On stochastic heat equation with measure initial data

The aim of this short note is to obtain the existence, uniqueness and moment upper bounds of the solution to a stochastic heat equation with measure initial data, without using the iteration method in Chen and Dalang(2015), Chen and Kim(2016).


Introduction
Consider the stochastic heat equation for (t, x) ∈ (0, ∞) × R d (d 1) where L is the generator of a Lévy process X = {X t } t 0 .Ẇ is a centered Gaussian noise with covariance formally given by where f is some nonnegative and nonnegative definite function whose Fourier transform is denoted byf in distributional sense, and δ denotes the Dirac delta function at 0. For some technical reasons, we will assume that f is lower semicoutinous (see Lemma 4 below). Let Φ be the Lévy exponent of X t , we will assume that exp(−ReΦ) ∈ L t (R d ) for all t > 0 .
Thus according to Proposition 2.1 in [5], X t has a transition function p t (x) and we can (and will) find a version of p t (x) which is continuous on (0, ∞) × R d and uniformly continuous for all (t, x) ∈ [η, ∞) × R d for every η > 0, and that p t vanishes at infinity for all t > 0.
The initial condition u(0, ·) is assumed to be a (positive) measure µ(·) such that To avoid trivialities, we assume that µ(·) ≡ 0. Using iteration method, the existence, uniqueness and some moment bounds of the solution have been obtained in [1,2,3] for the case b ≡ 0 and for some specific choice of L. However, these approaches rely on the structure (or asymptotic structure) of p t (x). In this article, we will study the equation (1) with also a Lipschitz drift term b and establish the existence, uniqueness and p-th moment upper bound, without using the iteration method in [1,2,3], also, our cribteria only need some integrability of the Lévy exponent.
To state the result, let us recall that by a solution u to (1) we mean a mild solution. That is, (i) u is a predictable random field on a complete probability space {Ω, F , P }, with respect to the Brownian filtration generated by the cylindrical Brownian motion defined by B t (φ) := [0,t]×R d φ(y)W (ds, dy), for all t 0 and measurable φ : and (ii) for any (t, x) ∈ (0, ∞) × R d , the following equation holds a.s.
where p t (x) is the transition function for X t and the stochastic integral above is in the sense of Walsh [6]. The following theorem is the main result of this paper.
Theorem 1. Assume that the initial condition satisfies (3) and assume that for any β > 0. And assume that σ and b are Lipschitz functions with Lipschitz coefficients L σ , L b > 0 respectively. Then there exists a unique mild solution to equation (1). Moreover, defineγ where where and z p denotes the largest positive zero of the Hermite polynomial He p .

Proof of Theorem 1
In the proof of Theorem 1 we will need two results about taking Fourier transforms, which we now state.
Lemma 4 (Corollary 3.4 in [5]). Assume that f is lower semicontinuous, then for all Borel probability measures ν on R d , Lemma 5. If f is lower semicontinuous, then Proof. We begin by noting that and as a function of y, the quotient p t−s (x−y)ps(y−z) is the probability density of the Lévy bridgeX z,x,t = {X z,x,t (s)} 0 s t which is at z when s = 0 and at x when s = t. Actually, X z,x,t (s) can be written as hence by the independence of increment of Lévy process, we have Thus, an application of Lemma 4 to ν j (dy) = which proves the lemma.
To prove Theorem 1, we first define a norm for all β, p > 0 and all predictable random Let B β,p denote the collection of all predictable random fields v := {v(t, x)} t 0,x∈R d such that v β,p < ∞. We note that after the usual identification of a process with its modifications, B β,p is a Banach space (see Section 5 in [5]).
Note that by the dominated convergence theorem, the condition B(β, p) < 1 can be achieved if β is sufficiently large.
(13) is clearly true for n = 0. Using induction, assume (13) is true for some n, using Burkholder inequality (see [4]) and the assumption on σ and b , we obtain , multiplying both sides by e −βt and applying Minkowski's inequality to the third summand above we obtain where in obtaining I 2 and I 3 above, we have used the bound We will estimate I 1 , I 2 , I 3 separately. For I 1 , the semigroup property of p t (x) yields For I 2 , an application of Lemma 4 to ν(dy) = p t−s (x − y)dy yields thus we obtain Finally, an application of Lemma 5 yields Combining the estimates for I 1 , I 2 , I 3 , we arrive at where B(β, p) is defined in (10). Using the iteration, we see that (13) holds for all n 1 if B(β, p) < 1. The same technique applied to u n+1 (t,x)−u n (t,x) τ +pt * µ(x) yields that u n+1 − u n τ + p * µ β,p B(β, p) u n − u n−1 τ + p * µ β,p , and if β is chosen such that B(β, p) < 1, we will obtain that ∞ n=1 u n − u n−1 τ + p * µ β,p < ∞ .
Therefore, we can find a predictable random field u ∞ ∈ B β,p such that lim n→∞ u n = u ∞ in B β,p . It is easy to see that this u ∞ is a solution to equation (4), and uniqueness is checked by a standard argument. To prove (9), we note that since u ∈ B β,p for those β such that B(β, p) < 1, sup x∈R d u(t, x) τ + p t * µ(x) L p (Ω) sup x∈R d τ τ + p t * µ(x) + Ce βt for some C > 0 which does not depend on t, thus (9) is proved and the proof of Theorem 1 is complete.

Acknowledgement
The author thanks Davar Khoshnevisan and David Nualart for stimulating discussions and encouragement.