Leonid Mytnik
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Leonid Mytnik has a total of 39 co-authors. Here are the top co-authors by number of shared publications:
- Vitali Wachtel
Universität Augsburg
8 shared publications - Edwin Perkins
Department of Mathematics The University of British Columbia
6 shared publications - Lenya Ryzhik
Department of Mathematics, Stanford University, Stanford, USA
5 shared publications - Achim Klenke
Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Mainz, Germany
4 shared publications - Oleg Butkovsky
Weierstrass Institute, Berlin, Germany
4 shared publications - Siva Athreya
International Centre for Theoretical Sciences, TIFR, Bengaluru, India
4 shared publications
Latest Publications
- Open Access
- 3 January 2026
- Published by Springer Nature in Stochastics and Partial Differential Equations: Analysis and Computations
- p. 1-2
Publisher Correction: Analytically weak and mild solutions to stochastic heat equation with irregular drift
- 3 January 2026
- Published by Springer Nature in Stochastics and Partial Differential Equations: Analysis and Computations
- p. 1-2
Open AccessPublisher Correction: Analytically weak and mild solutions to stochastic heat equation with irregular drift
- Open Access
- 10 October 2025
- Published by Springer Nature in Stochastics and Partial Differential Equations: Analysis and Computations
- p. 1-44
Analytically weak and mild solutions to stochastic heat equation with irregular drift
Consider the stochastic heat equation $$\begin{aligned} \partial _t u_t(x)=\frac{1}{2} \partial ^2_{xx}u_t(x) +b(u_t(x))+\dot{W}_{t}(x),\quad t\in (0,T],\, x\in D, \end{aligned}$$ where b is a generalized function, D is either [0, 1] or $$\mathbb {R}$$ , and $$\dot{W}$$ is space-time white noise on $$\mathbb {R}_+\times D$$ . If the drift b is a sufficiently regular function, then it is well-known that any analytically weak solution to this equation is also analytically mild, and vice versa . We extend this result to drifts that are generalized functions, with an appropriate adaptation of the notions of mild and weak solutions. As a corollary of our results, we show that for $$b\in L_p(\mathbb {R})$$ , $$p\geqslant 1$$ , this equation has a unique analytically weak and mild solution, thus extending the classical results of Gyöngy and Pardoux (Probab Theory Related Fields 94(4):413–425, 1993).- 10 October 2025
- Published by Springer Nature in Stochastics and Partial Differential Equations: Analysis and Computations
- p. 1-44
Open AccessAnalytically weak and mild solutions to stochastic heat equation with irregular drift
Consider the stochastic heat equation $$\begin{aligned} \partial _t u_t(x)=\frac{1}{2} \partial ^2_{xx}u_t(x) +b(u_t(x))+\dot{W}_{t}(x),\quad t\in (0,T],\, x\in D, \end{aligned}$$ where b is a generalized function, D is either [0, 1] or $$\mathbb {R}$$ , and $$\dot{W}$$ is space-time white noise on $$\mathbb {R}_+\times D$$ . If the drift b is a sufficiently regular function, then it is well-known that any analytically weak solution to this equation is also analytically mild, and vice versa . We extend this result to drifts that are generalized functions, with an appropriate adaptation of the notions of mild and weak solutions. As a corollary of our results, we show that for $$b\in L_p(\mathbb {R})$$ , $$p\geqslant 1$$ , this equation has a unique analytically weak and mild solution, thus extending the classical results of Gyöngy and Pardoux (Probab Theory Related Fields 94(4):413–425, 1993). - 17 July 2025
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 194 (3-4) , 1711-1786
Wright–Fisher stochastic heat equations with irregular drifts
- 17 July 2025
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 194 (3-4) , 1711-1786
Wright–Fisher stochastic heat equations with irregular drifts
- Open Access
Strong Existence and Uniqueness for Singular SDEs Driven by Stable Processes
We consider the one-dimensional stochastic differential equation \begin{equation*} X_t = x_0 + L_t + \int_0^t μ(X_s)ds, \quad t \geq 0, \end{equation*} where $μ$ is a finite measure of Kato class $K_η$ with $η\in (0,α-1]$ and $(L_t)_{t \geq 0}$ is a symmetric $α$-stable process with $α\in (1,2)$. We derive weak and strong well posedness for this equation when $η\leqα-1$ and $η< α-1$, respectively, and show that the condition $η\leq α-1$ is sharp for weak existence. We furthermore reformulate the equation in terms of the local time of the solution $(X_{t})_{t \geq 0}$ and prove its well posedness. To this end, we also derive a Tanaka-type formula for a symmetric, $α$-stable processes with $α\in (1,2)$ that is perturbed by an adapted, right-continuous process of finite variation.Open AccessStrong Existence and Uniqueness for Singular SDEs Driven by Stable Processes
