We consider the stochastic partial differential equation (SPDE) <Equation ID="Equ35"> <EquationSource Format="TEX">$\begin{aligned} \partial _t u = \tfrac{1}{2} \partial ^2_x u + b(u) + \sigma (u) \dot{W}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>=</mo> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <msubsup> <mi>∂</mi> <mi>x</mi> <mn>2</mn> </msubsup> <mi>u</mi> <mo>+</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mover accent="true"> <mi>W</mi> <mo>˙</mo> </mover> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u=u(t,x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is defined for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(t,x)\in (0,\infty )\times \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\dot{W}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>W</mi> <mo>˙</mo> </mover> </math></EquationSource> </InlineEquation> denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition u(0) is bounded and measurable, and b and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> are locally Lipschitz continuous functions having at most linear growth with regularly behaved local Lipschitz constants. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The novelty of our method is in the pointwise nature of the truncation argument.

On the Well-Posedness of Stochastic Partial Differential Equations with Locally Lipschitz Coefficients

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2026年2月23日

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Mohammud Foondun,
Davar Khoshnevisan,
Eulalia Nualart

摘要

We consider the stochastic partial differential equation (SPDE) $\begin{aligned} \partial _t u = \tfrac{1}{2} \partial ^2_x u + b(u) + \sigma (u) \dot{W}, \end{aligned}$ $\begin{matrix} \partial_{t} u = \frac{1}{2} \partial_{x}^{2} u + b (u) + σ (u) \dot{W}, \end{matrix}$ where $u=u(t,x)$ $u = u (t, x)$ is defined for $(t,x)\in (0,\infty )\times \mathbb {R}$ $(t, x) \in (0, \infty) \times R$ and $\dot{W}$ $\dot{W}$ denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition u(0) is bounded and measurable, and b and $\sigma $ $σ$ are locally Lipschitz continuous functions having at most linear growth with regularly behaved local Lipschitz constants. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The novelty of our method is in the pointwise nature of the truncation argument.