Initial measures for the stochastic heat equation

www.imstat.org/aihp 2014, Vol. 50, No. 1, 136–153

DOI:10.1214/12-AIHP505

© Association des Publications de l’Institut Henri Poincaré, 2014

Initial measures for the stochastic heat equation

Daniel Conus

, Mathew Joseph

, Davar Khoshnevisan

and Shang-Yuan Shiu

aDepartment of Mathematics, Lehigh University, Christmas–Saucon Hall, 14 East Packer Avenue, Bethlehem, PA 18015, USA.

E-mail:[email protected]

bDepartment of Mathematics, University of Utah, 155 South 1400 East JWB 233, Salt Lake City, UT 84112-0090, USA.

E-mail:[email protected];[email protected]

cInstitute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan.

E-mail:[email protected]

Received 18 October 2011; revised 13 June 2012; accepted 19 June 2012

Abstract. We consider a family of nonlinear stochastic heat equations of the form∂tu=Lu+σ (u)W˙, whereW˙ denotes space–

time white noise,Lthe generator of a symmetric Lévy process onR, andσ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measureu₀. Tight a priori bounds on the moments of the solution are also obtained.

In the particular case thatLf =cffor somec >0, we prove that ifu₀is a finite measure of compact support, then the solution is with probability one a bounded function for all timest >0.

Résumé. Nous considérons une famille d’équations de la chaleur stochastique de la forme∂_tu=Lu+σ (u)W, où˙ W˙ est un bruit- blanc espace–temps,Lest le générateur d’un processus de Lévy symétrique surR, etσest une fonction lipschizienne s’annulant en 0. Nous montrons que cette équation aux dérivées partielles stochastique a une solution de type champ aléatoire pour toute mesure initiale finieu₀. Nous obtenons également des bornes a priori sur les moments de la solution.

Dans le cas particulier oùLf=cfpour unc >0, nous montrons que siu₀est une mesure finie à support compact, la solution est presque sûrement une fonction bornée pour toutt >0.

MSC:Primary 60H15; secondary 35R60

Keywords:The stochastic heat equation; Singular initial data

1. Introduction

Consider the stochastic heat equation

∂

∂tu_t(x)=κ 2

∂²

∂x²u_t(x)+σ

u_t(x)W˙_t(x), (1.1)

whereκ>0 is a constant,σ:R→Ris a Lipschitz function that satisfies

σ (0)=0, (1.2)

andW˙ denotes space–time white noise. In other words,W˙ is a mean-zero generalized Gaussian random field [17], Ch. 2, §2.4, with covariance measure Cov(W˙_t(x),W˙_s(y)):=δ₀(x−y)δ₀(t−s)for alls, t≥0 andx, y∈R.

1Supported in part by the NSF Grant DMS-07-47758.

2Supported in part by the NSF Grant DMS-10-06903.

The solution to (1.1) represents the density of heat in an idealized thin metal rod that is placed in a homogeneous medium, the white noise represents a nonlinear source/sink of heat, and the constantκ/2>0 – the so-calledviscosity coefficient – denotes the viscosity of the medium. It is well known that (1.1) has a random-field solution if, for example, the initial heat profileu₀is a bounded and measurable function [23], Ch. 3.

Now suppose thatu₀:R→R₊ is in fact bounded uniformly away from zero, as well as infinity; i.e., that 0<

infu₀≤supu₀<∞. We have shown recently [10] that, in that case,x→u_t(x)is a.s. unbounded for allt >0 under various conditions on σ. In particular, ifσ (x)=cx for a constantc >0 – this is the so calledparabolic Anderson model[7] – then our results [10] imply that

0<lim sup

|x|→∞

logu_t(x)

(log|x|)^2/3 <∞ a.s. (1.3)

Eq. (1.3) holds, for instance, for the parabolic Anderson model in the important [flat] case thatu₀(x)≡1.

Another well-studied case is the parabolic Anderson model when u₀=δ₀is point mass at 0 [the narrow-wedge case]. This case arises in the study of directed random polymers [20]. Gérard Ben Arous, Ivan Corwin, and Jeremy Quastel have independently asked us whether (1.3) continues to hold in that case (private communications). One of the goals of the present articles is to prove that the answer to this question is “no.” In fact, we have the following much more general fact, which is a corollary to the development of this paper.

Theorem 1.1. Ifσ (0)=0andu₀is a finite measure of compact support,thensup_x∈Ru_t(x)=sup_x∈R|u_t(x)|<∞ a.s.for allt >0.

