Exponential Convergence of Non-Linear Monotone SPDEs

For a Markov semigroup $P_t$ with invariant probability measure $\mu$, a constant $\ll>0$ is called a lower bound of the ultra-exponential convergence rate of $P_t$ to $\mu$, if there exists a constant $C\in (0,\infty)$ such that $$ \sup_{\mu(f^2)\le 1}\|P_tf-\mu(f)\|_\infty \le C \e^{-\ll t},\ \ t\ge 1.$$ By using the coupling by change of measure in the line of [F.-Y. Wang, Ann. Probab. 35(2007), 1333--1350], explicit lower bounds of the ultra-exponential convergence rate are derived for a class of non-linear monotone stochastic partial differential equations. The main result is illustrated by the stochastic porous medium equation and the stochastic $p$-Laplace equation respectively. Finally, the $V$-uniformly exponential convergence is investigated for stochastic fast-diffusion equations.


Introduction
It is well known that the solution to the porous medium equation (1.1) dX t = ∆X r t dt decays at the algebraic rate t − 1 r−1 as t → ∞, where ∆ is the Dirichlet Laplacian on a bounded domain in R d , r > 1 is a constant and X r t := |X t | r−1 X t , see [1]. This type algebraic convergence has been extended in [3] to stochastic generalized porous media equations. When r ∈ (0, 1), (1.1) is called the fast-diffusion equation.
Consider, for instance, the stochastic porous medium equation where ∆ is the Dirichlet Laplacian on (0, l) for some l > 0, and W t is the cylindrical Brwonian motion on L 2 (m), where m is the normalized Lebesgue measure on (0, l). By [3,Theorem 1.3], for any x ∈ H := H −1 (the duality of the Sobolev space w.r.t. L 2 (m), see Section 3), the equation has a unique solution starting at x, and the associated Markov semigroup P t has a unique invariant probability measure µ such that On the other hand, by using the dimension-free Harnack inequality and a result due to [5], the uniform exponential convergence is proved in [8] for some constants C, λ > 0. Since, according to [17,Theorem 1.2(4)] (see also [8,Theorem 1.4(iv)]) P t is ultrabounded, i.e. P t L 2 (µ)→L ∞ (µ) < ∞ for t > 0, this implies the ultra-exponential convergence for some constant C, λ > 0. To see that (1.3) improves (1.2) for large time, we note that with constant C ′ ∈ (0, ∞) since µ( · 2 H ) < ∞, see for instance [3,Theorem 1.3]. However, explicit estimates on the ultra-exponential convergence rate λ is not yet available. We note that in [6] an lower bound estimate of exponential convergence rate is presented for a class of semi-linear SPDEs (stochastic partial differential equations). But the main result in [6] does not apply to the present non-linear model, since both [6, Hypothesis 2.4(a)] (i.e. F is a Lipschitz map from H to H) and [6, Hypothesis 2.4(b)] (i.e. Im(F ) ⊂ L 2 (m)) are not satisfied for the present F (x) := ∆x r , which is not a well defined map from H to H.
In this paper, we aim to present explicit lower bound estimates for the ultra-exponential convergence rate λ in (1.3). In the next section, we prove a general result for a class of non-linear SPDEs considered in [8]. The main tool in the study is the coupling by change of measure constructed in [17] (see also [8]). A general theory on this kind of couplings and applications has been addressed in the recent monograph [18]. The main result is applied to the stochastic porous medium equation and the stochastic p-Laplace equation in Section 3 and Section 4 respectively. Finally, in Section 5 we investigate the exponential convergence for stochastic fast-diffusion equations.