We consider the one-dimensional stochastic differential equation \begin{equation*} X_t = x_0 + L_t + \int_0^t μ(X_s)ds, \quad t \geq 0, \end{equation*} where $μ$ is a finite measure of Kato class $K_η$ with $η\in (0,α-1]$ and $(L_t)_{t \geq 0}$ is a symmetric $α$-stable process with $α\in (1,2)$. We derive weak and strong well posedness for this equation when $η\leqα-1$ and $η< α-1$, respectively, and show that the condition $η\leq α-1$ is sharp for weak existence. We furthermore reformulate the equation in terms of the local time of the solution $(X_{t})_{t \geq 0}$ and prove its well posedness. To this end, we also derive a Tanaka-type formula for a symmetric, $α$-stable processes with $α\in (1,2)$ that is perturbed by an adapted, right-continuous process of finite variation. - Open Access
Effect of small noise on the speed of reaction-diffusion equations with non-Lipschitz drift
We consider the $[0,1]$-valued solution $(u_{t,x}:t\geq 0, x\in \mathbb R)$ to the one dimensional stochastic reaction diffusion equation with Wright-Fisher noise \[\partial_t u= \partial_x^2 u + f(u) + \epsilon \sqrt{u(1-u)} \dot W.\] Here, $W$ is a space-time white noise, $\epsilon > 0$ is the noise strength, and $f$ is a continuous function on $[0,1]$ satisfying $\sup_{z\in [0,1]}|f(z)|/ \sqrt{z(1-z)} < \infty.$ We assume the initial data satisfies $1 - u_{0,-x} = u_{0,x} = 0$ for $x$ large enough. Recently, it was proved in (Comm. Math. Phys. \textbf{384} (2021), no. 2) that the front of $u_t$ propagates with a finite deterministic speed $V_{f,\epsilon}$, and under slightly stronger conditions on $f$, the asymptotic behavior of $V_{f,\epsilon}$ was derived as the noise strength $\epsilon$ approaches $\infty$. In this paper we complement the above result by obtaining the asymptotic behavior of $V_{f,\epsilon}$ as the noise strength $\epsilon$ approaches $0$: for a given $p\in [1/2,1)$, if $f(z)$ is non-negative and is comparable to $z^p$ for sufficiently small $z$, then $V_{f,\epsilon}$ is comparable to $\epsilon^{-2\frac{1-p}{1+p}}$ for sufficiently small $\epsilon$.Open AccessEffect of small noise on the speed of reaction-diffusion equations with non-Lipschitz drift
We consider the $[0,1]$-valued solution $(u_{t,x}:t\geq 0, x\in \mathbb R)$ to the one dimensional stochastic reaction diffusion equation with Wright-Fisher noise \[\partial_t u= \partial_x^2 u + f(u) + \epsilon \sqrt{u(1-u)} \dot W.\] Here, $W$ is a space-time white noise, $\epsilon > 0$ is the noise strength, and $f$ is a continuous function on $[0,1]$ satisfying $\sup_{z\in [0,1]}|f(z)|/ \sqrt{z(1-z)} < \infty.$ We assume the initial data satisfies $1 - u_{0,-x} = u_{0,x} = 0$ for $x$ large enough. Recently, it was proved in (Comm. Math. Phys. \textbf{384} (2021), no. 2) that the front of $u_t$ propagates with a finite deterministic speed $V_{f,\epsilon}$, and under slightly stronger conditions on $f$, the asymptotic behavior of $V_{f,\epsilon}$ was derived as the noise strength $\epsilon$ approaches $\infty$. In this paper we complement the above result by obtaining the asymptotic behavior of $V_{f,\epsilon}$ as the noise strength $\epsilon$ approaches $0$: for a given $p\in [1/2,1)$, if $f(z)$ is non-negative and is comparable to $z^p$ for sufficiently small $z$, then $V_{f,\epsilon}$ is comparable to $\epsilon^{-2\frac{1-p}{1+p}}$ for sufficiently small $\epsilon$. - Open Access
On the coming down from infinity of coalescing Brownian motions
Consider a system of Brownian particles on the real line where each pair of particles coalesces at a certain rate according to their intersection local time. Assume that there are infinitely many initial particles in the system. We give a necessary and sufficient condition for the number of particles to come down from infinity. We also identify the rate of this coming down from infinity for different initial configurations.
Open AccessOn the coming down from infinity of coalescing Brownian motions
Consider a system of Brownian particles on the real line where each pair of particles coalesces at a certain rate according to their intersection local time. Assume that there are infinitely many initial particles in the system. We give a necessary and sufficient condition for the number of particles to come down from infinity. We also identify the rate of this coming down from infinity for different initial configurations.
