| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>=</mo><mi>U</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>∈</mo><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub><mo>×</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>d</mi></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∂</mo><mi>x</mi></msub><mi>U</mi><mo>=</mo><msub><mo>∂</mo><mi>x</mi></msub><mi>U</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>∈</mo><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub><mo>×</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> be the solution and gradient solution of the fourth order linearized Kuramoto–Sivashinsky (L-KS) SPDE driven by the space-time white noise in one-to-three dimensional spaces, respectively. We use the underlying explicit kernels and symmetry analysis, yielding exact, dimension-dependent, and temporal moduli of non-differentiability for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mo>·</mo><mo>,</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∂</mo><mi>x</mi></msub><mi>U</mi><mrow><mo>(</mo><mo>·</mo><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. It has been confirmed that almost all sample paths of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mo>·</mo><mo>,</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∂</mo><mi>x</mi></msub><mi>U</mi><mrow><mo>(</mo><mo>·</mo><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, in time, are nowhere differentiable.
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