Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values

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Abstract

In this article we develop a framework for studying parabolic semilinear stochastic evolution equations (SEEs) with singularities in the initial condition and singularities at the initial time of the time-dependent coefficients of the considered SEE. We use this framework to establish existence, uniqueness, and regularity results for mild solutions of parabolic semilinear SEEs with singularities at the initial time. We also provide several counterexample SEEs that illustrate the optimality of our results.

Introduction

There are a number of existence, uniqueness, and regularity results for mild solutions of semilinear stochastic evolution equations (SEEs) in the literature; see, e.g., [10], [11], [4], [28], [16], [18], [21], [27] and the references mentioned therein. In this work we extend the above cited results by adding singularities in the initial condition and by introducing singularities at the initial time of the time-dependent coefficients of the considered SEE; see also Chen & Dalang [7], [8] for related results. To be more specific, in the first main result of this work (see Proposition 2.7) we establish a general perturbation estimate (see (3) below) for a general class of stochastic processes which allows us to derive a priori bounds (see, e.g., (5) below) for solutions and numerical approximations of SEEs with singularities at the initial time. This perturbation estimate, in turn, is used to prove the second main result of this article (see Theorem 2.9) which establishes existence, uniqueness, and regularity properties for solutions of SEEs with singularities at the initial time (see (4) and (5) below). As an application of our perturbation estimate and this second main result of our article, we reveal a regularity barrier (see (8) below) for the initial condition of the considered SEE under which the considered SEE has a unique solution which is Lipschitz continuous with respect to initial values (see Corollary 2.10). By means of several counterexamples (see Proposition 3.2, Proposition 3.4, Proposition 3.5) we also demonstrate that this regularity barrier can in general not essentially be improved (cf. (10) and (11) below). We illustrate the above findings in the case of possibly nonlinear stochastic heat equations on an interval such as the continuous version of the parabolic Anderson model on an interval (cf. Corollary 3.1, Proposition 3.2, and Proposition 3.3). Existence, uniqueness, and regularity results for possibly nonlinear stochastic heat equations on the whole real line with rough initial values, that is, signed Borel measures with exponentially growing tails over R as initial values can be found in Chen & Dalang [7], [8] (see Theorem 2.4 in Chen & Dalang [8] for an existence and uniqueness result and a priori estimates and see Theorem 3.1 in Chen & Dalang [7] for a Hölder regularity result). Moreover, Proposition 2.11 in Chen & Dalang [8] disproves the existence of a solution of the considered stochastic heat equation in the case of a specific rough initial value, that is, the derivative of the Dirac delta measure at zero as the initial value.

To illustrate the results of this article in more details, we assume the following setting throughout this introductory section. Let (H,H,,H) and (U,U,,U) be nontrivial separable R-Hilbert spaces. Let T(0,), ηR, p[2,), α[0,1), αˆ(,1), β[0,12), βˆ(,12), L0,Lˆ0,L1,Lˆ1[0,), κ=1(0,)(L1) satisfy κ(α+αˆ)<32. Let (Ω,F,P,(Ft)t[0,T]) be a stochastic basis. Let (Wt)t[0,T] be an IdU-cylindrical (Ft)t[0,T]-Wiener process. Let A:D(A)HH be a generator of a strongly continuous analytic semigroup with spectrum(A){zC:Re(z)<η}. Let (Hr,Hr,,Hr), rR, be a family of interpolation spaces associated to ηA (cf., e.g., [26, Section 3.7]). Let F:[0,T]×Ω×HHα be a (Pred((Ft)t[0,T])B(H))/B(Hα)-measurable mapping, let B:[0,T]×Ω×HHS(U,Hβ) be a (Pred((Ft)t[0,T]) B(H))/B(HS(U,Hβ))-measurable mapping, and assume for all t(0,T], X,YLp(P;H) thatF(t,X)F(t,Y)Lp(P;Hα)L0XYLp(P;H),F(t,0)Lp(P;Hα)Lˆ0tαˆ,B(t,X)B(t,Y)Lp(P;HS(U,Hβ))L1XYLp(P;H),B(t,0)Lp(P;HS(U,Hβ))Lˆ1tβˆ.

