Let U, H be two separable Hilbert spaces. The main goal of this paper is to study weak uniqueness of a Stochastic Differential Equation evolving in H of the form <Equation ID="Equ59"> <EquationSource Format="TEX">$\begin{aligned} dX(t)=AX(t)dt+VB(X(t))dt+GdW(t), \quad t>0, \quad X(0)=x \in H, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>d</mi> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>+</mo> <mi>V</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>+</mo> <mi>G</mi> <mi>d</mi> <mi>W</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>X</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>∈</mo> <mi>H</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{W(t)\}_{t\ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <mi>W</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> is a U-cylindrical Wiener process, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A:D(A)\subseteq H\rightarrow H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>:</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>H</mi> <mo stretchy="false">→</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> is the infinitesimal generator of a strongly continuous semigroup, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$V,G:U\rightarrow H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo>,</mo> <mi>G</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> are linear bounded operators and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$B:H\rightarrow U$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">→</mo> <mi>U</mi> </mrow> </math></EquationSource> </InlineEquation> is a locally uniformly continuous function. The abstract result in the paper gives weak uniqueness for a large class of heat and damped equations in any dimension without any Hölder continuity assumption on B.

Weak uniqueness for stochastic partial differential equations in Hilbert spaces

期刊论文
发表日期：

2026年2月23日

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Davide Addona,
Davide Augusto Bignamini

摘要

Let U, H be two separable Hilbert spaces. The main goal of this paper is to study weak uniqueness of a Stochastic Differential Equation evolving in H of the form $\begin{aligned} dX(t)=AX(t)dt+VB(X(t))dt+GdW(t), \quad t>0, \quad X(0)=x \in H, \end{aligned}$ $\begin{matrix} d X (t) = A X (t) d t + V B (X (t)) d t + G d W (t), t > 0, X (0) = x \in H, \end{matrix}$ where $\{W(t)\}_{t\ge 0}$ ${W (t)}_{t \geq 0}$ is a U-cylindrical Wiener process, $A:D(A)\subseteq H\rightarrow H$ $A : D (A) \subseteq H \to H$ is the infinitesimal generator of a strongly continuous semigroup, $V,G:U\rightarrow H$ $V, G : U \to H$ are linear bounded operators and $B:H\rightarrow U$ $B : H \to U$ is a locally uniformly continuous function. The abstract result in the paper gives weak uniqueness for a large class of heat and damped equations in any dimension without any Hölder continuity assumption on B.