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Nyström, Kaj

Open this publication in new window or tab >>*L*^{2} Solvability of boundary value problems for divergence form parabolic equations with complex coefficients### Nyström, Kaj

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##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-268050 (URN)10.1016/j.jde.2016.11.011 (DOI)000392463200026 ()
#####

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Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-12-01Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}=-\mbox{div}\, A(X,t)\nabla,$$ in$\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We assume that $A$ is a $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_{n+1}$ as well as of the time coordinate $t$. For such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on $L^2(\mathbb R^{n+1},\mathbb C)=L^2(\partial\mathbb R^{n+2}_+,\mathbb C)$ under complex, $L^\infty$ perturbations of the coefficient matrix. Subsequently, using this general result, we establish solvability of the Dirichlet, Neumann and Regularity problems for $\partial_t+\mathcal{L}$, by way of layer potentials and with data in $L^2$, assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix.

Open this publication in new window or tab >>The A∞-Property of the Kolmogorov Measure### Nyström, Kaj

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_1_j_idt185_some",{id:"formSmash:j_idt181:1:j_idt185:some",widgetVar:"widget_formSmash_j_idt181_1_j_idt185_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_1_j_idt185_otherAuthors",{id:"formSmash:j_idt181:1:j_idt185:otherAuthors",widgetVar:"widget_formSmash_j_idt181_1_j_idt185_otherAuthors",multiple:true}); 2017 (English)In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 10, no 7, p. 1709-1756Article in journal (Refereed) Published
##### Abstract [en]

##### Keyword

Kolmogorov equation, ultraparabolic, hypoelliptic, Lipschitz domain, doubling measure, parabolic measure, Kolmogorov measure, A(infinity)
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-335534 (URN)10.2140/apde.2017.10.1709 (DOI)000409094100005 ()
#####

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##### Funder

Göran Gustafsson Foundation for Research in Natural Sciences and Medicine
Available from: 2017-12-08 Created: 2017-12-08 Last updated: 2017-12-08Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala Univ, Dept Math, Uppsala, Sweden..

We consider the Kolmogorov-Fokker-Planck operator K := Sigma(m)(i=1) partial derivative(xixi) + Sigma(m)(i=1) x(i)partial derivative(yi) - partial derivative(t) in unbounded domains of the form Omega={(x,x(m),y,y(m),t) RN+1 vertical bar x(m)>psi(x,y,t)}. Concerning and psi, we assume that Omega is what we call an (unbounded) admissible Lip(K)-domain: psi satisfies a uniform Lipschitz condition, adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator K, as well as an additional regularity condition formulated in terms of a Carleson measure. We prove that in admissible Lip(K)-domains the associated parabolic measure is absolutely continuous with respect to a surface measure and that the associated Radon-Nikodym derivative defines an A(infinity) weight with respect to this surface measure. Our result is sharp.

Open this publication in new window or tab >>The weak-A∞ PROPERTY OF HARMONIC AND p-HARMONIC MEASURES IMPLIES UNIFORM RECTIFIABILITY### Hofmann, Steve

### Le, Phi

### Maria Martell, Jose

### Nyström, Kaj

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_2_j_idt185_some",{id:"formSmash:j_idt181:2:j_idt185:some",widgetVar:"widget_formSmash_j_idt181_2_j_idt185_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_2_j_idt185_otherAuthors",{id:"formSmash:j_idt181:2:j_idt185:otherAuthors",widgetVar:"widget_formSmash_j_idt181_2_j_idt185_otherAuthors",multiple:true}); 2017 (English)In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 10, no 3, p. 513-558Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

MATHEMATICAL SCIENCE PUBL, 2017
##### Keyword

harmonic measure and p-harmonic measure, Poisson kernel, uniform rectifiability, Carleson measures, Green function, weak-A(infinity)
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-328276 (URN)10.2140/apde.2017.10.513 (DOI)000403096000001 ()
#####

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Available from: 2017-11-13 Created: 2017-11-13 Last updated: 2017-11-13Bibliographically approved

Univ Missouri, Dept Math, Columbia, MO 65211 USA..

Syracuse Univ, Math Dept, 215 Carnegie Bldg, Syracuse, NY 13244 USA..

CSIC, CSIC, Inst Ciencias Matemat, UAM,UC3M,UCM, C Nicolas Cabrera 13-15, E-28049 Madrid, Spain..

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Let E subset of Rn+1, n >= 2, be an Ahlfors-David regular set of dimension n. We show that the weak- A 1 property of harmonic measure, for the open set Omega: =Rn+1 \ E, implies uniform rectifiability of E. More generally, we establish a similar result for the Riesz measure, p-harmonic measure, associated to the p-Laplace operator, 1 < p < infinity.

