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Register a new userOn the fundamental solution of heat and stochastic heat equations
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Authors
Marina Kleptsyna
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We consider the generic divergence form second order parabolic equation with coefficients that are regular in the spatial variables and just measurable in time. We show that the spatial derivatives of its fundamental solution admit upper bounds that agree with the Aronson type estimate and only depend on the ellipticity constants of the equation and the L $infty$ norm of the spatial derivatives of its coefficients. We also study the corresponding stochastic partial differential equations and prove that under natural assumptions on the noise the equation admits a mild solution, given by anticipating stochastic integration.
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Let $(Omega, mu)$ be a probability space endowed with an ergodic action, $tau$ of $( {mathbb R} ^n, +)$. Let $H(x,p; omega)=H_omega(x,p)$ be a smooth Hamiltonian on $T^* {mathbb R} ^n$ parametrized by $omegain Omega$ and such that $ H(a+x,p;tau_aomega)=H(x,p;omega)$. We consider for an initial condition $fin C^0 ( {mathbb R}^n)$, the family of variational solutions of the stochastic Hamilton-Jacobi equations $$left{ begin{aligned} frac{partial u^{ varepsilon }}{partial t}(t,x;omega)+Hleft (frac{x}{ varepsilon } , frac{partial u^varepsilon }{partial x}(t,x;omega);omega right )=0 & u^varepsilon (0,x;omega)=f(x)& end{aligned} right .$$ Under some coercivity assumptions on $p$ -- but without any convexity assumption -- we prove that for a.e. $omega in Omega$ we have $C^0-lim u^{varepsilon}(t,x;omega)=v(t,x)$ where $v$ is the variational solution of the homogenized equation $$left{ begin{aligned} frac{partial v}{partial t}(x)+{overline H}left (frac{partial v }{partial x}(x) right )=0 & v (0,x)=f(x)& end{aligned} right.$$
This work is concerned with model reduction of stochastic differential equations and builds on the idea of replacing drift and noise coefficients of preselected relevant, e.g. slow variables by their conditional expectations. We extend recent results by Legoll & Leli`evre [Nonlinearity 23, 2131, 2010] and Duong et al. [Nonlinearity 31, 4517, 2018] on effective reversible dynamics by conditional expectations to the setting of general non-reversible processes with non-constant diffusion coefficient. We prove relative entropy and Wasserstein error estimates for the difference between the time marginals of the effective and original dynamics as well as an entropy error bound for the corresponding path space measures. A comparison with the averaging principle for systems with time-scale separation reveals that, unlike in the reversible setting, the effective dynamics for a non-reversible system need not agree with the averaged equations. We present a thorough comparison for the Ornstein-Uhlenbeck process and make a conjecture about necessary and sufficient conditions for when averaged and effective dynamics agree for nonlinear non-reversible processes. The theoretical results are illustrated with suitable numerical examples.
Consider the following equation $$partial_t u_t(x)=frac{1}{2}partial _{xx}u_t(x)+lambda sigma(u_t(x))dot{W}(t,,x)$$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if $lambda$ is large enough. But if $lambda$ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what $lambda$ is. We also provide various extensions.
In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of $mathbb{R}^d$ and a complete Riemannian manifold $mathrm{M}$. On one hand, in $mathbb{R}^d$, we prove that any solution $u=u(t,x)$ to $u_t(t,x)-mathrm{L}_alpha^{kappa} u(t,x)=0$, where $mathrm{L}_alpha^{kappa}$ is a nonlocal operator of order $alpha$, is time analytic in $(0,1]$ if $u$ satisfies the growth condition $|u(t,x)|leq C(1+|x|)^{alpha-epsilon}$ for any $(t,x)in (0,1]times mathbb{R}^d$ and $epsilonin(0,alpha)$. We also obtain pointwise estimates for $partial_t^kp_alpha(t,x;y)$, where $p_alpha(t,x;y)$ is the fractional heat kernel. Furthermore, under the same growth condition, we show that the mild solution is the unique solution. On the other hand, in a manifold $mathrm{M}$, we also prove the time analyticity of the mild solution under the same growth condition and the time analyticity of the fractional heat kernel, when $mathrm{M}$ satisfies the Poincare inequality and the volume doubling condition. Moreover, we also study the time and space derivatives of the fractional heat kernel in $mathbb{R}^d$ using the method of Fourier transform and contour integrals. We find that when $alphain (0,1]$, the fractional heat kernel is time analytic at $t=0$ when $x eq 0$, which differs from the standard heat kernel. As corollaries, we obtain sharp solvability condition for the backward fractional heat equation and time analyticity of some nonlinear fractional heat equations with power nonlinearity of order $p$. These results are related to those in [8] and [11] which deal with local equations.
We construct a radially smooth positive ancient solution for energy critical semi-linear heat equation in $mathbb{R}^n$, $ngeq 7$. It blows up at the origin with the profile of multiple Talenti bubbles in the backward time infinity.
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