In displays (3)–(11) below we illustrate the above framework through several examples and applications. Our first result is a suitable perturbation estimate for predictable stochastic processes. We employ the following additional notation to formulate this perturbation estimate. For every δR and every sufficiently regular predictable stochastic process Y:[0,T]×ΩHδ let I(,Y)=(I(t,Y))t[0,T]:[0,T]×ΩH be a predictable stochastic process which satisfies for all t[0,T] P-a.s. that I(t,Y)=Yt0te(ts)AF(s,Ys)ds0te(ts)AB(s,Ys)dWs. Proposition 2.7 below then proves that there exists a function Θ=(Θλ)λR:RR such that for all δR, λ(,12[1+1{0}(L1)]) and a wide class of (Ft)t[0,T]-predictable stochastic processes Y1,Y2:[0,T]×ΩHδ it holds thatsupt(0,T][tλYt1Yt2Lp(P;H)]Θλ(supt(0,T][tλI(t,Y1)I(t,Y2)Lp(P;H)]). We also note that we explicitly specify the function Θ=(Θλ)λR:RR in Proposition 2.7 below. Estimate (3) follows from an appropriate application of a generalized Gronwall-type inequality (see the proof of Proposition 2.7 below for details).

We use inequality (3) to establish an existence, uniqueness, and regularity result for SEEs with singularities at the initial time. More precisely, in Theorem 2.9 below we prove that there exists a function Θ=(Θλ)λR:RR such that for all suitable δ,λR, ξLp(P|F0;Hmax{δ,0}) it holds (i) that there exists a suitable up-to-modifications unique (Ft)t[0,T]-predictable stochastic process X:[0,T]×ΩHmax{δ,0} which satisfies for all t[0,T] P-a.s. thatXt=etAξ+0te(ts)AF(s,Xs)ds+0te(ts)AB(s,Xs)dWs and (ii) thatsupt(0,T][tλXtLp(P;H)]Θλ[1+supt(0,T](tδetAξLp(P;H))Tδ]<. In Theorem 2.9 we also explicitly specify the function Θ=(Θλ)λR:RR. We would like to point out that inequality (5) under the generality of (1) and (2) is a crucial ingredient to establish essentially sharp weak convergence rates for numerical approximations of SEEs with possibly smooth initial values (see the last paragraph in this introductory section for more details). Inequality (5) follows from the perturbation estimate (3) (with Y1=X and Y2=0 in the notation of (3)).

We now illustrate Theorem 2.9 and (4)–(5), respectively, by some examples. In particular, in Corollary 2.10 below we prove by an application of Theorem 2.9 that for all FLip(H,Hα), BLip(H,HS(U,Hβ)), δˆ=12[1+1{0}(|B|Lip(H,HS(U,Hβ)))] it holds (i) that there exist up-to-modifications unique (Ft)t[0,T]-predictable stochastic processes Xx:[0,T]×ΩHδ, xHδ, δ[0,δˆ), which fulfill for all q[2,), δ[0,δˆ), xHδ, t[0,T] that Xx((0,T]×Ω)H, that sups(0,T]sδXsxLq(P;H)<, and P-a.s. thatXtx=etAx+0te(ts)AF(Xsx)ds+0te(ts)AB(Xsx)dWs and (ii) thatδ[0,δˆ),q[2,):supx,yHδ,xysupt(0,T]max{tδXtxLq(P;H)max{1,xHδ},tδXtxXtyLq(P;H)xyHδ}<. Here and below for two R-Banach spaces (V,V) and (W,W) we denote by Lip(V,W) the set of all Lipschitz continuous functions from V to W and for two R-Banach spaces (V,V) and (W,W) and a function fLip(V,W) we denote by |f|Lip(V,W)[0,) the Lipschitz semi-norm associated to f (see (13) in Subsection 1.1 below for details). The finiteness of the second element in the set in the maximum in (7) follows from the perturbation estimate (3) (with Y1=Xx and Y2=Xy for x,yHδ, δ[0,δˆ) in the notation of (3)) and the finiteness of the first element in the set in the maximum in (7) is a consequence from (5), which, in turn, also follows from the perturbation estimate (3) (see above and the proof of Corollary 2.10 for details). Roughly speaking, Corollary 2.10 establishes the existence of mild solutions of the SEE (6) and also establishes the Lipschitz continuity of the solutions with respect to the initial conditions for any initial condition in Hδ and any δ<δˆ=12[1+1{0}(|B|Lip(H,HS(U,Hβ)))] (see (7)). In Corollary 3.1, Proposition 3.2, Proposition 3.4, and Proposition 3.5 below we demonstrate that the regularity barrierδˆ=12[1+1{0}(|B|Lip(H,HS(U,Hβ)))]={12:B is not a constant function1:B is a constant function for the regularity of the initial conditions revealed in Corollary 2.10 (and Proposition 2.7 and Theorem 2.9, respectively) can, in general, not essentially be improved. In particular, Corollary 3.1 and Proposition 3.2 below prove in the case where H=U=L2((0,1);R), where β(14,12), where A:D(A)HH is the Laplacian with periodic boundary conditions on H, and where BL(H,HS(H,Hβ)) satisfies u,vH:B(v)u=vu (B is not a constant function) that it holds (i) that there exist up-to-modifications unique (Ft)t[0,T]-predictable stochastic processes Xx:[0,T]×ΩHδ, xHδ, δ[0,12), which fulfill for all q[2,), δ[0,12), xHδ, t[0,T] that Xx((0,T]×Ω)H, that sups(0,T]sδXsxLq(P;H)<, and P-a.s. thatXtx=etAx+0te(ts)AB(Xsx)dWs, (ii) thatδ[0,12),q[2,),t(0,T]:supx,yH,xy[XtxXtyLq(P;H)xyHδ]<, and (iii) thatδ(12,),q[2,),t(0,T]:supx,yH,xy[XtxXtyLq(P;H)xyHδ]=. The SEE (9) is sometimes referred to as a continuous version of the parabolic Anderson model in the literature (see, e.g., Carmona & Molchanov [6]). In addition, Proposition 3.2 below disproves the existence of square integrable solutions of the SEE (9) with initial conditions in (δRHδ)\H1/2. The noise in the counterexample SEE (9) is spatially very rough and one might question whether the regularity barrier (8) can be overcome in the case of more regular spatially smooth noise. In Proposition 3.4 below we answer this question to the negative by presenting another counterexample SEE with a non-constant diffusion coefficient but a spatially smooth noise for which we disprove the existence of square integrable solutions with initial conditions in (δRHδ)\H1/2 (cf., however, also Proposition 3.3 below). Proposition 3.5 below also provides a further counterexample SEE which illustrates the sharpness of the regularity barrier (8) in the case where B is a constant function.