Open this publication in new window or tab >>Tug-of-war, market manipulation, and option pricing### Nyström, Kaj

### Parviainen, Mikko

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_3_j_idt185_some",{id:"formSmash:j_idt181:3:j_idt185:some",widgetVar:"widget_formSmash_j_idt181_3_j_idt185_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_3_j_idt185_otherAuthors",{id:"formSmash:j_idt181:3:j_idt185:otherAuthors",widgetVar:"widget_formSmash_j_idt181_3_j_idt185_otherAuthors",multiple:true}); 2017 (English)In: Mathematical Finance, ISSN 0960-1627, E-ISSN 1467-9965, Vol. 27, no 2, p. 279-312Article in journal (Refereed) Published
##### Abstract [en]

##### Keyword

infinity Laplace, nonlinear parabolic partial differential equation, option pricing, stochastic differential game, tug-of-war
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-214209 (URN)10.1111/mafi.12090 (DOI)000397560600001 ()
#####

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Available from: 2014-01-08 Created: 2014-01-08 Last updated: 2017-05-16Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.

University of Jyväskylä, Jyväskylä, Finland.

We develop an option pricing model based on a tug-of-war game. This two-playerzero-sum stochastic differential game is formulated in the context of a multidimen-sional ﬁnancial market. The issuer and the holder try to manipulate asset price pro-cesses in order to minimize and maximize the expected discounted reward. We provethat the game has a value and that the value function is the unique viscosity solutionto a terminal value problem for a parabolic partial differential equation involving thenonlinear and completely degenerate inﬁnity Laplace operator.

Open this publication in new window or tab >>Boundary value problems for parabolic systems via a first order approach### Auscher, Pascal

### Egert, Moritz

### Nyström, Kaj

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##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-300486 (URN)
#####

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Available from: 2016-08-09 Created: 2016-08-09 Last updated: 2017-02-21Bibliographically approved

Univ. Paris-Sud, CNRS, Universit´e Paris-Saclay.

Univ. Paris-Sud, CNRS, Universit´e Paris-Saclay.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We introduce a first order strategy to study boundary value problems of parabolic systems with second order elliptic part in the upper half-space. This involves a parabolic Dirac operator at the boundary. We allow for measurable time dependence and some transversal dependence in the coefficients. We obtain layer potential representations for solutions in some classes and prove new well-posedness and perturbation results. As a byproduct, we prove for the first time a Kato estimate for the square root of parabolic operators with time dependent coefficients. This considerably extends prior results obtained by one of us under time and transversal independence. A major difficulty compared to a similar treatment of elliptic equations is the presence of non-local fractional derivatives in time.

Open this publication in new window or tab >>Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients### Castro, Alejandro

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Nyström, Kaj

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Sande, Olow

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_5_j_idt185_some",{id:"formSmash:j_idt181:5:j_idt185:some",widgetVar:"widget_formSmash_j_idt181_5_j_idt185_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_5_j_idt185_otherAuthors",{id:"formSmash:j_idt181:5:j_idt185:otherAuthors",widgetVar:"widget_formSmash_j_idt181_5_j_idt185_otherAuthors",multiple:true}); 2016 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 55, no 5, article id 124Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-266835 (URN)10.1007/s00526-016-1058-8 (DOI)000386708700016 ()
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Available from: 2015-11-11 Created: 2015-11-11 Last updated: 2017-12-01Bibliographically approved

We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}:=-\mbox{div}\, A(X,t)\nabla,$$ in$\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We assume that $A$ is an $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_{n+1}$ as well as of the time coordinate $t$. We prove that the boundedness of associated single layer potentials, with data in $L^2$, can be reduced to two crucial estimates (Theorem \ref{th0}), one being a square function estimate involving the single layer potential. By establishing a local parabolic Tb-theorem for square functions we are then able to verify the two crucial estimates in the case of real, symmetric operators (Theorem \ref{th2}). Our results are crucial when addressing the solvability of the classical Dirichlet, Neumann and Regularity problems for the operator $\partial_t+\mathcal{L}$ in $\mathbb R_+^{n+2}$, with $L^2$-data on $\mathbb R^{n+1}=\partial\mathbb R_+^{n+2}$, and by way of layer potentials.

Open this publication in new window or tab >>Extension properties and boundary estimates for a fractional heat operator### Nyström, Kaj

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Sande, Olow

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_6_j_idt185_some",{id:"formSmash:j_idt181:6:j_idt185:some",widgetVar:"widget_formSmash_j_idt181_6_j_idt185_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_6_j_idt185_otherAuthors",{id:"formSmash:j_idt181:6:j_idt185:otherAuthors",widgetVar:"widget_formSmash_j_idt181_6_j_idt185_otherAuthors",multiple:true}); 2016 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 140, p. 29-37Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-266836 (URN)10.1016/j.na.2016.02.027 (DOI)000375484000003 ()
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Available from: 2015-11-11 Created: 2015-11-11 Last updated: 2017-12-01Bibliographically approved

The square root of the heat operator $\sqrt{\partial_t-\Delta}$, can be realized as the Dirichlet to Neumann map of the heat extension of data on $\mathbb R^{n+1}$ to $\mathbb R^{n+2}_+$. In this note we obtain similar characterizations for general fractional powers of the heat operator, $(\partial_t-\Delta)^s$, $s\in (0,1)$. Using the characterizations we derive properties and boundary estimates for parabolic integro-differential equations from purely local arguments in the extension problem.

