<p>We establish a large deviation principle for the Stratonovich stochastic nonlinear heat equation posed on a smooth bounded domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathscr {O} \subset \mathbb {R}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> with Dirichlet boundary conditions. The dynamics evolve on the Hilbert manifold <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathscr {M} = \{u \in L^{2}(\mathscr {O}): |u|_{{L}^{2}} = 1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mi>u</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">O</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo stretchy="false">|</mo> <mi>u</mi> <msub> <mo stretchy="false">|</mo> <msup> <mrow> <mi>L</mi> </mrow> <mn>2</mn> </msup> </msub> <mo>=</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and are driven by multiplicative noise that is tangent to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathscr {M}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation>. Adopting the Budhiraja–Dupuis weak convergence framework, we verify the required compactness and stability conditions for the associated controlled skeleton system, and derive the corresponding good rate function in the space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$X_{T} = C([0,T];\textrm{V}) \cap {L}^{2}(0,T;\textrm{E})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mi>T</mi> </msub> <mo>=</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <mtext>V</mtext> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msup> <mrow> <mi>L</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <mtext>E</mtext> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Large Deviations for the Stochastic Nonlinear Heat Equation on a Hilbert Manifold

期刊论文
发表日期：

2026年1月22日

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Z. Brzeźniak,
J. Hussain

摘要

We establish a large deviation principle for the Stratonovich stochastic nonlinear heat equation posed on a smooth bounded domain $\mathscr {O} \subset \mathbb {R}^{d}$ $O \subset R^{d}$ with Dirichlet boundary conditions. The dynamics evolve on the Hilbert manifold $\mathscr {M} = \{u \in L^{2}(\mathscr {O}): |u|_{{L}^{2}} = 1\}$ $M = {u \in L^{2} (O) : | u |_{L^{2}} = 1}$ and are driven by multiplicative noise that is tangent to $\mathscr {M}$ $M$ . Adopting the Budhiraja–Dupuis weak convergence framework, we verify the required compactness and stability conditions for the associated controlled skeleton system, and derive the corresponding good rate function in the space $X_{T} = C([0,T];\textrm{V}) \cap {L}^{2}(0,T;\textrm{E})$ $X_{T} = C ([0, T]; V) \cap L^{2} (0, T; E)$ .