Regularity of the Mild Solution of a Parabolic Equation with Stochastic Measure

Strong Convergence Rate in Averaging Principle for the Heat Equation Driven by a General Stochastic Measure

“…At first, we will study the Hölder regularity of the stochastic integrals in our equations. The following statement is very similar to Lemma 2 [2]. But here we obtain that coefficient C(ω) in (4.1) does not depend on x.…”

“…Theorem of [2] and A1-A3 give that solutions of (3.1) and (3.4) exist and are unique, and have a Hölder continuous version on [δ, T ] × [−K, K] for each δ, K > 0 (a similar result is given by Theorem [17]). Therefore, u ε andū have a continuous versions on (0, T ] × R.…”

“…Thus, (4.12) implies (4.1). □ For solutions of (3.1) and (3.4), Theorem of [2] states the Hölder regularity for t ∈ [δ, T ] for any δ > 0 (a similar result is given by Theorem [17]). But we need an estimate of increments of the solutions for t ∈ (0, T ], and will prove the following statement.…”

“…is an SM, as follows from [15, Theorem 1.1]. An α-stable random measure defined on B is an SM too for α ∈ (0, 1) ∪ (1,2], see [22,Chapter 3]. Some other examples are given in [20].…”

“…whereū is the solution of the averaged equation { dū(t, x) = 1 4 ∂ 2ū (t,x) ∂x 2 dt +f (x,ū(t, x)) dt +σ(x) dµ(x) , (t, x) ∈ (0, T ] × R, u(0, x) = u 0 (x) .…”