2. Some background material

We begin by recalling some well-known facts; also, we use this opportunity to set forth some notation that will be used consistently in the sequel.

2.1. White noise

Throughout letW := {W_t(x)}t≥0,x∈R denote a two-parameter Brownian sheet indexed byR₊×R; that is,W is a two-parameter mean-zero Gaussian process with covariance

Cov

Wt(x), Ws(y)

=min(s, t)min

|x|,|y|

1(0,∞)(xy) (2.1)

for alls, t≥0 andx, y∈R. The space–time mixed derivative ofW_t(x)is denoted byW˙_t(x):=∂²W_t(x)/(∂t ∂x)and is calledspace–time white noise. Space–time white noise is a generalized Gaussian random field with mean zero and covariance measure Cov(W˙_t(x),W˙_s(y))=δ0(x−y)δ0(t−s).

2.2. Lévy processes

LetX:= {Xt}t≥0denote a symmetric Lévy process onR. That is,t→Xtis [almost surely] a right-continuous random function with left limits at everyt >0 whose increments are independent, identically distributed and symmetric. It is well known that X is a strong Markov process; see Jacob [19] for this and all of the analytic theory of Lévy processes that we will require here and throughout. We denote the infinitesimal generator ofXbyL. According to the Lévy–Khintchine formula, the law of the processXis characterized by itscharacteristic exponent; that is a function Ψ:R→Cthat is determined via the identity E exp(iξ·X_t)=exp(−tΨ (ξ )), valid for allt≥0 andξ∈R. Elementary arguments show that, because X is assumed to be symmetric, the characteristic exponentΨ is a nonnegative – in particular real valued – symmetric function. For reasons that will become apparent later on, we will be interested only in symmetric Lévy processes that satisfy the following:

_∞

−∞e^{−t Ψ (ξ )}dξ <∞ for allt >0. (2.2)

In such a case, the inversion formula for Fourier transforms applies and tells us thatXhas transition densitiesp_t(x) that can be defined by

pt(x):= 1 2π

_∞

−∞e^{−ix·ξ−t Ψ (ξ )}dξ (t >0, x∈R). (2.3)

Note that the function(t, x)→p_t(x)is continuous uniformly on(η,∞)×Rfor everyη >0.

Let us note two important consequences of the preceding formula for transition densities:

1. pt(x)≤pt(0)for allt >0 andx∈R; and 2. t→p_t(0)is nonincreasing.

We will appeal to the these properties without further mention.

Throughout we assume also that the transition densities of the Lévy processX satisfy the following regularity condition:

Θ:=sup

t >0

p_{t /2}(0) pt(0)

<∞. (2.4)

Becausept(0)≥pt(x)and_∞

−∞pt(x)dx=1, it followspt(0) >0 and henceΘis well defined [though it could in principle be infinity whenXis a general symmetric Lévy process].

Let us mention one example very quickly before we move on.

Example 2.1. LetXdenote a one-dimensional standard Brownian motion.Then,Xis a symmetric Lévy process with transition densities given byp_t(x):=(2πt )^−1/2exp{−x²/(2t )}fort >0andx∈R.In this case,we may note also thatLf =(1/2)f,Ψ (ξ )= ξ²/2,andΘ=√

2.

3. The main result

Our main goal is to study the nonlinear stochastic heat equation

∂

∂tu_t(x)=(Lu_t)(x)+σ

u_t(x)W˙_t(x) fort >0, x∈R, (3.1)

where:

1. Lis the generator of a symmetric Lévy process{X_t}t≥0that satisfies (2.2);

2. σ:R→Ris Lipschitz continuous with Lipschitz constant Lip_σ; and 3. σ (0)=0.

As regards the initial data, we will assume here and throughout that

u₀is a nonrandom, finite Borel measure onR. (3.2)

We recall from Walsh [23] that a solution to (3.1) amild solutionif it solves the following random integral equation:

ut(x)=(pt∗u0)(x)+

(0,t )×R

pt−s(y−x)σ us(y)

W (dsdy). (3.3)

It is known that a mild solution to (3.1) exists, under the above assumptions, provided that u₀ is a bounded and measurable, nonrandom function [23], Ch. 3. We will soon see that the same fact remains to hold under the less restrictive condition (3.2). Since we will never need another notion of a solution to (1.1), from now on we will mean

“mild solution” when we refer to a “solution” to (1.1).

The best-studied special case of the random heat equation (3.1) is whenLf =νf is a constant multiple of the Laplacian. In that case, Eq. (3.1) arises for several reasons that include its connections to the stochastic Burgers’

equation (see Gyöngy and Nualart [18]), the parabolic Anderson model (see Carmona and Molchanov [7]) and the KPZ equation (see Kardar, Parisi and Zhang [21]).