A general result
Let V ⊂ H ⊂ V * be a Gelfand triple, i.e. (H, ·, · H , | · | H ) is a separable Hilbert space, V is a reflexive Banach space continuously and densely embedded into H, and V * is the duality of V with respect to H. Let V * ·, · V be the dualization between V and V * . We Let W = (W t ) t≥0 be a cylindrical Brownian motion on a (possibly different) Hilbert space (E, ·, · E , | · | E ), i.e. W t := ∞ i=1 B i t e i for an orthonormal basis {e i } i≥1 of E and a sequence of independent one-dimensional Brownian motions {B i t } i≥1 on a complete filtered probability space (Ω, F , {F t } t≥0 , P). Consider the following stochastic equation: where b : V → V * is measurable and Q ∈ L HS (E; H), the space of all Hilbert-Schmidt linear operators from E to H, such that the following assumptions hold for some constants r > 1, and C 1 , C 2 > 0: and P-a.s.
where the Bochner integral Let P t be the associated Markov semigroup, i.e.
For any u ∈ H, let where we set inf ∅ = ∞ by convention. The study of (2.1) with the above type assumptions goes back to [12,13] for non-linear monotone SPDEs. Extensions to stochastic equations with "local conditions" as well as to non-monotone stochastic equations have been made in [9,11,15]. As mentioned in the Introduction that in this paper we aim to estimate the ultra-convergence rate of P t . The following is the main result of the paper.
H holds for all u, v ∈ V, then P t has a unique invariant probability measure µ and (1.3) holds for some constant C > 0 and Moreover, Proof. By [8, Theorem 1.4] with α = 1 + r (see also [18,Corollary 2.2.4] with α = r), P t has a unique invariant probability measure µ of full support on H. Moreover, P t is strong holding for some constant c > 0. So, Therefore, it suffices to prove for some constant C ∈ (0, ∞) and the desired constant λ, and to verify the claimed lower bounds of λ. We shall complete the proof by four steps.
(a) We first construct a coupling by change of measure using the idea of [17]. For fixed T > 0 and x, y ∈ H, let X t = X x t solve (2.1) for X 0 = x, and let Y t solve the equation As shown in [17, Theorem A.2] (see also [8]) that the equation (2.6) has a unique solution Then Combining this with T ≤ τ and θ ≥ 2, and using the Jensen inequality, we see that . Moreover, since θ ≥ 2, by (2.9) and the Hölder inequality we obtain (2.10) Then by the Girsanov theorem, is a well defined probability density of P, and the process is a cylindrical Brownian motion on E under the weighted probability measure dQ := RdP. Now, rewrite (2.6) by From the weak uniqueness of the solution to (2.1) and X T = Y T , we conclude that This together with P T f (x) = Ef (X T ) yields that (c) By (2.10) we have t and X y t solve the equation (2.1) starting at x and y respectively. Thus, Substituting this and (2.12) into (2.11) and using the Markov property, we arrive at where (2.14) For fixed t > 0, by taking s ∈ (0, t) and T = t − s in (2.13) we obtain (2.15) (d) To calculate the inf in (2.15), let We have α 1 + α 2 = r+1 r−1 , and by (2.14), Then it follows from (2.15) that Obviously, there exists t 0 ∈ (0, ∞) such that Letting i(t) = sup{n ∈ Z + : n ≤ t t 0 } be the integer part of t t 0 , combining this with the semigroup property and the L 2 (µ)-contraction of P t , we obtain Thus, (2.5) holds for C := e λt 0 .
Finally, to derive the desired explicit lower bounds of λ, we take , where the last step is due to the fact that To conclude this section, we indicate that P t is ultra-exponential convergent provided

Stochastic porous medium equation
Let ∆ be the Dirichlet Laplacian on the interval (0, l) for some l > 0, and let σ > 0, r > 1 be two constants. Let W t be the cylindrical Brownian motion on L 2 (m), where m(dx) := l −1 dx is the normalized Lebesgue measure on (0, l). Consider the following stochastic porous medium equation We first verify assumptions (A1)-(A4) for an appropriate choice of (H, V). It is well known that the spectrum of −∆ consists of simple eigenvalues {λ k := π 2 k 2 l 2 } k≥1 . Let {e k } k≥1 be the corresponding eigenbasis. Then Q := σI is Hilbert-Schmidt from L 2 (m) to H := H −1 , the completion of L 2 (m) under the inner product This implies (A3) and (A4) for C 1 = C 2 = 1. Moreover, for any v 1 , v 2 , v ∈ V, is continuous in s ∈ R; that is, (A1) holds. Finally, we have (see the proof of Proposition 3.1 below) Then that (A2) holds for any positive constant C 1 . Therefore, for any initial data x ∈ H the equation (3.2) has a unique solution starting at x. Let P t be the associated Markov semigroup.
Proof. We first prove (3.2). Obviously, we may assume that s∨t ≥ 0, otherwise simply use −s, −t to replace s, t respectively. Moreover, since the positions of s and t are symmetric, we may assume further that s > t (hence, s ≥ 0). Assuming s > t and s ≥ 0, we prove (3.2) by considering the following two situations respectively.
(ii) s ≥ 0 > t. By the Jensen inequality we have Thus, (3.2) holds. Now, let b(x) = ∆x r , x ∈ V := L r+1 (m). Since Q = σI, we have · Q = 1 σ · 2 . Combining this with · r+1 ≥ · 2 , λ 1 = π 2 l 2 and the definition of | · | H , we obtain Then, due to (3.2), for any θ ∈ (r − 1, r + 1] ∩ [2, r + 1], holds for η := 2 2−r σ θ π l r+1−θ , δ := 2 2−r π l r+1 . Therefore, by Theorem 2.1, (1.3) holds for some constant C ∈ (0, ∞) and Noting that r ≥ 1 implies θ + 1 − r ≤ θ, so that α θ is decreasing in θ, we obtain inf θ∈(r−1,r+1]∩ [2,r+1] α θ = α r+1 = l 4 r−1 (3 + r)(r + 1) So, (1.3) holds for some C ∈ (0, ∞) and the desired λ. Moreover, as in the proof of Theorem 2.1 that the desired lower bound estimates follows by taking in (2.3) To conclude this section, let us recall a corresponding result in the linear case, i.e. r = 1. Let R = σI and T t = e t∆ . In this case, for any p > 2 there exist constants C p , t p ∈ (0, ∞) such that To see this, we observe that σW t is a Wiener process on H with variance operator Qe i := σ 2 λ i , i ≥ 1. Taking M = 0, R = Q and T t = e t∆ , we see that assumptions in [16, Coroolary 1.4] hold for h 1 (t) = e −λ 1 t/2 and h 2 (t) = 0, so that Moreover, according to [2, Theorem 4 c)], P t is hypercontractive, i.e. for any p > 2 there exists a constant t p > 0 such that P t L 2 (µ)→L p (µ) = 1 holds for t ≥ t p . Combining this with (3.4) we prove (3.3). Note that in this linear case P t is not ultra-bounded, so that we do not have the ultra-exponential convergence as in (1.3). A feature in the linear case is that the exponential convergence rate λ 1 is independent of σ. Note that for r > 1 the lower bound estimates of λ presented in Proposition 3.1 are increasing to ∞ as σ ↑ ∞. But if we let r ↓ 1 in these estimates, the lower bounds of λ tend to 2λ 1 e (of course, the other constant C will tend to ∞ since P t is not ultracontractive for r = 1), which is also independent of σ. This indicates that the power of σ included in the lower bound estimates of λ presented in Proposition 3.1 is suitable when r goes down to 1.

