Regularity theory for a new class of fractional parabolic stochastic evolution equations

Abstract

Abstract A new class of fractional-order parabolic stochastic evolution equations of the form $$(\partial _t + A)^\gamma X(t) = {\dot{W}}^Q(t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>γ</mml:mi> </mml:msup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mi>Q</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , $$t\in [0,T]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , $$\gamma \in (0,\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , is introduced, where $$-A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> generates a $$C_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -semigroup on a separable Hilbert space H and the spatiotemporal driving noise $${\dot{W}}^Q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mi>Q</mml:mi> </mml:msup> </mml:math> is the formal time derivative of an H -valued cylindrical Q -Wiener process. Mild and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process X are investigated, the former being measured by mean-square or pathwise smoothness and the latter by using domains of fractional powers of A . In addition, the covariance of X and its long-time behavior are analyzed. These abstract results are applied to the cases when $$A:= L^\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>β</mml:mi> </mml:msup> </mml:mrow> </mml:math> and $$Q:={\widetilde{L}}^{-\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mover> <mml:mi>L</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle–)Matérn fields to space–time.

Locations

• Stochastic Partial Differential Equations Analysis and Computations - View - PDF