One can think of the solutionu_t(x)to (3.1) as the expected density of particles, at placex∈Rand timet >0, for a system of interacting branching random walks in continuous time: The particles move as independent Lévy processes onR; and the particles move through an independent external random environment that is space–time white noiseW˙. The mutual interactions of the particles occur through a possibly-nonlinear birthing mechanismσ. The special case Lf=νfdeals with the case that the mentioned particles move as independent Brownian motions.

The most special example of (3.1) is whenσ (x)≡0; that is the linear heat [Kolmogorov] equation forL, whose [weak] solution isu_t(x)=(p_t∗u₀)(x). It is a simple exercise in harmonic analysis that whenσ (x)≡0, the solution to (3.1) exists, is unique, and is a bounded function for all timet >0. Indeed,

(p_t∗u₀)(x)=

R

p_t(y−x)u₀(dy)≤p_t(0)

R

u₀(dy)=p_t(0)u0(R) <∞. (3.4) Hence, sup_x∈R(p_t∗u₀)(x)is finite, as was asserted.

Consider the case where the characteristic exponentΨ of our Lévy processXsatisfies the following condition: For some [hence all]β >0,

Υ (β):= 1 2π

_∞

−∞

dξ

β+2Ψ (ξ )<∞. (3.5)

It is well known that if, in addition,u0is a bounded and measurablefunction, then (3.1) has a solution that is a.s.

unique among all possible “natural” candidates. This statement follows easily from the theory of Dalang [11], for instance. Moreover, Dalang’s theory shows also that (3.5) is necessary as well as sufficient for the existence of a random-field solution to (3.1) whenσ is a constant function. This is why we assume (3.5) per force. The technical condition (2.2) in fact follows as a consequence of (3.5); see Foondun et al. [16], Lemma 8.1.

Dalang’s method proves also the following without any extra effort; for close variations on this, see also Foondun and Khoshnevisan [15]:

Theorem 3.1. Supposeu₀ is a random field,independent of the white noiseW˙,such thatsup_x∈RE(|u₀(x)|^k) <∞ for somek∈ [2,∞).Then(3.1)has a mild solution{u_t(x)}t >0,x∈R that solves the random integral equation(3.3).

Furthermore,{u_t(x)}t >0,x∈Ris a.s.-unique in the class of all predictable random fields{v_t(x)}t >0,x∈Rthat satisfy:

sup

t∈(0,T )

sup

x∈R

Ev_t(x)^k

<∞ for allT >0. (3.6)

Finally,the random field(t, x)→u_t(x)is continuous in probability.

We will not describe the proof, since all of the requisite ideas are already in the paper [11]. However, we mention that the reference to “predictable” assumes tacitly that the Brownian filtration of Walsh [23] has been augmented with the sigma-algebra generated by the random field{u₀(x)}x∈R. The mentioned stochastic integrals are also as defined in [23]. We will need the following variation of a theorem of Foondun and Khoshnevisan [15] also:

Theorem 3.2 (Foondun and Khoshnevisan [15]). Supposeu0is a random field,independent of the white noiseW˙, such thatsup_x∈RE(|u0(x)|^k) <∞for everyk∈ [2,∞).Then the mild solution{ut(x)}t >0,x∈R to(3.1)satisfies the following:For allε >0there exists a finite and positive constantCεsuch that for allt >0andk∈ [2,∞),

sup

x∈R

Eu_t(x)^k

≤C_ε^ke^{(1+ε)γ (k)t}, (3.7)

whereγ (k)is defined by:

γ (k):=inf β >0:Υ (2β/k) < 1 4kLip²_σ

. (3.8)

Once again, we omit the proof, since it follows closely the ideas of the paper [15] without making novel alterations.

The existence and uniqueness of the solution to (3.1) under a measure-valued initial condition has been studied earlier in various papers. For example, Bertini and Cancrini [1] obtain moment formulas for the parabolic Anderson model [that is,σ (x)=cx] in the special case thatLf =(κ/2)fandu₀=δ₀is the Dirac point mass at zero. They also give sense to what a “solution” might mean. The most-recent word on this topic can be found in Borodin and Corwin [2]. Weak solutions to the fully-nonlinear equation (3.1) have been studied in Conus and Khoshnevisan [9]. An independent work in preparation by Chen and Dalang [8] establishes, in the framework of Walsh [23], the existence of random field solutions to (3.1) in the case whereLf =f, and derives very precise information about the moments of the solution.

