) together with (105) that almost surelyp max (·,T extinct ) ≡ 0. In addition, we note thatp max is almost surely a subsolution of (79)-(80) on R × [0, ∞) with initial pressure p 0 (x) := m m−1 u 0 (x) m−1 in the sense of Definition 13. For a proof, the only non-trivial case is that of a lateral boundary point (x, t) ∈ t∈(0,∞) (νB t +I)×{t} such that C(p max (x, t)) =p max (x, t) ≤ 0. Sincep max (x, t) ≥ 0 by construction, it follows thatp max (x, t) = 0. From this point onwards one may argue as forp ε in the proof of Lemma 15.
However, once a subsolution became trivial it stays trivial (by comparison with the trivial viscosity solution). We deduce thatp max (·, T ) ≡ 0 is satisfied almost surely on {T ≥T extinct }. The claim thus follows from another application of (104). This concludes the proof of Theorem 3.