- 16 February 2022
- Published by IOP Publishing in Nonlinearity
- Vol. 35 (4) , 1558-1609
The distance between the two BBM leaders
We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit d 12 of this random variable can be expressed in terms of PDEs related to the Fisher–KPP equation. We use such a representation to determine the sharp asymptotics of P ( d 12 > a ) as a → +∞. These tail asymptotics were previously known to ‘exponential order;’ we discover an algebraic correction to this behavior.
- 16 February 2022
- Published by IOP Publishing in Nonlinearity
- Vol. 35 (4) , 1558-1609
The distance between the two BBM leaders
We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit d 12 of this random variable can be expressed in terms of PDEs related to the Fisher–KPP equation. We use such a representation to determine the sharp asymptotics of P ( d 12 > a ) as a → +∞. These tail asymptotics were previously known to ‘exponential order;’ we discover an algebraic correction to this behavior.
- Open Access
Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation
We study stochastic reaction--diffusion equation $$ \partial_tu_t(x)=\frac12 \partial^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D $$ where $b$ is a generalized function in the Besov space $\mathcal{B}^\beta_{q,\infty}({\mathbb R})$, $D\subset{\mathbb R}$ and $\dot W$ is a space-time white noise on ${\mathbb R}_+\times D$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever $\beta-1/q\ge-1$, $\beta>-1$ and $q\in[1,\infty]$. This class includes equations with $b$ being measures, in particular, $b=\delta_0$ which corresponds to the skewed stochastic heat equation. For $\beta-1/q > -3/2$, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalizes the results of Gy\"ongy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in L\^e~(2020).Open AccessWell-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation
We study stochastic reaction--diffusion equation $$ \partial_tu_t(x)=\frac12 \partial^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D $$ where $b$ is a generalized function in the Besov space $\mathcal{B}^\beta_{q,\infty}({\mathbb R})$, $D\subset{\mathbb R}$ and $\dot W$ is a space-time white noise on ${\mathbb R}_+\times D$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever $\beta-1/q\ge-1$, $\beta>-1$ and $q\in[1,\infty]$. This class includes equations with $b$ being measures, in particular, $b=\delta_0$ which corresponds to the skewed stochastic heat equation. For $\beta-1/q > -3/2$, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalizes the results of Gy\"ongy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in L\^e~(2020). - Open Access
Fisher-KPP equation with small data and the extremal process of branching Brownian motion
We consider the limiting extremal process ${\mathcal X}$ of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale $Z$, the rescaled "density" of particles, which are at distance $n+x$ from a position close to the tip of ${\mathcal X}$, converges in probability to a multiple of the exponential $e^x$ as $n\to+\infty$. We also show that the fluctuations of the density, after another scaling and an additional random but explicit shift, converge to a $1$-stable random variable. Our approach uses analytic techniques and is motivated by the connection between the properties of the branching Brownian motion and the Bramson shift of the solutions to the Fisher-KPP equation with some specific initial conditions initiated in \cite{BD1,BD2} and further developed in the present paper. The proofs of the limit theorems for ${\mathcal X}$ rely crucially on the fine asymptotics of the behavior of the Bramson shift for the Fisher-KPP equation starting with initial conditions of "size" $0<\varepsilon\ll 1$, up to terms of the order $[{(\log \varepsilon^{-1})]^{-1-\gamma}}$, with some $\gamma>0$.Open AccessFisher-KPP equation with small data and the extremal process of branching Brownian motion
We consider the limiting extremal process ${\mathcal X}$ of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale $Z$, the rescaled "density" of particles, which are at distance $n+x$ from a position close to the tip of ${\mathcal X}$, converges in probability to a multiple of the exponential $e^x$ as $n\to+\infty$. We also show that the fluctuations of the density, after another scaling and an additional random but explicit shift, converge to a $1$-stable random variable. Our approach uses analytic techniques and is motivated by the connection between the properties of the branching Brownian motion and the Bramson shift of the solutions to the Fisher-KPP equation with some specific initial conditions initiated in \cite{BD1,BD2} and further developed in the present paper. The proofs of the limit theorems for ${\mathcal X}$ rely crucially on the fine asymptotics of the behavior of the Bramson shift for the Fisher-KPP equation starting with initial conditions of "size" $0<\varepsilon\ll 1$, up to terms of the order $[{(\log \varepsilon^{-1})]^{-1-\gamma}}$, with some $\gamma>0$. - 1 May 2021
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 384 (2) , 699-732
The Speed of a Random Front for Stochastic Reaction–Diffusion Equations with Strong Noise
- 1 May 2021
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 384 (2) , 699-732
The Speed of a Random Front for Stochastic Reaction–Diffusion Equations with Strong Noise