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‎vault/.obsidian/workspace.json‎

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"title": "2017-representation-complex-cauchy"
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‎vault/_publications/2017-representation-complex-cauchy.md‎

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---
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layout: publication
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title: Probabilistic representation of the Cauchy problem solution for the multidimensional Schrödinger equation
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date: 2025-07-20
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year: 2017
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authors:
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- Pavel Ievlev
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venue: Zap. Nauchn. Sem. POMI
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type: academic
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status: published
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abstract: We construct a probabilistic representation of the Cauchy problem solution for the Shrödinger equation $2 i \partial_t u = -\Delta u$. The result is an extension to a mulidimensional case of the previous results by I. Ibragimov, N. Smorodina and M. Faddeev.
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keywords:
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bibtex: |-
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@article {MR3760049,
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AUTHOR = {Ievlev, P. N.},
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TITLE = {Probabilistic representation of the solution of the {C}auchy problem for the multidimensional {S}chr\"odinger equation},
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JOURNAL = {Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)},
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FJOURNAL = {Rossi\u iskaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheski\u i\ Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI)},
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VOLUME = {466},
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YEAR = {2017},
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PAGES = {145--158},
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ISSN = {0373-2703},
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MRCLASS = {60H30 (35C05 35C15 35Q41)},
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MRNUMBER = {3760049}
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}
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---
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# Untitled
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## Overview
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Detailed description of the publication, methodology, results, and impact.
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## Key Contributions
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- Contribution 1
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- Contribution 2
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- Contribution 3
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## Methodology
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Brief description of the approach used.
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## Results
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Summary of key findings.
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## Impact
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Discussion of the paper's significance and citations.

‎vault/_publications/2018-representation-boundary-value.md‎

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---
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layout: publication
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title: Probabilistic representations of the initial-boundary value problem solutions for the Schrödinger equation in a $d$-dimensional ball
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date: 2025-07-20
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year: 2018
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authors:
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- Pavel Ievlev
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venue: Zap. Nauchn. Sem. POMI
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type: academic
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status: published
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abstract: We extend the construction of probabilistic representations for initial-boundary value problem solutions to the non-stationary Schrödinger equation in $d$-hyperball first obtained in the works by I. Ibragimov, N. Smorodina and M. Faddeev to a multidimensional case. Further on, we show that in these representations the Wiener process could be replaced by a random walk approximation. The $L^2$-convergence rates are obtained.
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keywords:
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- Schrödinger equation
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- Boundary value problem
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- Probabilistic representations of PDE solutions
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bibtex: |
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@article{author2024title,
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title={Paper Title},
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author={Author, Name and Co-author, Name},
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journal={Journal Name},
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year={2024},
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publisher={Publisher}
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}
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---
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# Untitled
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## Overview
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Detailed description of the publication, methodology, results, and impact.
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## Key Contributions
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- Contribution 1
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- Contribution 2
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- Contribution 3
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## Methodology
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Brief description of the approach used.
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## Results
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Summary of key findings.
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## Impact
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Discussion of the paper's significance and citations.

‎vault/_publications/2019-reflecting-bm-in-d-ball.md‎

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---
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layout: publication
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title: Reflecting Brownian motion in $d$-ball
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date: 2025-07-20
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year: 2019
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authors:
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- Pavel Ievlev
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venue: Zap. Nauchn. Sem. POMI
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type: academic
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status: published
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abstract: Following the works of I. A. Ibrahimov, N. V. Smorodina and M. M. Faddeev, we develop a new construction of the Brownian motion with reflection in $d$-ball. The main advantage of our new approach is that it allows one to construct reflecting Lévy processes, whereas previous constructions are limited to diffusion processes. In our upcoming work we shall extend the results to the symmetric Levy processes in a smooth domain.
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keywords:
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- Brownian motion with reflection
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- Markov semigroup
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bibtex: |-
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@article {MR4060234,
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AUTHOR = {Ievlev, P. N.},
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TITLE = {Reflecting {B}rownian motion in {$d$}-ball},
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JOURNAL = {Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)},
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FJOURNAL = {Rossi\u iskaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheski\u i\ Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI)},
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VOLUME = {486},
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YEAR = {2019},
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PAGES = {158--177},
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ISSN = {0373-2703},
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MRCLASS = {60J65 (60G51)},
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MRNUMBER = {4060234}
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}
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---
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# Untitled
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## Overview
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Detailed description of the publication, methodology, results, and impact.
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## Key Contributions
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- Contribution 1
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- Contribution 2
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- Contribution 3
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## Methodology
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Brief description of the approach used.
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## Results
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Summary of key findings.
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## Impact
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Discussion of the paper's significance and citations.

‎vault/_publications/2021-symmetric-levy.md‎

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---
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layout: publication
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title: Symmetric Lévy processes with reflection
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date: 2025-07-20
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year: 2021
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authors:
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- Pavel Ievlev
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venue: Global and Stochastic Analysis
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type: academic
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status: published
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abstract: In this paper we give meaning to the notion of reflecting Levy process in a smooth domain $D$ in terms of the domain of its generator. In contrast with other approaches, we work with the classical Neumann conditions. The main idea is to construct a suitable continuation of an initial function $f$ defined on $D$ to $\mathbb{R}^d$, and then associate with it some Markov semi-group.
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keywords:
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- Reflected processes
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- Symmetric Lévy processes
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- Markov semigroup
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- Lévy operator
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---
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# Untitled
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## Overview
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Detailed description of the publication, methodology, results, and impact.
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## Key Contributions
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- Contribution 1
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- Contribution 2
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- Contribution 3
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## Methodology
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Brief description of the approach used.
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## Results
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Summary of key findings.
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## Impact
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Discussion of the paper's significance and citations.

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