We are now ready to state one of the main results of this paper, which extends the previous two results to the case when the initial data is a nonrandom, finite Borel measure.

Theorem 3.3. IfΘ <∞and (3.2)holds,then(3.1)has a mild solutionu that satisfies the following for all real numbersx∈R,ε, t >0,andk∈ [2,∞):There exists a positive and finite constantC_ε:=C_ε(Θ)– depending only on εandΘ– such that

Eu_t(x)^k

≤C^k_εe^{(1+ε)γ (k)t}

1+p_t(0)(pt∗u₀)(x)k/2

, (3.9)

where

γ (k):=inf β >0 :Υ (2β/k) < 1 4kLip²_σ

. (3.10)

Moreover,the solution is almost-surely unique among all predictable random fieldsvthat solve(3.1)and satisfy sup

t >0

sup

x∈R

E(|vt(x)|²)

e^{(1+ε)γ (2)t}{1∨p_t(0)(pt∗u₀)(x)}

<∞ for someε >0. (3.11)

From this we shall see that, in the particular case whereLis a multiple of the Laplacian, the solution remains bounded for every finite timet >0, as long as the finite initial measureu0has compact support. This verifies Theo- rem1.1.

4. An example

Consider (3.1) whereL= −κ(−Δ)^α/2 is the fractional Laplacian of indexα∈(0,2], where κ>0 is a viscosity parameter. The operatorL is the generator of a symmetric stable-α Lévy process with Ψ (ξ )∝κ|ξ|^α, where the constant of proportionality does not depend on(κ, ξ ). It is possible to check directly thatΥ (1) <∞if and only if α >1. Let us restrict attention to the case thatα∈(1,2], and recall that we consider only the case thatu₀is a finite measure.

A computation shows thatΥ (β)∝κ^−1/αβ^{−(α−1)/α} uniformly for allβ >0. Moreover,p_t(x)is a fundamental solution of the heat equation forL, andΘ=2^1/α sincep_t(0)∝(κt )^−1/αuniformly for allt >0. Theorem3.3then tells us that (3.1) has a unique mild solution which satisfies the following for all real numberst >0 andk∈ [2,∞):

sup

x∈R

Eu_t(x)^k

≤C₁^k

1+(κt )⁻¹k/α

exp

C2

t k^{(2α−1)/(α−1)} κ^1/(α−1)

, (4.1)

whereC₁andC₂are positive and finite constants that do not depend on(t, k,κ). In other words, the large-tbehavior of thekth moment of the solution is, as in [15], the same as it would be hadu₀been a bounded measurable function;

that is, lim sup

t→∞

1 t log sup

x∈R

u_t(x)

k≤const·

k^α/κ1/(α−1)

. (4.2)

However, we also observe the small-testimate, lim sup

t↓0

t^1/αsup

x∈R

u_t(x)

k≤const·κ^−1/α, (4.3)

which is a new property. Moreover, the preceding estimate is tight. Indeed, it is not hard to see that u_t(x)

k≥u_t(x)

2≥(p_t∗u₀)(x). (4.4)

Therefore, in the case thatu₀is apositive-definitefinite measure, sup

x∈R

u_t(x)

k≥(p_t∗u0)(0)= 1 2π

_∞

−∞e^{−const·κt|ξ|α}uˆ0(ξ )dξ

= 1

2π(tκ)^1/α _∞

−∞e^{−const·|ξ|α}uˆ0

ξ (tκ)^1/α

dξ. (4.5)

The second inequality follows from applying Parseval’s identity to(pt+ε∗u0)(x)and then lettingε↓0 using Fatou’s lemma. Another application of Fatou’s lemma then shows that

lim inf

t↓0 t^1/αsup

x∈R

ut(x)

k≥const·κ^−1/α, (4.6)

as long asu₀is a positive-definite finite measure such that lim_|z|→∞uˆ₀(z) >0; that is, as long as the conclusion of the Riemann–Lebesgue lemma doesnotapply tou₀. Thus, (4.3) is tight, as was claimed. There are many examples of such measureu₀. For instance, we can chooseu₀=aδ₀+μ, wherea >0 andμis any given positive-definite finite Borel measure onR.

