Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution

A mild formulation for stochastic parabolic Anderson model with time-homogeneous Gaussian potential suggests a way of defining a solution to obtain its optimal regularity. Two different interpretations in the equation or in the mild formulation are possible with usual pathwise product and the Wick product: the usual pathwise interpretation is mainly discussed. We emphasize that a modified version of parabolic Schauder estimates is a key idea for the existence and uniqueness of a mild solution. In particular, the mild formulation is crucial to investigate a relation between the equation with usual pathwise product and the Wick product.


Introduction
We start with the stochastic parabolic Anderson model (PAM) ∂u(t, x) ∂t = ∆u(t, x) + u(t, x)Ẇ (x) (1.1) driven by multiplicative Gaussian white noise potential in d space dimensions, where ∆ is the Laplacian operator, andẆ (x) is time-homogeneous Gaussian white noise with mean zero and covariance E Ẇ (x)Ẇ (y) = δ(x − y), where δ is the Dirac-delta function. Note that the white noise is a generalized process, and we need to make sense of the multiplication uẆ in (1.1).
Since the notationẆ (x) stands for the formal derivative of a Brownian sheet W (x), the equation (1.1) may be written as ∂u(t, x) ∂t = ∆u(t, x) + u(t, x) ∂ d ∂x 1 · · · ∂x d W (x). (1.2) Depending on the multiplication between u and ∂ d ∂x 1 · · · ∂x d W , the equation (1.2) may be classified as follows: • Stochastic PAM with usual pathwise product (Stratonovich interpretation): • Stochastic PAM with Wick product ⋄ (Wick-Itô-Skorokhod interpretation): As the idea of Stratonovich integral suggests, the equation (1.3) is equivalently interpreted as a pathwise limit of approximated equations (1.5) where W ε are smooth approximations of each sample path of W for ε > 0.
If d ≥ 2, the usual pathwise product of u · ∂ d ∂x 1 · · · ∂x d W is classically not well-defined, and constructing a solution of (1.3) presents a major challenge in the standard theory of stochastic partial differential equations. Indeed, it is known that d-dimensional Brownian sheet W has regularity 1/2 − ε in each parameter for any ε > 0. Therefore, ∂ d ∂x 1 · · · ∂x d W can be understood to have regularity −d/2 − ε in total. Then, the solution u is expected to have regularity 2−d/2−ε so that the sum of the regularity of u and ∂ d ∂x 1 · · · ∂x d W is strictly less than 0. Hence, unfortunately, classical integration theory cannot be applied to the product u · ∂ d ∂x 1 · · · ∂x d W , and advanced techniques are inevitably required.
When d = 2 and on the whole space R 2 , the paper [6] introduces a particular renormalization procedure and constructs a solution to (1.3) by subtracting a divergent constant from the equation. On a torus of R 2 , a solution of (1.3) is constructed independently using paracontrolled distributions in [3] and using the theory of regularity structures in [4]. When d = 3, using the theory of regularity structures, the paper [7] carries out the construction of (1.3) on the whole space R 3 . An alternative construction of solution to (1.3) on a torus of R 3 is also established in [5].
It turns out that for the model (1.3) when d = 1, there are several ways to define a solution in the Stratonovich sense, and the regularity of solution is worth attention.
In [9], the Feynman-Kac solution for (1.3) in the Stratonovich sense is introduced. The paper proves that the Feynman-Kac solution is almost Hölder 3/4 continuous in time and almost Hölder 1/2 continuous in space. Here, "almost" Hölder continuity of order γ means Hölder continuity of any order less than γ. However, the standard parabolic theory implies that the spatial regularity can be improved. Indeed, consider the additive model as a reference for optimal regularity. It is known that the explicit solution of (1.6) is almost Hölder 3/4 continuous in time and almost Hölder 3/2 continuous in space. In that sense, very recently, the paper [11] shows that a solution defined using change of variables is almost Hölder 3/4 continuous in time and almost Hölder 3/2 continuous in space as desired.
On the other hand, the Wick-Itô-Skorokhod interpretation (1.4) draws attention from several authors. For example, the paper [16] by Uemura when d = 1, and the paper [8] by Hu when d < 4, define the chaos solution using multiple Itô-Wiener integrals. Also, the paper [16] gives some regularity results: The chaos solution is almost Hölder 1/2 continuous in both time and space. Recently, the paper [10] pays attention to the Wick-Itô-Skorokhod interpretation of (1.4) in one space dimension and the optimal space-time regularity of solution. Using the chaos expansion (or Fourier expansion), the paper proves that the chaos solution is almost Hölder 3/4 continuous in time and almost Hölder 3/2 continuous in space.
The objectives of this paper are: • to introduce a new pathwise solution using the mild formulation (called a mild solution) of time-homogeneous parabolic Anderson model driven by Gaussian white noise on an interval with Dirichlet boundary condition u(t, 0) = u(t, π) = 0, u(0, x) = u 0 (x). (1.7) • to obtain the optimal space-time Hölder regularity of the mild solution. That is, the mild solution is almost Hölder 3/4 continuous in time and almost Hölder 3/2 continuous in space.
The main results are extended to any Hölder continuous function W on [0, π] of order γ ∈ (0, 1), which implies the derivative of W is a generalized function. Note that our results can be applied to (1.7) on the whole line R as long as an appropriate norm of W on R is bounded. We also show the importance of mild formulation: the mild formulation suggests a way of finding relations between Stratonovich interpretation and Wick-Itô-Skorokhod interpretation.
Section 2 discusses the classical Schauder estimate and a modified version of parabolic Schauder estimates. In Section 3, a mild solution of (1.7) is defined and the optimal space-time regularity of the mild solution is obtained. Section 4 gives conclusion and suggests further directions of research.

