Heat equation with general stochastic measure colored in time

A stochastic heat equation on $[0,T]\times{\mathbb{R}}$ driven by a general stochastic measure $d\mu(t)$ is investigated in this paper. For the integrator $\mu$, we assume the $\sigma$-additivity in probability only. The existence, uniqueness, and H\"{o}lder regularity of the solution are proved.


Introduction
In this paper, we consider a stochastic heat equation that can formally be written as    du(t, x) = a 2 ∂ 2 u(t, x) ∂x 2 dt + f t, x, u(t, x) dt + σ(t, x) dµ(t), u(0, x) = u 0 (x), (1) where (t, x) ∈ [0, T ] × R, a ∈ R, a = 0, and µ is a stochastic measure (SM) defined on the Borel σ-algebra of [0, T ]. We consider a solution to the formal equation (1) in the mild sense (see Eq. (5)). We prove the existence and uniqueness of the solution and obtain Hölder regularity of its paths under some general conditions for the stochastic part of equation.
A similar problem for µ dependent on the spatial variable x was considered in [5]. The stochastic heat equation on fractals was studied in [7], and a review of results on equations driven by SMs is given in [6].
For equations driven by white noise, the regularity of paths of solutions was considered in [10,Chapter 3]. Equations driven by fractional noise were studied in [9,Chapter 2]. In many papers, the regularity of solutions was considered in appropriate function spaces; see, for example, [2] and references therein.

Preliminaries
Let L 0 = L 0 (Ω, F , P) be the set of (equivalence classes of) all real-valued random variables defined on a complete probability space (Ω, F , P). The convergence in L 0 is understood as the convergence in probability. Let X be an arbitrary set, and B be a σ-algebra of subsets of X. Definition 1. Any σ-additive mapping µ : B → L 0 is called a stochastic measure (SM).
In other words, µ is a vector measure with values in L 0 . In [3], such µ is called a general SM.
Examples of SMs are the following. Let X = [0, T ] ⊂ R + , B be the σ-algebra of Borel subsets of [0, T ], and N (t) be a square-integrable martingale. Then is also an SM, as follows from [4,Theorem 1.1]. An α-stable random measure defined on a σ-algebra is an SM [8,Chapter 3]. Theorem 8.3.1 of [3] states the conditions under which the increments of a real-valued Lévy process generate an SM.
For a deterministic measurable function g : X → R and SM µ, an integral of the form X g dµ is defined and studied in [3,Chapter 7]; see also [1]. In particular, every bounded measurable g is integrable w.r.t. any µ. An analogue of the Lebesgue dominated convergence theorem holds for this integral [3, Proposition 7.1.1].
We consider the Besov spaces B α 22 ([c, d]). Recall that the norm in this classical space for 0 < α < 1 may be introduced by where For all n ≥ 1, 1 ≤ k ≤ 2 n , put ∆ (t) kn = ((k − 1)2 −n t, k2 −n t]. The following estimate is a key tool for the proof of Hölder regularity of the stochastic integral. In our estimates, C and C(ω) will denote a constant and a random constant, respectively, which may be different from formula to formula. Lemma 1 (Lemma 3.2 [5]). Let SM µ be defined on the Borel σ-algebra of [0, t],Z be an arbitrary set, and q(z, s) : Z × [0, t] → R be a function such that for some 1/2 < α < 1 and for each z ∈ Z, q(z, ·) ∈ B α 22 ([0, t]). Then the random function has a versionη(z) such that for some constant C (independent of z, ω) and each ω ∈ Ω, From Lemma 3.1 [5] it follows that, for ε > 0,

The problem
Consider equation (1) in the following mild sense: Here is the Gaussian heat kernel, u(t, x) = u(t, x, ω) : [0, T ] × R × Ω → R is an unknown measurable random function, and µ is an SM defined on the Borel σ-algebra of [0, T ].
The integrals of random functions w.r.t. dy and ds are taken for each fixed ω ∈ Ω. Throughout this paper, we will use the following assumptions.

Hölder continuity in x
Consider the regularity of paths of the stochastic integral from (5).

Lemma 2.
Let Assumptions A5 and A6 hold. Then, for any fixed t ∈ [0, T ] and has a Hölder continuous version with exponent γ 1 .

Proof.
Denote and apply (3) to η(z) = ϑ(x 1 )−ϑ(x 2 ). We will estimate the Besov space norm in (3). Consider the difference Using (6) and the change of variables we get By a similar way, we can estimate |D 2 | and obtain Further, consider

Equation (5) has a solution u(t, x).
If v(t, x) is another solution to (5), then for all t and x, u(t, x) = v(t, x) a.s.

Remark 1.
For u, we obtained less regularity than for elements of equation (5). However, a solution to a heat equation usually has the same regularity or even more regular than the coefficients. One may expect that using other methods gives (17) with exponents γ 2 ≤ β(µ) ∧ γ 2 < β(σ) and γ 1 < β(σ).