Counterexamples to regularities for the derivative processes associated to stochastic evolution equations

In the recent years there has been an increased interest in studying regularity properties of the derivatives of semilinear parabolic stochastic evolution equations (SEEs) with respect to their initial values. In particular, in the scientific literature it has been shown for every natural number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi></mml:mrow> </mml:math> that if the nonlinear drift coefficient and the nonlinear diffusion coefficient of the considered SEE are <i>n</i>-times continuously Fréchet differentiable, then the solution of the considered SEE is also <i>n</i>-times continuously Fréchet differentiable with respect to its initial value and the corresponding derivative processes satisfy a suitable regularity property in the sense that the <i>n</i>-th derivative process can be extended continuously to <i>n</i>-linear operators on negative Sobolev-type spaces with regularity parameters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>δ</mml:mi> <mml:mn>1</mml:mn></mml:msub> <mml:mo>,</mml:mo> <mml:msub><mml:mi>δ</mml:mi> <mml:mn>2</mml:mn></mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub><mml:mi>δ</mml:mi> <mml:mi>n</mml:mi></mml:msub> <mml:mo>∈</mml:mo> <mml:mrow><mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> provided that the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msubsup><mml:mo>∑</mml:mo> <mml:mrow><mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn></mml:mrow> <mml:mi>n</mml:mi></mml:msubsup> <mml:msub><mml:mi>δ</mml:mi> <mml:mi>i</mml:mi></mml:msub> <mml:mo><</mml:mo> <mml:mfrac><mml:mn>1</mml:mn> <mml:mn>2</mml:mn></mml:mfrac> </mml:mrow> </mml:math> is satisfied. The main contribution of this paper is to reveal that this condition can essentially not be relaxed.