5. Preliminaries

5.1. Some inequalities

We recall from Foondun and Khoshnevsian [15] that Υ (β):= 1

2π _∞

−∞

dξ β+2Ψ (ξ )=

_∞

0

e^−βtp_t²_L2(R)dt. (5.1)

This is merely a consequence of Plancherel’s theorem. Because Xis symmetric we can describeΥ in terms of the resolvent ofX. To this end define

Υ (β):=

_∞

0

e^−βtp_t(0)dt. (5.2)

This is the resolvent density, at zero, of the Lévy processX. Because of symmetry,

p_t²_L2(R)=(p_t∗p_t)(0)=p_2t(0). (5.3)

Therefore, it follows thatΥ (β)=¹₂Υ (β/2). In particular, Dalang’s condition [11], (26), thm. 2,Υ (1) <∞is equiv- alent to the condition thatΥ (β) <∞for some, hence all,β >0.

We close this subsection with some convolution estimates.

Lemma 5.1. For allt >0, p_t(0)·

t 0

p_r(0)dr≤ t

0

p_t−s(0)ps(0)ds≤2Θ·p_t(0)· t

0

p_r(0)dr. (5.4)

As it turns out, the preceding simple-looking result is the key to our analysis of existence and uniqueness, because it tells us that

_t

0

p_t−s(0)ps(0)

p_t(0) ds↓0 ast↓0, (5.5)

at sharp rate_t

0p_r(0)dr.

Proof of Lemma5.1. The first inequality holds simply becauseps(0)≥pt(0)for alls∈(0, t ). For the second one, we splitt

0pt−s(0)ps(0)ds into two parts:t /2 0 andt

t /2. Note thatpt−s(0)≤pt /2(0)≤Θpt(0)whens∈(0, t /2);

andp_s(0)≤p_{t /2}(0)≤Θp_t(0)whens∈(t/2, t ). The lemma follows from these observations.

Letdenote space–time convolution; that is, (fg)t(x):=

_t

0

ds _∞

−∞dy ft−s(x−y)gs(y), (5.6)

wheneverf, g:(0,∞)×R→R₊are both measurable.

Lemma 5.2. For allt >0,x∈R,andn≥1, p²· · ·p²

ntimes

t(x)≤

2Θ t

0

p_s(0)ds n−1

·p_t(0)pt(x). (5.7)

Proof. The result holds trivially whenn=1. Let us suppose that (5.7) is valid forn=m; we prove that (5.7) is valid also forn=m+1. Note that

Tm+1:=

p²· · ·p²

m+1 times

t(x)

= _t

0

ds _∞

−∞dy

p²· · ·p²

mtimes

t−s(y)p_s²(x−y)

≤ _t

0

ds

2Θ _t−s

0

p_r(0)dr _m−1

p_t−s(0) _∞

−∞dy pt−s(y)p²_s(x−y)

≤

2Θ _t

0

p_r(0)dr

m−1 _t

0

ds pt−s(0) _∞

−∞dy pt−s(y)p²_s(x−y). (5.8)

Sincep_s²(x−y)≤ps(0)ps(x−y), the Chapman–Kolmogorov equation implies that Tm+1≤

2Θ

t 0

p_r(0)dr m−1

· t

0

p_t−s(0)ps(0)ds·p_t(x), (5.9)

and the result follows from this, Lemma5.1, and induction.

Lemma 5.3. For allt >0,x∈R,andn≥1, p²· · ·p²

ntimes

(p_•∗u₀)²

t(x)

≤u₀(R)

2Θ _t

0

p_s(0)ds n

·p_t(0)(pt∗u₀)(x). (5.10)

Proof. For every nonnegative functionf, (5.6) and Lemma5.2together imply that p²· · ·p²

ntimes

f

t(x)

= _t

0

ds

R

dy

p²· · ·p²

ntimes

s(y)f_t−s(x−y)

≤ _t

0

ds

2Θ _s

0

p_r(0)dr n−1

p_s(0)

R

dy ps(y)f_t−s(x−y)

≤

2Θ _t

0

p_r(0)dr

n−1 _t

0

ds ps(0)(ps∗f_t−s)(x). (5.11)

We setf_t(x):=p_t(0)(pt∗u₀)(x)and appeal the Chapman–Kolmogorov property [ps∗p_t−s=p_t] in order to obtain the following:

p²· · ·p²

ntimes

(p_•∗u₀)²

t(x)

≤u₀(R)

p²· · ·p²

ntimes

p_•(0)(p_•∗u₀)

t(x)

≤u₀(R)

2Θ _t

0

p_r(0)dr

n−1 _t

0

p_s(0)pt−s(0)ds·(p_t∗u₀)(x). (5.12)

An application of Lemma5.1completes the proof.