The Schauder Estimates
In this section, we establish a modified version of the Schauder estimates for parabolic type on an interval in some spaces.
We start with the Hölder spaces on R d . Denote by ∂ ∂z i the differentiation operator with respect to z i , and for a multi-index α = (α 1 , · · · , α d ) with α i ∈ N 0 and |α| = We say that u is Hölder continuous with Hölder exponent γ (or Hölder γ continuous) The collection of Hölder γ continuous functions on G is denoted by C γ (G) with the norm We say that u is a k times continuously differentiable function on G if ∂ α z u exists and is continuous for all |α| ≤ k. The collection of k times continuously differentiable functions on G such that ∂ α z u ∈ C γ (G), |α| = k is denoted by C k+γ (G) with the norm In particular, due to the presence of time and space variables, we often write with the norm We restrict the space domain by G = (0, π) and let us denote the Dirichlet heat kernel on (0, π) by Define a convolution ⋆ for a function f by Let 0 < γ / ∈ N and T > 0 be given. The classical parabolic type of Schauder's estimate in Hölder spaces (Theorem 5.2 of Chapter IV in [15]) says that the convolution mapping ). More specifically, there exists a Schauder constant C T > 0 such that C T remains bounded as T → 0 and Unfortunately, the classical Schauder constant C T does not give a good estimate for our purpose. We now give a relaxed version of Schauder's estimate in fractional Sobolev spaces instead of in Hölder spaces. The modified results will be useful for the existence of mild solution in the next section.
Denote by H s p the fractional Sobolev space as the collection of all functions f such that Γ is the gamma function, and e t∆ is the semigroup of Laplacian operator on (0, π) with zero boundary conditions.
for some C(T, θ, p) > 0 depending on T , θ and p.
for some C > 0 depending only on p. Indeed, (2.1) follows from the fact and [2, Lemma 2.5].

