Stochastics and Partial Differential Equations-Analysis and Computations最新文献

{"title":"Norm inflation for a non-linear heat equation with gaussian initial conditions","authors":"Ilya Chevyrev","doi":"10.1007/s40072-023-00317-6","DOIUrl":"https://doi.org/10.1007/s40072-023-00317-6","url":null,"abstract":"Abstract We consider a non-linear heat equation $$partial _t u = Delta u + B(u,Du)+P(u)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>B</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> posed on the d -dimensional torus, where P is a polynomial of degree at most 3 and B is a bilinear map that is not a total derivative. We show that, if the initial condition $$u_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is taken from a sequence of smooth Gaussian fields with a specified covariance, then u exhibits norm inflation with high probability. A consequence of this result is that there exists no Banach space of distributions which carries the Gaussian free field on the 3D torus and to which the DeTurck–Yang–Mills heat flow extends continuously, which complements recent well-posedness results of Cao–Chatterjee and the author with Chandra–Hairer–Shen. Another consequence is that the (deterministic) non-linear heat equation exhibits norm inflation, and is thus locally ill-posed, at every point in the Besov space $$B^{-1/2}_{infty ,infty }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> ; the space $$B^{-1/2}_{infty ,infty }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> is an endpoint since the equation is locally well-posed for $$B^{eta }_{infty ,infty }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> <mml:mi>η</mml:mi> </mml:msubsup> </mml:math> for every $$eta &gt;-frac{1}{2}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> .","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136184881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions","authors":"Yassine Tahraoui, Fernanda Cipriano","doi":"10.1007/s40072-023-00314-9","DOIUrl":"https://doi.org/10.1007/s40072-023-00314-9","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135093199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On partially observed jump diffusions II: the filtering density","authors":"Alexander Davie, Fabian Germ, István Gyöngy","doi":"10.1007/s40072-023-00311-y","DOIUrl":"https://doi.org/10.1007/s40072-023-00311-y","url":null,"abstract":"Abstract A partially observed jump diffusion $$Z=(X_t,Y_t)_{tin [0,T]}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> given by a stochastic differential equation driven by Wiener processes and Poisson martingale measures is considered when the coefficients of the equation satisfy appropriate Lipschitz and growth conditions. Under general conditions it is shown that the conditional density of the unobserved component $$X_t$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> given the observations $$(Y_s)_{sin [0,t]}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> </mml:math> exists and belongs to $$L_p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> if the conditional density of $$X_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> given $$Y_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> exists and belongs to $$L_p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> .","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135739399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multilevel Monte Carlo FEM for elliptic PDEs with Besov random tree priors","authors":"Christoph Schwab, Andreas Stein","doi":"10.1007/s40072-023-00313-w","DOIUrl":"https://doi.org/10.1007/s40072-023-00313-w","url":null,"abstract":"Abstract We develop a multilevel Monte Carlo (MLMC)-FEM algorithm for linear, elliptic diffusion problems in polytopal domain $${mathcal {D}}subset {mathbb {R}}^d$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> , with Besov-tree random coefficients. This is to say that the logarithms of the diffusion coefficients are sampled from so-called Besov-tree priors, which have recently been proposed to model data for fractal phenomena in science and engineering. Numerical analysis of the fully discrete FEM for the elliptic PDE includes quadrature approximation and must account for (a) nonuniform pathwise upper and lower coefficient bounds, and for (b) low path-regularity of the Besov-tree coefficients. Admissible non-parametric random coefficients correspond to random functions exhibiting singularities on random fractals with tunable fractal dimension, but involve no a-priori specification of the fractal geometry of singular supports of sample paths. Optimal complexity and convergence rate estimates for quantities of interest and for their second moments are proved. A convergence analysis for MLMC-FEM is performed which yields choices of the algorithmic steering parameters for efficient implementation. A complexity (“error vs work”) analysis of the MLMC-FEM approximations is provided.","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136248793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds","authors":"David Candil, Le Chen, Cheuk Yin Lee","doi":"10.1007/s40072-023-00310-z","DOIUrl":"https://doi.org/10.1007/s40072-023-00310-z","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135246029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An $$L_q(L_p)$$-theory for space-time non-local equations generated by Lévy processes with low intensity of small jumps","authors":"Jaehoon Kang, Daehan Park","doi":"10.1007/s40072-023-00309-6","DOIUrl":"https://doi.org/10.1007/s40072-023-00309-6","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86882898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the wellposedness of periodic nonlinear Schrödinger equations with white noise dispersion","authors":"Gavin Stewart","doi":"10.1007/s40072-023-00306-9","DOIUrl":"https://doi.org/10.1007/s40072-023-00306-9","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91227135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotic stability of evolution systems of probability measures of stochastic discrete modified Swift–Hohenberg equations","authors":"Fengling Wang, T. Caraballo, Yangrong Li, Renhai Wang","doi":"10.1007/s40072-023-00307-8","DOIUrl":"https://doi.org/10.1007/s40072-023-00307-8","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87944234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A branching particle system approximation for solving partially observed stochastic optimal control problems via stochastic maximum principle","authors":"Hexiang Wan, Guangchen Wang, Jie Xiong","doi":"10.1007/s40072-023-00294-w","DOIUrl":"https://doi.org/10.1007/s40072-023-00294-w","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83414119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Correction to: Uniqueness for nonlinear Fokker–Planck equations and weak uniqueness for McKean-Vlasov SDEs","authors":"V. Barbu, M. Röckner","doi":"10.1007/s40072-021-00223-9","DOIUrl":"https://doi.org/10.1007/s40072-021-00223-9","url":null,"abstract":"","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76965863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}