6. Finite-horizon estimates

We first define a sequence{u⁽ⁿ⁾}n∈Nof random fields by:u⁽⁰⁾_t (x):=0 for allt >0 andx∈R. Then, for everyn≥0, we set

u⁽ⁿ_t ⁺¹⁾(x):=(p_t∗u₀)(x)+

(0,t )×R

p_t−s(y−x)σ u⁽ⁿ⁾_s

W (dsdy). (6.1)

Thus,u⁽ⁿ⁾denotes simply thenth stage of a Picard iteration approximation to a reasonable candidate for a solution to (3.1).

Proposition6.1below will show that the random variables{u⁽ⁿ⁾_t (x)}n∈Nare well-defined with values inL^k(P), for x∈Rand all timest that are “reasonably small.”

For eacha >0, let g(a):=inf t >0:

_t

0

pr(0)dr≥a

, (6.2)

where inf∅:= ∞. Clearly,g(a) >0 whena >0.

Proposition 6.1. For all integersn≥0,real numbersk∈ [2,∞)andx∈R, u⁽ⁿ_t ⁺¹⁾(x)

k≤4

1∨k^1/2Lip_σ

u₀(R)pt(0)(pt∗u₀)(x) (6.3)

for0< t≤g((32Θ[1∨kLip²_σ])⁻¹).

Proof. Define

μ^(n,k)_t (x):=u⁽ⁿ⁾_t (x)²

k. (6.4)

Clearly,μ^(0,k)_t (x)≡0. Then, we have

μ⁽ⁿ_t ^+1,k)(x)

≤(p_t∗u₀)(x)+

(0,t )×R

p_t−s(y−x)σ u⁽ⁿ⁾_s (y)

W (dsdy) k

≤(p_t∗u0)(x)+

4k· t

0

ds _∞

−∞dy p²_t−s(y−x)σ

u⁽ⁿ⁾_s (y)²

k

1/2

≤(pt∗u0)(x)+

4kLip²_σ · _t

0

ds _∞

−∞dy p²_t−s(y−x)μ^(n,k)_s (y) 1/2

=(p_t∗u₀)(x)+

4kLip²_σ·

p²μ^(n,k)

t(x)1/2

. (6.5)

Notice, that (3.4) implies that |(p_t∗u₀)(x)|²≤u₀(R)pt(0)(pt∗u₀)(x). Now, defineC_k:=8(1∨kLip²_σ), and note that

μ⁽ⁿ_t ^+1,k)(x)≤2(p_t∗u₀)(x)²+8kLip²_σ·

p²μ^(n,k)

t(x)

≤C_k(p_t∗u₀)(x)²+

p²μ^(n,k)

t(x) ...

≤ n

j=0

C_k^j+1

p²· · ·p²

jtimes

(p_•∗u0)²

t(x). (6.6)

We apply Lemma5.3to find that

μ⁽ⁿ_t ^+1,k)(x)≤Cku0(R)pt(0)(pt∗u0)(x)· n

j=0

2CkΘ·

_t

0

ps(0)ds j

≤2Cku₀(R)pt(0)(pt∗u₀)(x), (6.7)

provided thatt≤g((4CkΘ)⁻¹). This is another way to state the lemma.

Proposition 6.2. If0< t≤g((16kΘLip²_σ)⁻¹),then for all integersn≥0and for allx∈R, u⁽ⁿ_t ⁺¹⁾(x)−u⁽ⁿ⁾_t (x)²

k≤2⁻ⁿu₀(R)pt(0)(pt∗u₀)(x). (6.8)

Proof. Define for alln≥0,t >0, andx∈R, D⁽ⁿ_t ^+1,k)(x):=Eu⁽ⁿ_t ⁺¹⁾(x)−u⁽ⁿ⁾_t (x)^k2/ k

=u⁽ⁿ_t ⁺¹⁾(x)−u⁽ⁿ⁾_t (x)²

k. (6.9)

Clearly,

D⁽ⁿ_t ^+1,k)(x)≤4k _t

0

ds _∞

−∞dy p_t²_−s(y−x)Eσ u⁽ⁿ⁾_s (y)

−σ

u⁽ⁿ_s ⁻¹⁾(y)^k2/ k

≤4kLip²_σ· _t

0

ds _∞

−∞dy p_t²_−s(y−x)D_s^(n,k)(y)

=4kLip²_σ ·

p²D^(n,k)

t(x)

≤ · · · ≤

4kLip²_σn

·

p²· · ·p²

ntimes

D^(1,k)

t(x). (6.10)

BecauseD_s^(1,k)(y)= |(ps∗u0)(y)|²for allk∈ [2,∞), Lemma5.3yields D_t^(n+1,k)(x)≤u0(R)

8kLip²_σΘ t

0

pr(0)dr n

pt(0)(pt∗u0)(x), (6.11)

and hence the result follows.

