A NON-NONSTANDARD PROOF OF REIMERS ’ EXISTENCE RESULT FOR HEAT

In 1989, Reimers gave a nonstandard proof of the existence of a solution to heat SPDEs, driven by space-time white noise, when the diffusion coefficient is continuous and satisfies a linear growth condition. Using the martingale problem approach, we give a non-nonstandard proof of this fact, and with the aid of Girsanov’s theorem for continuous orthogonal martingale measures (proved in a separate paper by the author), the result is extended to the case of a measurable drift.


Introduction
We consider the SPDE OU Ot 02U + qr 1 20x2 JOtOx' U(O,) h(), (1.1) where Y: [0, T] for some 0 < T < ee, W(t,x) is the Brownian sheet corresponding to the driving space-time white noise 02W/OtOx, with intensity Lebesgue measure (see [15]). Our main result in this paper is Theorem 1.1: Suppose that a E C(N;N) (continuous real-valued function on N) and h is a deterministic function in Cc(; (continuous real-valued function on N with compact support). Suppose further that there exists a constant K > 0 such that a2(x) _< K(1 + x2), (1.2) for all x R. Then, there exists a solution to the heat SPDE (1.1).
Our approach will be as follows: We approximate the SPDE in (1.1) by a sequence of Stochastic Differential-Difference Equations (SDDEs) associated with interacting diffusion models and 30 HASSAN ALLOUBA solve the SDDEs; We then derive bounds on moments of spatial and temporal differences of the solutions obtained in the previous step; From those bounds, we conclude the tightness of the sequence of solutions; We then extract a subsequential limit, which solves a martingale problem that is equivalent to (1.1). Girsanov's theorem for space-time white noise may then be used (see [1]  where a and h are as above, and the drift b is a Borel measurable real-valued function on R such that the random field X(t,x)b(U(t,x))/a(U(t,x))satisfies the Novikov condition: where U is a solution to the SPDE with no drive (1.1). For the convenience of the reader, we restate below Girsanov's theorem for white noise as well as the existence theorem relating (1.1) to (1.3). The reader is referred to [1] where Z is some predictable random field (see [15] [15] pp. 312-321), that the formulations in (1.6) and (1.7) are equivalent, provided the random field a(U)is locally bounded, which we will assume throughout this article. Remark 1.4: We will sometimes place a superscript Ri, i-1,2,.., above a R mathematical relation; e.g., _<. This makes it easy to refer to the relation in question and renders our explanations more concise. Also, throughout this article, K will denote a constant that may change its value from line to line.

The Interacting Diffusion Models
Consider the sequence of sets (Xn)= 1 defined by where 5 n > 0 for all n and 5 n n__+-+0. Then the SPDE in (1.1) may be approximated by the following sequence of stochastic differential-difference equations (SDDEs)" where t Y and x Xn, and Anf(x is the nth approximate Laplacian given by We think of W(t) as a sequence of standard Brownian motions indexed by x and we assume that, for each n-1,2,...,U(0)h(x) for all x e X n. It follows from the boundedness of h that sup ]Ur(0) _< K.
(2.2) xEX n By a straightforward adaptation of Reimers' observations ( [11], pp. 325-326) we get where Q (t) is the density of a random walk on the lattice Xn, in which the times n between transitions are exponentially distributed with mean 2p52n, where p is the pro-32 HASSAN ALLOUBA bability of a transition to the right (or to the left) and 1-2p is the probability of no transition at a transition epoch. The subscript 5 n in Q (t) is to remind us that the size of each step is 5 n. The second term on the r.h.s, of (2.3) is deterministic and will henceforth be denoted by U(t). From (2.2), it follows that u(t) K.

Some Bounds
Here, we glove bounds on the moments of spatial and temporal differences of the sequence {Un(t))n= ix x) that are used to conclude tightness for our approximating sequence, along with some inequalities related to Q (t) and some bounds on the moments of U(t) that are useful in proving these spatial and temporal bounds.
Since all the results in this section hold for all n, we will suppress the dependence on n to simplify the notation. This section is a simple adaptation of Reimers' corresponding results to our setting, and most of the proofs will be omitted.

Bounds related to Q(t)
Lemma 3.1: There is a constant K such that (Q(t)): <_ si5lv .

The Sequence of SDDEs Solutions is Tight
Let U n(t,x) be the extension of U(t) to 3]-x N obtained by linear interpolation of the U(t)'s between the lattice points of N n. The following is Kolmogorov's continuity criterion for random fields. (See [9] pp. 53-55 and p. 118; see also Corollary 1.   (see also the discussion on p. 97 in [a]). This is a routine argument, so we omit the details.

The Martingale Problem
)n Since the sequence U n(,X) 1 is tight, it follows that there exists a subsequence U nk which induces measures P nk on (C, C) that converge weakly to a limit P, where C: C(YxN;N) and C: %(C), where %(C)is the Borel r-field over C. Now, we can construct processes Yk d nk on some probability following Skorokhod [13], space (fS, S, pS) such that with probability 1, as k-,o, Yk(t,x) converges to a random field Y(t,x) uniformly on compact subsets of 3]-xN for any T < oc. We will show that Y(t,x) is a solution to the heat SPDE (1.1) by solving an equivalent martingale problem ( [8]).