Proposition 2.7, Theorem 2.9, and Corollary 2.10 outlined above (see (3)–(7)) are of particular importance for establishing regularity properties for Kolmogorov backward equations associated to parabolic semilinear SEEs and, thereby, for establishing essentially sharp probabilistically weak convergence rates for numerical approximations of parabolic semilinear SEEs (cf., e.g., Lemmas 4.4–4.6 in Debussche [12], Lemma 3.3 in Wang & Gan [30], (4.2)–(4.3) in Andersson & Larsson [1], Propositions 5.1–5.2 and Lemma 5.4 in Bréhier [2], Lemma 3.3 in Wang [29], (79) in Conus et al. [9], Proposition 7.1, Lemma 10.5, and Lemma 10.10 in Kopec [20], and (183)–(184) in Jentzen & Kurniawan [17]). The analytically weak norm for the initial condition in (7) as well as the singularities in the nonlinear coefficients of the SEE in (1) and (2) above translate in an analytically weak norm for the approximation errors in the probabilistically weak error analysis which, in turn, results in essentially sharp probabilistically weak convergence rates (cf., e.g., Theorem 2.2 in Debussche [12], Theorem 2.1 in Wang & Gan [30], Theorem 1.1 in Andersson & Larsson [1], Theorem 1.1 in Bréhier [2], Theorem 5.1 in Bréhier & Kopec [3], Corollary 1 in Wang [29], Corollary 5.2 in Conus et al. [9], Theorem 6.1 in Kopec [20], and Corollary 8.2 in [17]). The perturbation inequality in Proposition 2.7 (see (3) above) is also useful to establish essentially sharp probabilistically strong convergence rates for numerical approximations and perturbations of SEEs (cf., e.g., Proposition 4.1 in Conus et al. [9] and Proposition 4.3 in [17]).