Stochastic p-Laplace equation
Again let D = (0, l) for some l > 0 and m be the normalized volume measure. For p > 2, let H 1,p 0 be the closure of C ∞ 0 (D) with respect to the norm where, since D is one-dimensional, ∇f := f ′ . The p-Laplacian on D is defined by Consider the SPDE where W t is a cylindrical Brownian motion on L 2 (m), and Q ∈ L (H) is such that Thus, (A1) and (A4) with C 1 = 1 hold. Next, since for any f ∈ C ∞ 0 (D) we have it follows that From this and (4.3) it is easy to see that (A3) holds for C 1 = 0 and some C 2 > 0. Moreover, according to [8,Example] we have Then (A2) holds for C 1 = 0. Therefore, for any x ∈ H the equation (4.1) has a unique solution starting at x. Let P t be the Markov semigroup associated to (4.1).
From now on, we let r ∈ (0, 1) and consider the equation (2.1) such that assumptions (A1)-(A4) hold. Let P t be the associated Markov semigroup. We aim to investigate the V -uniformly exponential convergence for some constants C, λ > 0, where µ is the invariant probability measure of P t and V ≥ 1 is a continuous function on H. Obviously, (5.1) is equivalent to hold for some constants α, η > 0 and θ ≥ 4 1+r . Then P t has a unique invariant probability measure µ, and for any γ > 0, there exist two constants C, λ > 0 such that (5.1) holds for Proof. By (5.2) and the Itô formula, we see that Since h has relatively compact level sets in H, this implies that the sequence { 1 n n 0 δ 0 P t dt} n≥1 is tight and each of its weak limit point gives rise to an invariant probability measure of P t . Now, according to the proof of [5, Theorem 2.5(1)], it suffices to verify (i) (Assumption 2.1 in [5]): P t is strong Feller (i.e. P t B b (H) ⊂ C b (H), t > 0) and P t 1 U (x) > 0 holds for any t > 0, x ∈ H and non-empty open set U ⊂ H.
(ii) (Assumption 2.2 in [5]): For any r > 0 there exists t 0 > 0 and a compact subset K of H such that inf |x| H ≤r E1 K (X x t 0 ) > 0.
Firstly, according to [18,Theorem 2.3.1] (see also [10, Theorem 1.1] under a more specific framework), for any p > 1 there exists a continuous function Φ p on H×H×(0, ∞) with H(x, x, t) = 0 such that the Harnack inequality holds. By [18,Theorem 1.4.1] (see also [19,Proposition 3.1]), this implies that the invariant probability measure of P t is unique with full support on H, and P t is strong Feller and has a strictly positive density with respect to the unique invariant probability measure µ. Therefore, (i) holds. Next, since h has relatively compact level sets in H, it follows from (5.4) that P t 0 1 K (0) > 0 holds for some t 0 > 0 and compact set K in H. Indeed, (5.4) implies c 0 := Eh(X t 0 ) < ∞ for some t 0 > 0, so that we may take K being the closure of {h ≤ c 0 + 1}. Then it follows from (5.5) that for any r > 0, inf |x| H ≤r P t 0 1 K (x) ≥ (P t 0 1 K (0)) p inf |x| H ≤r e −Φp(0,x,t 0 ) > 0. dt + dM t ≤ C 1 − C 2 e γ(1+|Xt| 2 H ) (1−r)/2 dt + dM t for some constants C 1 , C 2 > 0 and some local martingale M t . This implies de C 2 t+γ(1+|Xt| 2 H ) (1−r)/2 ≤ C 1 e C 2 t dt + e C 2 t dM t .