Open this publication in new window or tab >>Kolmogorov-Fokker-Planck Equations: comparison Principles near Lipschitz type Boundaries### Nyström, Kaj

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Polidoro, Sergio

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_7_j_idt185_some",{id:"formSmash:j_idt181:7:j_idt185:some",widgetVar:"widget_formSmash_j_idt181_7_j_idt185_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_7_j_idt185_otherAuthors",{id:"formSmash:j_idt181:7:j_idt185:otherAuthors",widgetVar:"widget_formSmash_j_idt181_7_j_idt185_otherAuthors",multiple:true}); 2016 (English)In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 106, no 1, p. 155-202Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-238221 (URN)10.1016/j.matpur.2016.02.007 (DOI)000378189900004 ()
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Available from: 2014-12-10 Created: 2014-12-10 Last updated: 2017-12-05Bibliographically approved

Univ Modena & Reggio Emilia, Dipartimento Sci Fis Informat & Matemat, Via Campi 218-B, I-41125 Modena, Italy.

We prove several new results concerning the boundary behavior of non-negative solutions to the equation $\Ku=0$ where

\begin{eqnarray*}%\label{kolsim} \K:= \sum_{i=1}^m\partial_{x_i x_i}+\sum_{i=1}^m x_i\partial_{y_{i}}-\partial_t.\end{eqnarray*}

Our results are established near the non-characteristic part of the boundary of certain local $\MLip$-domains where the latter is a class of local Lipschitz type domains adapted to the geometry of $\K$. Generalizations to more general operators of Kolmogorov-Fokker-Planck type are also discussed.

Open this publication in new window or tab >>Reflected BSDE of Wiener-Poisson type in time-dependent domains### Nyström, Kaj

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Olofsson, Marcus

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_8_j_idt185_some",{id:"formSmash:j_idt181:8:j_idt185:some",widgetVar:"widget_formSmash_j_idt181_8_j_idt185_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_8_j_idt185_otherAuthors",{id:"formSmash:j_idt181:8:j_idt185:otherAuthors",widgetVar:"widget_formSmash_j_idt181_8_j_idt185_otherAuthors",multiple:true}); 2016 (English)In: Stochastic Models, ISSN 1532-6349, E-ISSN 1532-4214, Vol. 32, no 2, p. 275-300Article in journal (Refereed) Published
##### Abstract [en]

##### Keyword

Backward stochastic differential equation, convex domain, reflected backward stochastic differential equation, time-dependent domain, 60H10, 60H20
##### National Category

Mathematical Analysis Probability Theory and Statistics
##### Identifiers

urn:nbn:se:uu:diva-224258 (URN)10.1080/15326349.2015.1116011 (DOI)000377141900005 ()
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Available from: 2014-05-07 Created: 2014-05-07 Last updated: 2017-12-05Bibliographically approved

In the paper we study multi-dimensional reflected backward stochasticdifferential equations driven by Wiener-Poisson type processes. We prove existence and uniqueness of solutions, with reflection in the inward spatial normal direction, in the setting of certain time-dependent domains.

Open this publication in new window or tab >>Square function estimates and the Kato problem for second order parabolic operators in Rn+1### Nyström, Kaj

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_9_j_idt185_some",{id:"formSmash:j_idt181:9:j_idt185:some",widgetVar:"widget_formSmash_j_idt181_9_j_idt185_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt181_9_j_idt185_otherAuthors",{id:"formSmash:j_idt181:9:j_idt185:otherAuthors",widgetVar:"widget_formSmash_j_idt181_9_j_idt185_otherAuthors",multiple:true}); 2016 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 293, p. 1-36Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-263917 (URN)10.1016/j.aim.2016.02.006 (DOI)000373093200001 ()
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Available from: 2015-10-04 Created: 2015-10-04 Last updated: 2017-12-01Bibliographically approved

In [4] the Kato square root problem for second order complex uniformly elliptic operators of the form L = -div(A del f), with only bounded and measurable coefficients, was solved. The solution is a consequence of a square function estimate for the operator (1 + lambda L-2)(-1)lambda L. This and related square function estimates have recently spurred a wave of new and ground breaking results in the area of elliptic PDEs. In this paper we establish similar square function estimates for second order parabolic operators in Rn+1 of the form at partial derivative(t) + L paving the way for important developments in the area of parabolic PDEs.