Let us conclude this section by making a few remarks about the predictability of the Picard iteratesu⁽¹⁾, u⁽²⁾, . . .. [We thank Dr. Le Chen and Professor Robert Dalang for correctly pointing out to us that this issue requires an explanation.] In the case thatLf=f, a detailed proof can be found in Chen and Dalang [8].

We wish to demonstrate that ifZis a predictable random field and satisfies the integrability condition _t

0

ds _∞

−∞dy

pt−s(y−x)2

EZ_s(y)²

<∞, (6.12)

then(t, x)→

(0,t )×Rp_t−s(y−x)Z_s(y)W (dsdy)defines a predictable random field. Thanks to the construction of space–time stochastic integrals, due to Walsh [23], it suffices to consider only the case thatZis an “elementary random field” [23], (2.5), p. 292, and this reduces our problem to one about showing that the Gaussian field

(t, x)→Γ_t(x):=

(0,t )×R

p_t−s(y−x)W (dsdy) (6.13)

is itself a predictable random field. It is not hard to verify the following “stochastic Fubini theorem”:

(Γ_t∗ϕ)(x)=

(0,t )×R

(p_t−s∗ϕ)(y)W (dsdy) a.s., (6.14)

valid for every nonrandom rapidly-decreasing test functionϕ:R→R. A variant of this can be found in Walsh [23], Theorem 2.6, p. 296; the present formulation can be proved in a similar way.

Standard facts about Gaussian random fields and Lévy processes imply that(t, x)→(Γt∗ϕ)(x)is continuous a.s.

[up to a modification], and this implies that(t, x)→(Γ_t ∗ϕ)(x)is a predictable random field. Finally, letϕ_ε denote the probability density function of a mean-zero normal distribution onRwith varianceε. Then, one can check directly that

limε↓0 sup

t∈[0,T]sup

x∈R

E(Γ_t∗ϕε)(x)−Γt(x)²

=0 (6.15)

for allT ∈(0,∞). In accord with the Walsh theory [23], this is sufficient for the predictability of the Gaussian random fieldΓ.

7. Proof of Theorem3.3

We prove Theorem3.3in two parts: First we show that there exists a solution up to time T:=g

64Θ

1∨Lip²_σ−1

. (7.1)

From there on, it is easy to produce an all-time solution, starting from timet=T.

Let{u⁽ⁿ⁾}^∞_n=0be the described Picard iterates defined in (6.1). Since 0<T≤g

32ΘLip²_σ−1

, (7.2)

Proposition6.2implies that the sequence of random variables{u⁽ⁿ⁾_t (x)}n∈N converge inL²(Ω)for every 0< t≤T andx∈R.³

DefineU_t(x):=limn→∞u⁽ⁿ⁾_t (x), where the limit is taken inL²(Ω). By default, {U_t(x);x∈R, t∈(0,T]}is a predictable random field such that

nlim→∞Eu⁽ⁿ⁾_t (x)−U_t(x)^k

=0 for allx∈R, (7.3)

provided thatt∈(0,T_k], whereT_k:=g((32kΘ[1∨Lip²_σ])⁻¹). (Notice thatT₂=T.) Moreover, Proposition6.1tells us that

Ut(x)

k≤4k^1/2[1∨Lip_σ]

u0(R)pt(0)(pt∗u0)(x) (7.4)

for allx∈Randt∈(0,T_k]. Finally, these remarks readily imply that Ut(x)=(pt∗u0)(x)+

(0,t )×R

pt−s(y−x)σ Us(y)

W (dsdy) (7.5)

for allx∈Randt∈(0,T]. In other words,Uis a mild solution to (3.1) up to the nonrandom timeT.

Next we define a space–time white noiseW˙ by defining its Wiener integrals as follows: For allh∈L²(R₊×R),

R₊×R

h_s(y)W(dsdy):=

(T,∞)×R

h_s−T(y)W (dsdy). (7.6)

In other words,W˙ is obtained fromW˙ by shifting the timeTsteps. Induction reveals that every{u⁽ⁿ⁾_t (x);x∈R, t∈ (0,T]}is independent of the space–time white noiseW. Therefore, so is the short-time solution˙ {U_t(x);x∈R, t∈ (0,T]}.