The Mild Solution and its Regularity
Let {W (x)} x∈[0,π] be a standard Brownian motion on a probability space (Ω, F , P).
Here, F denotes the filtration generated by W . It is known that Brownian motion is in C γ (0, π) for any 0 < γ < 1/2. Let W ε be smooth approximations of W such that Consider the approximated equations of (1.7) for ε > 0: Since W ε is smooth, the equation (3.1) has the classical solution u ε . Denote Note that as t → 0 + , we see that P 0 (t, x) → u 0 (x) for each x ∈ [0, π]. Then, the mild formulation for the equation (3.1) is given by By integration by parts with respect to y in the last term above, we rewrite (3.2) as Definition 3.1. We say that u is a mild solution of (1.7) if • for every 0 < t < T and 0 < x < π, u is continuous in t, continuously differentiable in x, and it satisfies the equation We first prove the well-posedness of (3.4).
Proof. It is clear from Lemma 2.1. π) is well-defined. Also, there exists a fixed point of M. That is, the fixed point is the unique mild solution of (1.7).
Proof. Since P D y (t, x, y) = −P N x (t, x, y) for each t > 0 and x, y ∈ [0, π], we have Then, we rewrite (3.4) by We show the well-definedness term by term. By Lemma 3.2, Fix t ∈ (0, T ). We have and similarly by (2.2), for some C > 0. Choose δ > 0 such that Cδ 1/2 W C γ (0,π) < 1. Then, clearly there exists a fixed point of M up to δ. Consider a time partition 0 = t 0 < · · · < t n = T such that t i+1 − t i ≤ δ for i = 0, · · · , n − 1. We now define u recursively on (t i , t i+1 ] with the initial condition u(t i , x) for i = 0, · · · , n − 1: The existence of the fixed point to (3.5) is guaranteed by the above arguments since t i+1 − t i ≤ δ. Since n is finite, we obtain the fixed point solution over the whole time interval (0, T ). We note that the fixed point solution satisfies the mild formulation (3.3). Since the fixed point solution is unique, the uniqueness of mild solution clearly holds.
Remark 3.4. From Theorem 3.3, we have that for all t ∈ [0, T ], By the Sobolev embedding theorem, we have for any p ≥ 1. This shows that the mild solution u is indeed almost Hölder 3/2 continuous in space.
Remark 3.8. Consider the following equation on the whole line R Theorem 3.3, 3.5 and 3.6 will also work on the whole line R if the Hölder γ norm of W on R is bounded with 0 < γ < 1; • The modified Schauder estimate result in Theorem 2.2 is sharper in Hölder spaces if P D is replaced by the Gaussian heat kernel P (t, x − y) on R: for any f ∈ (0, T ; C γ (R)), we have for some C > 0, which is independent of T . Note that a Brownian motion on R do not have a sample trajectory that has a bounded Hölder norm. • Clearly, P y (t, x − y) = P x (t, x − y) holds for t > 0.

Conclusion and Further Directions
4.1. Spatial Optimal Regularity. The paper [9] gives the spatial (Hölder) regularity only less than 1/2. However, Theorem 3.3 using the fixed point argument and [11,Theorem 2.3] show that the optimal spatial (Hölder) regularity of the solution u is 3/2 − ε for any ε > 0 as long as u 0 ∈ C 3/2 (0, π). The result implies several important remarks: • Achieve the spatial regularity higher than 1/2. Since the standard Brownian motion W is Hölder 1/2 − ε continuous almost surely, it is possible to apply Young's integral: For each s < t and x, π 0 P (t − s, x, y)u(s, y) ∂ ∂y W (y)dy := π 0 P (t − s, x, y)u(s, y)dW (y) appearing in the classical mild formulation of (1.7); • The regularity 3/4 − ε in time and 3/2 − ε in space for ε > 0 are indeed in line with the standard parabolic partial differential equation theory.
The natural further question is to find a meaningful relation between the usual solution of (1.7) and the Wick-Itô-Skorokhod solution of (1.4) with Dirichlet boundary condition in the mild formulation.