In this paper, we consider the linear evolution equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10220_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="255" /> </InlineMediaObject> <EquationSource Format="TEX">$dy(t)\!=\!Ay(t)dt+\sum _{i=1}^dG_iy(t)dx_i(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mi>y</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mi>A</mi> <mi>y</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>+</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>d</mi> </msubsup> <msub> <mi>G</mi> <mi>i</mi> </msub> <mi>y</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <msub> <mi>x</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where A is a closed operator, associated to a semigroup, with good smoothing effects in a Banach space E, x is a nonsmooth <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10220_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>-path, which is <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10220_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-Hölder continuous for some <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10220_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$\eta \in \left( \frac{1}{3},\frac{1}{2}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>η</mi> <mo>∈</mo> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo>,</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10220_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$G_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11118_2025_10220_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$i=1,\ldots ,d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation>) is a non-smoothing linear operator on E. We prove that the Cauchy problem associated with the previous equation admits a unique mild solution and we also show that the solution increases the regularity of the initial datum as soon as time evolves. Then, we show that the mild solution is also an integral solution and this allows us to prove an Itô formula.

Space Regularity of Evolution Equations Driven by Rough Paths

期刊论文
发表日期：

2025年5月20日

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Davide Addona,
Luca Lorenzi,
Gianmario Tessitore

摘要

In this paper, we consider the linear evolution equation $dy(t)\!=\!Ay(t)dt+\sum _{i=1}^dG_iy(t)dx_i(t)$ $d y (t) = A y (t) d t + \sum_{i = 1}^{d} G_{i} y (t) d x_{i} (t)$ , where A is a closed operator, associated to a semigroup, with good smoothing effects in a Banach space E, x is a nonsmooth $\mathbb {R}^d$ $R^{d}$ -path, which is $\eta $ $η$ -Hölder continuous for some $\eta \in \left( \frac{1}{3},\frac{1}{2}\right) $ $η \in (\frac{1}{3}, \frac{1}{2})$ , and $G_i$ $G_{i}$ ( $i=1,\ldots ,d$ $i = 1, \dots, d$ ) is a non-smoothing linear operator on E. We prove that the Cauchy problem associated with the previous equation admits a unique mild solution and we also show that the solution increases the regularity of the initial datum as soon as time evolves. Then, we show that the mild solution is also an integral solution and this allows us to prove an Itô formula.