Throughout this article the following notation is used. For two measurable spaces (A,A) and (B,B) we denote by M(A,B) the set of all A/B-measurable functions. For a set A we denote by P(A) the power set of A and we denote by #A:P(A)[0,] the counting measure on A. For a Borel measurable set AB(R) we denote by μA:B(A)[0,] the Lebesgue-Borel measure on A. For a real number T(0,) and a probability space (Ω,F,P) with a normal filtration (Ft)t[0,T] (see, e.g., Definition 2.1.11 in [23]) we call the quadruple (Ω,F,P,(Ft)t[0,T]) a stochastic basis. For a real number T(0,) and a filtered probability space (Ω,F,P,(Ft)t[0,T]) we denote by Pred((Ft)t[0,T]) the sigma-algebra given byPred((Ft)t[0,T])=σ[0,T]×Ω({(s,t]×A:s,t[0,T],s<t,AFs}{{0}×A:AF0}) (the predictable sigma-algebra associated to (Ft)t[0,T]). We denote by h:RR, h(0,), the functions which satisfy for all h(0,), tR that th=min([t,){0,h,h, 2h,2h,}). For R-Banach spaces (V,V) and (W,W) we denote by ||Lip(V,W):C(V,W)[0,] and Lip(V,W):C(V,W)[0,] the functions which satisfy1 for all fC(V,W) that|f|Lip(V,W)=sup({f(x)f(y)WxyV:x,yV,xy}{0}),fLip(V,W)=f(0)W+|f|Lip(V,W) and we denote by Lip(V,W) the set given by Lip(V,W)={fC(V,W):|f|Lip(V,W)<}. We denote by B:(0,)2(0,) the function with the property that for all x,y(0,) it holds that B(x,y)=01t(x1)(1t)(y1)dt (Beta function). We denote by Eα,β:[0,)[0,), α,β(,1), the functions which satisfy for all α,β(,1), x[0,) thatEα,β[x]=1+n=1xnk=0n1B(1β,k(1β)+1α) (generalized exponential function; cf. Lemma 7.1.1 in Chapter 7 in Henry [14], (1.0.3) in Chapter 1 in Gorenflo et al. [13], and Lemma 2.6 below). For a separable R-Hilbert space (H,H,,H), real numbers T(0,), ηR, r[0,), s[0,1], p[1,), a,λ(,1), b(,12), and a generator A:D(A)HH of a strongly continuous analytic semigroup with spectrum(A){zC:Re(z)<η} we denote by χA,ηr,T,κA,ηs,T[0,) the real numbers given byχA,ηr,T=supt(0,T]tr(ηA)retAL(H) and κA,ηs,T=supt(0,T]ts(ηA)s(etAIdH)L(H) (cf., e.g., [24, Lemma 11.36]) and we denote by ΘA,η,p,Ta,b,λ:[0,)2[0,] the function which satisfies for all L,Lˆ[0,) thatΘA,η,p,Ta,b,λ(L,Lˆ)={2|E2λ,max{a,2b}[|χA,ηa,TL2T(1a)1a+χA,ηb,TLˆp(p1)T(12b)|2]|1/2:(λ,Lˆ)(,12)×(0,)Eλ,a[χA,ηa,TLT(1a)]:Lˆ=0:otherwise. For a measure space (Ω,F,μ), a measurable space (S,S), and an F/S-measurable function f:ΩS we denote by [f]μ,S the set given by[f]μ,S={gM(F,S):(AF:μ(A)=0 and {ωΩ:f(ω)g(ω)}A)} and, as usual, we often do not distinguish between an F/S-measurable function f:ΩS and its equivalence class [f]μ,S.

Throughout this article the following setting is frequently used. Consider the notation in Section 1.1, let (H,H,,H) and (U,U,,U) be separable R-Hilbert spaces with #H(H) >1, let T(0,), ηR, let (Ω,F,P,(Ft)t[0,T]) be a stochastic basis, let (Wt)t[0,T] be an IdU-cylindrical (Ft)t[0,T]-Wiener process, let A:D(A)HH be a generator of a strongly continuous analytic semigroup with spectrum(A){zC:Re(z)<η}, let (Hr,Hr,,Hr), rR, be a family of interpolation spaces associated to ηA.

Section snippets

Stochastic evolution equations (SEEs) with singularities at the initial time

In the main result of this section, see Theorem 2.9 in Subsection 2.4 below, we establish existence, uniqueness, and regularity properties for solutions of certain SEEs with time-dependent coefficients and singularities at the initial time. In Subsection 2.1 below we formulate the precise framework which we employ to state Theorem 2.9 in Subsection 2.4 below. The framework in Subsection 2.1 is similar to the hypothesis used in the introductory section above.

Examples and counterexamples for SEEs with irregular initial values

Corollary 2.10 in Subsection 2.4 above establishes existence, uniqueness, and regularity properties for solutions of parabolic SEEs. In this section we first illustrate the statement of Corollary 2.10 in the case of semilinear stochastic heat equations with space-time white noise and periodic boundary conditions; see Corollary 3.1 in Subsection 3.2 below. Roughly speaking, Corollary 3.1 shows existence and uniqueness of solutions of the considered stochastic heat equation provided that the

Acknowledgments

Stig Larsson and Christoph Schwab are gratefully acknowledged for some useful comments. This project has been supported through the SNSF-Research project 200021_156603 “Numerical approximations of nonlinear stochastic ordinary and partial differential equations”.

References (30)

  • L. Chen et al.

    Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions

    Ann. Probab.

    (2015)
  • D. Conus et al.

    Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

    Ann. Appl. Probab.

    (2019)
  • G. Da Prato et al.

    Stochastic Equations in Infinite Dimensions

    (1992)
  • G. Da Prato et al.

    Ergodicity for Infinite-Dimensional Systems

    (1996)
  • A. Debussche

    Weak approximation of stochastic partial differential equations: the nonlinear case

    Math. Comput.

    (2011)
  • Cited by (0)

    View full text