A Note on a Fenyman-Kac-Type Formula

In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation in $[0,1] \times \bR^d$, with multiplicative noise, which is fractional in time and colored in space. This representation is similar to the one given in Dalang, Mueller and Tribe (2008) in the case of an s.p.d.e. driven by a Gaussian noise, which is white in time. Unlike the formula of Dalang, Mueller and Tribe (2008), which is based on the usual Poisson process, our representation is based on the planar Poisson process, due to the fractional component of the noise.


Introduction
The classical Feynman-Kac (F-K) formula gives a stochastic representation for the solution of the heat equation with potential, as an exponential moment of a functional of Brownian paths (see e.g. [14]). This representation is a useful tool in stochastic analysis, in particular for the study stochastic partial differential equations (s.p.d.e.'s). We mention briefly several examples in this direction. The F-K formula lied at the origin of the existence, large time asymptotic and intermittency results of [4] and [5], for the solution of the heat equation with random potential on Z d , respectively R d . The same method, based on a discretized F-K formula, was used in [17] for obtaining an upper bound for the exponential behavior of the solution of the heat equation with random potential on a smooth compact manifold. The technique of [4] was further refined in [6] in the case of parabolic equations with Lévy noise, for proving the exponential growth of the solution. The F-K formula was used in [9] for solving stochastic parabolic equations, in the context of white noise analysis. A F-K formula for the solution of the stochastic KPP equation is used in [15] for examining the asymptotic behavior of the solution.
The present work has been motivated by the recent article [8], in which the authors obtained an alternative probabilistic representation for the solution of a deterministic p.d.e., as well as a representation for the moments of the (mildsense) solution of a s.p.d.e. perturbed by a Gaussian noiseḞ , with "formal" covariance: More precisely, in [8], } with respect to the inner product ·, · P given by In the particular case of the stochastic heat equation: the representation for the second-moments of the (mild-sense) solution u is: (with the convention that on {N t = 0}, the product is defined to be 1), where B 1 = (B 1 t ) t≥0 and B 2 = (B 2 t ) t≥0 are independent d-dimensional Brownian motions starting from x, respectively y, N = (N t ) t≥0 is an independent Poisson process with rate 1 and points τ 1 < τ 2 < . . ., and Note that the representation (2) does not rely on the entire Brownian path, but only on its values at the (random) points τ 1 , τ 2 , . . . , τ Nt . This property has allowed the authors of [8] to generalize the representation to a large class of s.p.d.e.'s, including the wave equation.
To see where the idea for this representation comes from, we recall briefly the salient points leading to (2). If the solution of (1) exists, then it is unique and admits the Wiener chaos expansion: (see e.g. Proposition 4.1 of [8]) where I n (f n (·, t, x)) = ([0,1]×R d ) n f n (t 1 , x 1 , . . . , t n , x n )dF t1,x1 . . . dF tn,xn , and f n ∈ P ⊗n is a symmetric function given by: By the orthogonality of the terms in the series (3), A crucial observation of [8] is that, for any F : [0, t] n → R + measurable, This is a key idea, which yields a probabilistic representation for J n (t, x, y), based on the jump times of the Poisson process. This idea is new in the literature, although it appeared implicitly in the earlier works [10], [13], [16]. Secondly, for any 0 < t 1 < . . . < t n < t, F (t − t n , . . . , t − t 1 ) is represented as: Relation (2) follows from these observations, using the independence between N and B 1 , B 2 .
In this article, we generalize these ideas to the case of the stochastic heat equation driven by a fractional-colored noise. More precisely, we consider the following equation: is non-random, ⋄ denotes the Wick product, andẆ is a Gaussian noise whose covariance is formally given by: Here, H ∈ (1/2, 1), α H = H(2H − 1) and µ is a tempered measure on R d . (Note that η(t, s) is the covariance kernel of the fractional Brownian motion.) The noiseẆ is defined rigourously as in [2], by considering a zero-mean Gaussian process W = {W (h); h ∈ HP} with covariance to the inner product ·, · HP given by: (See also [7] for a martingale treatment of the case η(t, s) = δ 0 (t − s), which corresponds to H = 1/2.) As in [3], the solution of equation (7) is interpreted in the mild sense, using the Skorohod integral with respect to W . More precisely, an adapted square- The existence of a solution u (in the space of square-integrable processes) depends on the roughness of the noise, introduced by H and the kernel f . We mention briefly several cases which have been studied in the literature. If f = δ 0 , the solution exists for d = 1, 2 (see [12]). [11]). Recently, it was shown in [3] that if f is the Riesz kernel of order α, or the Bessel kernel of order α < d, then the solution exists for d ≤ 2 + α, whereas if f is the heat kernel or the Poisson kernel, then the solution exists for any d.
Our main result establishes a probabilistic representation for the second moment of the solution, similar to (2), and based on the planar Poisson process.

Proof of Theorem 1.1
We begin by recalling some basic facts about the planar Poisson process. If We say that N = (N t,s ) (t,s)∈[0,1] 2 is a planar Poisson process of rate λ > 0, if it satisfies the following conditions: (i) N vanishes on the axes, i.e. N t,0 = N 0,t = 0 for all t ∈ [0, 1].
For any n ≥ 1 and for any distinct positive integers i 1 , . . . , i n , let A i1,...,in (t, s) be the event that N has points P i1 , . . . , P in in [0, t] × [0, s]. Then The following result is probably well-known. We include it for the sake of completeness.
The next result gives a probabilistic representation of the inner product appearing in (13). This result is the analogue of (6), needed for the treatment of the fractional case. Conclusion of the Proof of Theorem 1.1: Since equation (7) has a solution, this solution is unique and admits the Wiener chaos expansion (3). The result follows from (11), (13) and Lemma 2.2.