Optimal Strong Rates of Convergence for a Space-Time Discretization ofthe Stochastic Allen–Cahn Equation with Multiplicative Noise

The stochastic Allen–Cahn equation with multiplicative noise involves the nonlinear drift operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="script">𝒜</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>⁢</m:mo> <m:mi>x</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:math> {{A}(x)=Δ x-(|x|²-1)x} . We use the fact that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="script">𝒜</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:msup> <m:mi mathvariant="script">𝒥</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {{A}(x)=-{J}^{}(x)} satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:munder> <m:mo movablelimits="false">sup</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>j</m:mi> <m:mo>≤</m:mo> <m:mi>J</m:mi> </m:mrow> </m:munder> <m:mo>⁡</m:mo> <m:mrow> <m:mi>𝔼</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:msubsup> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:msub> <m:mi>t</m:mi> <m:mi>j</m:mi> </m:msub> </m:msub> <m:mo>-</m:mo> <m:msup> <m:mi>Y</m:mi> <m:mi>j</m:mi> </m:msup> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:msup> <m:mi>𝕃</m:mi> <m:mn>2</m:mn> </m:msup> <m:mn>2</m:mn> </m:msubsup> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi>δ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi>k</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mi>δ</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>h</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> 1≤ j≤ J{E}[\|X_{tⱼ}-Yʲ\|_{{L}²}²]% ≤ Cδ(k1-δ+h²) for all small <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>δ</m:mi>