Next letV := {Vt(x)}x∈R,t >0denote the mild solution to the stochastic heat equation

∂

∂tV_t(x)=(LV_t)(x)+σ

V_t(x)W˙t(x), (7.7)

subject toV₀(x)=U_T(x). SinceU_Tis independent of the noiseW, the preceding has a unique solution, thanks to˙ Dalang’s theorem (Theorem3.1). And since sup_x∈RU_T(x)2<∞for allε >0, there existsD_ε∈(0,∞)such that for allt >0, andx∈R,

EV_t(x)²

≤D_ε²e^{(1+ε)γ (2)t}, (7.8)

thanks to Theorem3.2. Finally, we define for allx∈R, u_t(x):= U_t(x) ift∈(0,T],

V_t−T(x) ift >T. (7.9)

Then it is easy to see that the random fielduis predictable, and is a mild solution to (3.1) for allt >0, subject to initial measure beingu0. Uniqueness is a standard consequence of (3.9).

Let us now considerk >2 and follow the same argument as above, but useTkinstead ofT; i.e., we run the solution Uup to timeT_konly and then keep going with the classical technique of Dalang. Then, a similar argument leads us to a moment estimate of orderkforu_t(x), thanks to another application of Theorem3.2. The solution obtained from T_kis the same as the one obtained when stopping atTby the uniqueness result.

We pause to state an immediate corollary of the proof of Theorem3.3, as it might be of some independent interest.

In words, the following shows that ifσ has truly-linear growth andΘ <∞, then the solution to (3.1) has nontrivial moment Liapounov exponents.

3As is customary,Ωdenotes the underlying sample space on which random variables are defined, andL²(Ω)denotes the collection of all random vaiables onΩthat have two finite moments.

Corollary 7.1. SupposeΘ <∞, Lσ:=infx∈R|σ (x)/x|>0,andu₀is a finite Borel measure onR.Then, 0<lim sup

t→∞

1

t log Eu_t(x)^k

<∞ (7.10)

for allk∈ [2,∞).

Proof. Let{V_t(x)}t >0,x∈Rdenote the post-Tkprocess used in the proof of Theorem3.3. It suffices to prove that 0<lim sup

t→∞

1

t log EV_t(x)^k

<∞ (7.11)

for allk∈ [2,∞). This follows from [15].

The proof of Theorem3.3is based on the idea that one can solve (3.1) up to timeT, using the method of the present paper; and then from timeTon we paste the more usual solution, shifted by timeTtime steps, in order to obtain a global solution to (3.1). But in fact since the pre-Tand the post-Tsolutions are unique [a.s.], we could replaceTby any other timeη(not necessarily one of theT_k) before it as well. The following merely enunciates these observations in the form of a proposition. The proof follows from the fact that the sequenceT_k goes to 0 askincreases. We omit the details. However, we state this simple result explicitly, as it will be central to our proof of Theorem1.1.

Proposition 7.2. Choose and fix someη∈(0,T),and let us define the predictable random field{ ¯V_t(x)}t >0,x∈Rexactly as we defined{V_t(x)}t >0,x∈R,except withTreplaced everywhere byη.Finally defineu¯_t(x)as we didu_t(x),except we replace(U, V ,T)by(U,V , η);¯ that is,

¯

u_t(x):= U_t(x) ift∈(0, η],

V¯_t−η(x) ift > η. (7.12)

Then,the random fieldu¯ is a modification of the random fieldu.

8. Stability and positivity

Letudenote the solution to (3.1), as defined in Theorem3.3, starting from a finite Borel measureu₀. We have seen that(pt∗u0)(x)is finite for allt >0 andx∈Rfixed. Also, forε >0, letU^(ε)denote the solution to (3.1), starting from the [bounded and measurable] initial functionpε∗u0.

Proposition 8.1 (Stability). For everyt, ε >0,u_t, U_t^(ε)∈L²(Ω×R).Moreover,the following bound is valid for all β such thatΥ (β)≥(2Lip²_σ)⁻¹:

_∞

0

e^−βtdt _∞

−∞dxEu_t(x)−U_t^(ε)(x)²

≤[u0(R)]² π

_∞

−∞

(1−e^{−εΨ (ξ )})²

β+2Ψ (ξ ) dξ. (8.1)

In particular,the left-hand side tends to zero asε↓0.

Proof. Letu⁽ⁿ⁾_t (x)be thenth Picard iterate, defined in (6.1). Then, u⁽ⁿ_t ⁺¹⁾(x)²

2

≤(pt∗u0)(x)²+Lip²_σ· _t

0

ds _∞

−∞dy p²_t−s(y−x)u⁽ⁿ⁾_s (y)²

2. (8.2)