We consider the SPDE<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$:dZ=(AZ+b(Z)) dt+dW(t),\,Z_0=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>:</mo> <mi>d</mi> <mi>Z</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>Z</mi> <mo>+</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>+</mo> <mi>d</mi> <mi>W</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.166667em" /> <msub> <mi>Z</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>, on a separable Hilbert space H, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A:H\rightarrow H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">→</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> is self-adjoint <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$b:H\rightarrow H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">→</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> is Lipschitz continuous and W is a cylindrical Wiener process on H. We determine, with the help of a well-known formula for nonlinear transformations of Gaussian integrals due to R. Ramer [<CitationRef CitationID="CR16">16</CitationRef>], an explicit representation for the law of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Z_x$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Z</mi> <mi>x</mi> </msub> </math></EquationSource> </InlineEquation> in C([0, T]; H), see Theorem <InternalRef RefID="FPar14">3.2</InternalRef> below. When b is, in addition, dissipative, we determine the invariant measure <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation> of the semigroup <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$P_t\varphi (x)=\mathbb {E}[\varphi (Z_x(t))]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mi>t</mi> </msub> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="double-struck">E</mi> <mrow> <mo stretchy="false">[</mo> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Z</mi> <mi>x</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the corresponding stationary process <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$Z_{\mathbb {R}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Z</mi> <mi mathvariant="double-struck">R</mi> </msub> </math></EquationSource> </InlineEquation>. The final Section <InternalRef RefID="Sec11">5</InternalRef> is devoted to colored noise.

A mild Girsanov formula

期刊论文
发表日期：

2025年8月19日

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Giuseppe Da Prato,
Enrico Priola,
Luciano Tubaro

摘要

We consider the SPDE $:dZ=(AZ+b(Z)) dt+dW(t),\,Z_0=x$ $: d Z = (A Z + b (Z)) d t + d W (t), Z_{0} = x$ , on a separable Hilbert space H, where $A:H\rightarrow H$ $A : H \to H$ is self-adjoint $b:H\rightarrow H$ $b : H \to H$ is Lipschitz continuous and W is a cylindrical Wiener process on H. We determine, with the help of a well-known formula for nonlinear transformations of Gaussian integrals due to R. Ramer [16], an explicit representation for the law of $Z_x$ $Z_{x}$ in C([0, T]; H), see Theorem 3.2 below. When b is, in addition, dissipative, we determine the invariant measure $\nu $ $ν$ of the semigroup $P_t\varphi (x)=\mathbb {E}[\varphi (Z_x(t))]$ $P_{t} φ (x) = E [φ (Z_{x} (t))]$ , the corresponding stationary process $Z_{\mathbb {R}}$ $Z_{R}$ . The final Section 5 is devoted to colored noise.